· iv Contents 6 Interactionsandcurvedmomentumspace73 6.0.1...

Theory and Phenomenology of Planck scaledeformed relativistic theories

Scuola di Dottorato in Scienze Astronomiche, Chimiche, Fisiche,Matematiche e della Terra “Vito Volterra”

Dottorato di Ricerca in Fisica – XXV Ciclo

Candidate

Giacomo RosatiID number 123456

Thesis Advisor

Prof. Giovanni Amelino-Camelia

A thesis submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics

October 2012

Thesis not yet defended

Theory and Phenomenology of Planck scale deformed relativistic theoriesPh.D. thesis. Sapienza – University of Rome

© 2012 Giacomo Rosati. All rights reserved

This thesis has been typeset by LATEX and the Sapthesis class.

Version: 19 October 2012

Author’s email: [email protected]

mailto:[email protected]

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Contents

1 Introduction 1

2 Free particle in Galilean and special relativity 92.1 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Galilean relativistic particle in covariant Hamiltonian formalism . . . 142.3 Special relativistic particle in covariant Hamiltonian formalism . . . 202.4 Galilei algebra as Inonu-Wigner contraction of

central-extended Poincaré algebra . . . . . . . . . . . . . . . . . . . . 232.5 Loss of simultaneity and synchronization of clocks . . . . . . . . . . 24

3 DSR theories and κ-Poincaré/κ-Minkowski 273.1 κ-Minkowski/κ-Poincaré properties description in terms of plane waves 293.2 Time-to-the-right DSR symplectic structure . . . . . . . . . . . . . . 32

4 Free particle in DSR: relativity of locality 354.1 Relative locality for translations: physical and coordinate speed of

particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.1 R-DSR particle in covariant Hamiltonian formalism . . . . . 374.1.2 Covariance of the worldlines . . . . . . . . . . . . . . . . . . . 394.1.3 The role of deformed translations: relative locality . . . . . . 40

4.2 Relative locality: analogy with relativity of simultaneity . . . . . . . 434.3 Canonical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Relative locality for boosts . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 canonical coordinates . . . . . . . . . . . . . . . . . . . . . . 484.4.2 Boosts in R-DSR coordinates . . . . . . . . . . . . . . . . . . 51

4.5 A general result: “Taming non locality” . . . . . . . . . . . . . . . . 53

5 Opportunities for DSR phenomenology 595.1 In vacuo dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Spacetime fuzziness and blurring of images from distant sources . . . 615.3 Absence of threshold anomalies . . . . . . . . . . . . . . . . . . . . . 63

5.3.1 DSR theories and deformed conservation laws . . . . . . . . . 645.3.2 Photon stability . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.4 Synchrotron radiated power . . . . . . . . . . . . . . . . . . . . . . . 67

iv Contents

6 Interactions and curved momentum space 736.0.1 Leading-order anatomy of relative-locality momentum spaces 766.0.2 curved κ- momentum space . . . . . . . . . . . . . . . . . . . 77

6.1 The example of κ-Poincaré-inspired momentum space with Majid-Ruegg connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.1.1 Distance from the origin in a de Sitter momentum space . . . 816.1.2 Momentum space with de Sitter metric and Majid-Ruegg

connection: torsion and (non)metricity . . . . . . . . . . . . . 826.2 Partial anatomy of distant relative-locality observers . . . . . . . . . 83

6.2.1 A starting point for the description of distant relative-localityobservers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2.2 Some properties of our conservation laws . . . . . . . . . . . 866.2.3 Boundary terms and conservation of momenta . . . . . . . . 866.2.4 A challenge for spacetime-translation invariance in theories on

a relative-locality momentum space . . . . . . . . . . . . . . . 886.3 A Lagrangian description of relative locality with interactions . . . . 91

6.3.1 R-DSR symplectic structure and translations generated bytotal momentum . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3.2 Causally connected interactions and translations generated bytotal momentum . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4 Implications for the times of arrival of simultaneously-emitted ultra-relativistic particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.4.1 Matching Lagrangian and Hamiltonian description of R-DSR

free particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.4.2 More on observations of distant bursts of massless particles . 1066.4.3 Nonmetricity, torsion and time delays . . . . . . . . . . . . . 109

6.5 A consistency check for translation invariance: the case of 3 connectedfinite worldlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7 DSR in expanding spacetime 1137.1 Covariant formulation of free particles motion in De Sitter spacetime 114

7.1.1 Worldlines in comoving-time coordinates . . . . . . . . . . . . 1157.1.2 Finite translations in comoving-time coordinates . . . . . . . 1167.1.3 Worldlines in conformal-time coordinates . . . . . . . . . . . 1177.1.4 Finite translations in conformal-time coordinates . . . . . . . 1207.1.5 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.2 A reformulation of the results of Sec. 4.5 . . . . . . . . . . . . . . . . 1227.3 DSR De Sitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.3.1 DSR-deformed de-Sitter relativistic symmetries and equationsof motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.3.2 Travel time of massless particles . . . . . . . . . . . . . . . . 1297.4 Implications for phenomenology . . . . . . . . . . . . . . . . . . . . . 133

8 Spacetime fuzziness 1398.1 Preliminaries on κ-Minkowski differential calculus and translation

generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Contents v

8.2 A novel pregeometric representation of κ-Minkowski coordinates,differential calculus and translations . . . . . . . . . . . . . . . . . . 143

8.3 Boosts and a fully pregeometric picture . . . . . . . . . . . . . . . . 1458.4 Fuzzy points, translation transformations and relative locality . . . . 148

8.4.1 Fuzzy points . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.4.2 Translations and Relative Locality . . . . . . . . . . . . . . . 150

8.5 From kinematical to physical κ-Minkowski Hilbert space . . . . . . . 152

9 Conclusions 157

Acknowledgements 159

1

Chapter 1

Introduction

The problem of quantum gravity, i.e. the attempt to find a unified descriptionof quantum mechanical and gravitational phenomena, has been on the agenda oftheoretical physics since the early 1930s [1]. The fact that after such a long time theproblem remains completely open is mostly due to the smallness of the scale at whichthe magnitudes of the effects of quantum mechanics and of gravitational effectsbecome comparable, the Planck length (Lp =

√~G/c3 ∼ 10−35m). We are unable

to gain direct access to this scale, and as a result quantum-gravity research has beenmissing the guidance of experimental data needed for the development of theoreticalmodels. It was only rather recently, as a result of a research programme that startedtoward the end of the 1990s, that we realized that at least indirect experimental accessto the Planck scale is not necessarily beyond our reach [2, 3, 4, 5, 6, 7, 8, 9, 10, 11].The objective of exploiting these however rare experimental opportunities plays asignificant role in the thesis work here reported.

We shall do this while exploring some of the conceptual issues that attracted mostinterest in the recent quantum-gravity literature. Through its long history interest inthe quantum-gravity problem also evolved through stages where the focus was placedon some or some other of the issues that could be relevant for the understandingof the problem. During the last decade a considerable effort was directed, amongother topics, toward investigating the fate of spacetime relativistic symmetries at thePlanck scale. Some studies have argued that at the Planck scale Lorentz symmetrycould be violated, with the associated emergence of a preferred reference frame (asort of “quantum-gravity aether”). We will refer to theories adopting this intuitionas Lorentz-symmetry-breaking theories (LSB). But the main focus of this thesis is onthe alternative possibility provided by the class of theories labeled as “DSR” (doubly-special, or, for some authors, deformed-special relativity) [12, 13, 14, 15, 16, 17, 18],in which the price to pay for introducing the Planck scale in the laws of relativistickinematics is not the emergence of a preferred reference frame, but the fact that thetransformation laws between inertial observers are “deformed” at the Planck scale(one therefore speaks of “deformed relativistic symmetries”).

DSR theories were first proposed [12] as an opportunity to introduce the Planckscale as an observer-independent scale in a relativistic theory, governing the formof relativistic transformation among observers. This renders DSR theories suitablefor introducing as observer-independent laws some of the properties for the Planck

2 1. Introduction

scale that have been most popular in the quantum-gravity literature, such as a rolefor the Planck length (the inverse of the Planck scale) as the minimum allowedvalue for wavelengths [12]. In order to accommodate an analogous Planck-lengthminimum-wavelength principle without resorting to the DSR proposal one couldonly enforce it as a law valid in one class of frames, since wavelengths transformunder standard Lorentz boosts (so that a wavelength of Planck-length magnitudefor one observer is of totally different wavelength for other observers; more on thedifferences between DSR and LSB in Ch. 5).

From a conceptual perspective, additional motivation comes from the fact thatthe DSR proposal reflects the content of results rigorously established for some 4Dnoncommutative spacetimes (see, e.g., Refs. [19, 16]), and it fits the indicationsemerging from some semi-heuristic arguments based on 4D Loop Quantum Grav-ity [20, 21] (though of course the loop-quantum-gravity case could be rigorouslyanalyzed from this perspective only once the “classical-limit problem” [22] will befully understood).

And it is also noteworthy that DSR theories are found to play a significant rolein exploiting some of the few opportunities for Planck-scale phenomenology. Forexample, one can accommodate within a DSR framework a modification of thedispersion relation of the kind

E2 − ~p2c2 + f (E, p; `)−m2c4 = 0 , (1.1)

where ` plays the role of dimensionful (length/inverse-momentum) deformationparameter, and is usually assumed to be of the order of the (inverse of-)Planck scale.It has been shown that a deformation of the dispersion relation of the kind (1.1)may lead to a momentum dependence in the speed of photons [19], with effects thatare observably large in the context of observations of gamma ray bursts [23, 24].Although the effect amounts (see later) at best to a correction of parts in 1019 forphotons of, say, GeV energies, for sources that are typically at cosmological distances,such as gamma-ray bursts, the long propagation times convert such tiny velocitydeformations into an accumulated time-of-arrival effect which is macroscopic [19, 10].On the basis of a similar mechanism of amplification due to long propagation times,it has been argued [5, 25, 26, 27, 28, 29, 30, 31, 32] that the presence of a minimumwavelength, thought to be approximately close to the Planck length, could producean observably-large contribution to the blurring of the images of distant astrophysicalsources, such as quasars.

It should be stressed however, that at the start of the PhD-thesis researchproject here reported the development of the DSR framework was still at a relativelyearly stage, with limitations that affected both its usefulness from the conceptualperspective and its applicability in phenomenology. At the time the only robustresults available for the DSR framework concerned exclusively the case of classicalpoint particles in flat spacetime. It is clear however that, since as mentioned large(cosmological) distances are needed in order to amplify the smallness of the Planck-scale effects, the relevant phenomenology cannot neglect the contribution of thecurvature/expansion of spacetime, and therefore the flat-spacetime analyses areinconclusive. Similarly one cannot go much further with the phenomenology ofworldline fuzziness (relevant for the mentioned studies of possible blurring of images

3

of distant quasars) without control over the quantum properties of DSR-relativisticparticles.

So, already at the start of this thesis research work, it was appreciated that thedevelopment of the DSR framework needed to be generalized for accommodatingspacetime curvature and quantum-mechanical effects. Moreover, we felt it wasimportant to address a challenge already present in DSR theories of classical particlesin flat spacetime: the fate of spacetime locality. In previous attempts of descriptionof particle propagation, and specifically when attempting to formalize the localizationof events of collisions among particles, it appeared that one would inevitably findthat the nonlinearities in momentum space characteristic of the most studied DSRsetups would result in paradoxical properties (see, e.g., Ref. [81, 33, 34, 35, 36, 37])for spacetime locality.

This thesis research work, whose results were published/announced in Refs.[11,38, 39, 40, 41, 42, 43, 44, 45, 46], has been mainly devoted to address the challengeswe have stressed:

1. The introduction of a scale of spacetime curvature in DSR theories.

2. The formalization of DSR theories in a quantum mechanical framework, i.e.the inclusion of ~ (the Planck constant) in the theory.

3. The clarification of the fate of locality in DSR theories.

Concerning the fate of locality it was only with our works [11, 38] (also see[47] and[48, 49]), that the relevant puzzles started to be understood as part of a logicallyconsistent relativistic picture, the “relative locality” scenario. Relative locality,as understood in the above mentioned works, is the spacetime counterpart of theDSR-deformation scale ` just in the same sense that relative simultaneity is thespacetime counterpart of the special-relativistic scale c (scale of deformation ofGalilean Relativity into Special Relativity). And one of the striking consequencesof relative locality is that events established to be coincident by nearby observersmay appear to be non-coincident in the description of those events given by distantobservers on the basis of their inferences about the events (such as their observationof particles originating from the events).

Understanding relative locality proved to be an important asset also for pursuingthe other objectives of this research project. In particular, relative locality alsoplayed a role in our study [44] establishing that one can formulate a DSR theory ina curved/expanding spacetime. Our result [44] focuses on the case of a (De Sitterlike-)constant rate of spacetime expansion, and is also significant as the first everexample1 of a relativistic theory of worldlines of particles with 3 nontrivial relativisticinvariants: a large speed scale c (“speed-of-light scale”), a large distance scale H−1

(inverse of the “expansion-rate scale”), and a large momentum scale `−1 (“Planckscale”).

There had been previous attempts of investigating the interplay between DSR-type deformation scales and spacetime expansion (see, e.g., Refs. [51, 52]), butwithout ever producing a fully satisfactory picture of how the worldlines of particles

1Studies such as those in Refs. [50, 51, 52] did contemplate the possibility of 3 invariants, butdid not go as far as giving a consistent relativistic picture of worldlines of particles.

4 1. Introduction

should be formalized and interpreted. In retrospect we can now see that theseprevious difficulties were due to the fact that the notion of relative locality had notyet been understood, and without that notion the interplay between DSR-deformationscale and expansion-rate scale remains unintelligible. Our results appear to carryrather strong significance for the phenomenology. In particular, our constructiveanalysis leads to a description of the dependence of the novel effects on redshiftthat is not like anything imagined in the previous DSR literature (relying thenonly on heuristic arguments for the interplay between DSR-deformation scale andexpansion-rate scale).

A formulation of quantum mechanics compatible with DSR-relativistic sym-metries is proposed in our work [43], where we implemented the DSR-relativisticsymmetries within a covariant formulation of quantum mechanics of the type de-veloped in Ref.[53]. The action of the deformed spacetime symmetries is definedon the “kinematical Hilbert space”, in terms of a “pregeometric” representation”.The empowerment provided by this kinematical-Hilbert-space representation, evenif restricted to a specific flat spacetime case of DSR, allows us to give a crisp charac-terization of the spacetime fuzziness, enriched by the features of relative locality. Inparticular we find that relative locality, for a quantum spacetime, takes the shapeof a dependence of the fuzziness of a spacetime point on the distance at which anobserver infers properties of the event that marks the point. And in Ref. [46] wereported preliminary results on the implications of these findings for the mentionedphenomenology looking for effects blurring images of distant quasars.

Several of the results of this thesis work where obtained assuming (through aparametrization) a rather general DSR setup. But when we do focus on a specificillustrative example, we consistently take as starting point previous studies onκ-Minkowski non commutative spacetime and the related κ-Poincaré descriptionof spacetime symmetries. Indeed, the κ-Minkowski/κ-Poincaré framework is oneof the formalisms which has provided indications in support of modified on-shellrelations and modified laws of composition of momenta. The κ-Minkowski/κ-Poincaréformalism has interesting structure also from a mathematical viewpoint, since itinvolves the use of Hopf algebras (indeed the κ-Poincaré Hopf algebra) for describingspacetime symmetries.The characterization of noncommutativity properties betweenspacetime coordinates such as the ones of κ-Minkowski2

[xj , x0] = i`xj , [xj , xk] = 0 , (1.2)

with ` ∼ Lp, had been already of sizable interest in the quantum gravity literature [55,56, 57, 58] as a way to introduce a nontrivial spacetime structure at the Planck scale.And the results we here report on this structure are likely to strengthen this interestfurther.

It has been conjectured [12], in analogy with the relation between space/momentumnoncommutativity and the Heisenberg uncertainty principle, that such a noncom-mutativity between spacetime coordinates should be linked with the existence of aminimum length (thought to be close to the Planck length) limiting the precision in

2We write the “hat” over the spacetime variables suggesting for the relation (1.2) to representcommutation relations between operators over a given Hilbert space. We will return in detail onthis argument in Ch. 8.

5

localizing the position of an events (possible measurements). We will come back indetail to this argument in Ch. 8 of this thesis. Moreover, it has been shown [59, 60]that the Hopf algebraic properties of κ-Poincaré/κ-Minkowski are related in a “dual”way to the curvature of momentum space, a characteristic which has been argued bysome authors [59, 60] to be of intrinsic interest for quantum gravity models in orderto have a quantum system combined in a consistent way with curvature.

A specific representation of κ-Poincaré/κ-Minkowski, named the “bicrossprod-uct” [55, 56] (or “time-to-the-right” [61]) basis, introduced firstly in [55], has been ofparticular interest in the literature due to its group algebraic properties. The classi-cal (non-quantum) DSR model inspired by this “basis” of κ-Poincaré/κ-Minkowski,which we will call R-DSR (R standing for (time-to-the-Right)) throughout thismanuscript, is defined in terms of the Poisson structure inherited from the bi-crossproduct representation of κ-Poincaré/κ-Minkowski.

We will refer to R-DSR in this thesis as starting point for the analysis of freeparticle worldlines in DSR. This will allow us to exhibit constructively the emergenceof relative locality within an illustrative example (see our work [38]). And the R-DSR set-up will also give us a useful example for illustrating how one can introduceinteractions in the classical (non-quantum) DSR theory (see our work [39]). The(time-to-the-right) bicrossproduct basis of κ-Poincaré/κ-Minkowski will be, in itsfull quantum nature, the framework in which we will develop (as reported in ourworks [43, 46]) a first example of DSR-compatible covariant quantum mechanics.

Among the results which instead we derived within more general parametrizationsof the DSR setup we should highlight our analysis (reported in [11]) establishingthat relative locality is an inevitable consequence of any DSR theory predictingmomentum-dependent velocity of light. And it is an equally general property that,in these relevant forms of relative locality, one does not loose the objectivity oflocality as assessed by nearby observers. Also the inclusion in the theory of acurvature/expansion observer-invariant scale, based on our work [44], is worked outadopting a rather general parametrization of the DSR setup.

In the classical (non-quantum) framework adopted in this thesis, except for theresults of chapter 8, in which we rely on its full quantum properties, the properties ofκ-Poincaré/κ-Minkowski reflect themselves in a `-deformed Poisson structure for thealgebra of spacetime symmetry generators. We describe then the motion of a particlein DSR through a covariant Hamiltonian formalism, in which the Casimir of thesymmetry algebra plays the role of Hamiltonian/constraint. The particle worldlinesare then derived by evolving the constrained Hamiltonian system, the evolution beingthe unfolding of a gauge transformation generated by the Hamiltonian constraint.

The outline of this thesis will be the following: we will start reviewing, inCh. 2, the motion of a classical particle in Galilean and special relativity, within acovariant Hamiltonian formalism, starting from the algebra of symmetries of therelativistic theory. The motion of the particle, as mentioned before, is derived fromthe Poisson structure of the Hamiltonian system, where its evolution is generatedby an Hamiltonian constraint coinciding with the Casimir of the symmetry algebra.This strategy of analysis will be the starting point for the characterization of themotion of a particle in DSR in the following chapters. We will also discuss, in thischapter, some of the relativistic features emerging in transitioning from Galilean to

6 1. Introduction

special relativity, like the loss of absolute simultaneity. We will stress moreover howone can understand special relativity as a c-deformation (c being the speed of light)of Galilean relativity, showing in particular how one can recover Galilean relativityfrom special relativity by Inönü-Wigner contraction, in the c → ∞ limit. Theseconcepts will allow us to stress the analogy with the characterization of DSR as a`-deformation of special relativity, and the consequent loss of absolute locality infavor of a relativity of locality.

In Ch. 3 we will introduce the specific R-DSR model which will be the startingpoint for most of our illustrative examples. We will start with a short review of themain DSR features, stressing in particular how, differently from LSB scenarios, the(Galilean) principle of relativity is not violated in DSR. We will then introduce themain features of κ-Poincaré/κ-Minkowski Hopf algebras which shape the structure ofthe R-DSR model. We will introduce the κ-Poincaré/κ-Minkowski properties relevantfor our analysis, without dwelling to much on their Hopf-algebraic demonstration,but relying directly on their description in terms of plane waves in κ-Minkowski [19].In particular we will rely, as mentioned, on the bicrossproduct (time-to-the-right)description of κ-Poincaré/κ-Minkowski firstly introduced in Ref. [55]. We will thendescribe the Poisson structure of the R-DSR model.

Ch. 4, based mostly on our works [11, 38], will be devoted to the characterizationof the main concepts of relative locality, investigated by analyzing the motion of afree particle in DSR. As a first result [38], we will address the long debated (see, e.g.,Refs. [19, 56, 62, 63, 64, 65, 67, 66, 68, 69, 70, 71, 72, 73]) question of the discrepancybetween wave velocity [19] and the velocity derived from Hamiltonian analysis [62]in R-DSR. Indeed it appears, by wave analysis [19], that photons in κ-Poincaré/κ-Minkowski (and thus in R-DSR) must travel with a momentum-dependent speed. Ithas been argued [62], however, that if one analyzes the R-DSR Hamiltonian system,the results should give an ordinary momentum-independent speed of photons. We willshow how the non-trivial role of deformed translations solves this conflict. This willa first opportunity to see at work relative locality produced by deformed translationtransformations: we will see indeed that while the photons “coordinate” velocitydescribed by a given observer is momentum-independent, a momentum dependenceof the physical velocity (operatively described in terms of travel times) is uncoveredupon investigating the different coordinatizations implemented by distant observersconnected by a deformed translation transformation.

In the remainder of the chapter, we will compare the description of free particle’sworldlines given by inertial observers connected both by translation and boosttransformations. We will finally exhibit a general result concerning the class of DSRtheories predicting a momentum-dependent photons velocity [11] of the kind (at firstorder in `)

v = 1− ` |p| , (1.3)

with p being the particle’s momentum. We find that in such class of DSR theoriesrelative locality is present as an unavoidable feature. And we also show how theeffects of relative locality for the characterization of an event are proportional to thedistance of the event from the observer. So that, in the DSR framework, “locality”,a coincidence of events, preserves its objectivity if assessed by local observers.

In the following Ch. 5, we will discuss in more detail the phenomenological

7

opportunities available to the DSR research program. We will focus on two topics:Planck-scale-induced in-vacuo dispersion amenable to study via observation of high-energy astrophysical sources [23, 24, 19, 10, 74], and Planck-scale-induced worldlinefuzziness, which can be investigated searching for blurring of images by distantquasars [5, 25, 26, 27, 28, 29, 30, 31, 32]. During this thesis we also contributedto clarify some of the differences between DSR and LSB approaches, and in theremainder of the chapter we will present a discussion on the different perspectiveof the two approaches towards the mentioned phenomenological effects. We willpoint out in particular that while these effects can be contemplated even withinLorentz symmetry breaking scenarios, a crucial difference with the latter is that thefull relativistic picture of DSR protects the theory against some of the most virulenteffects predicted by the LSB scenario [75, 76, 77, 78].

In Ch. 6, based on our work [39], we will show how, still at the level of aclassical (non-quantum) description of DSR particle kinematics, one can introduceinteractions between particles. We will follow the proposed “principle of relativelocality” approach [48, 49]. The starting point of [48], is the assumption that therelevant physical description of a system should be formulated most primitively onmomentum space, which is allowed to be curved. Then, interactions (“vertices”)are introduced in a Lagrangian formalism, in terms of constraints (boundary terms)in the Lagrangian, reproducing the (deformed) conservation laws of momenta.The main obstacle we contributed to overcome with our work [39], was to obtaina consistent picture of a relative locality framework (with interactions) wheretranslation invariance is ensured for the case of causally connected interactions(particle exchange). This allowed us to study, at a classical (non-quantum) level, howthe relativistic features of the particle’s kinematics (like the time-of-flight) in DSR,are determined by the characteristics of the process of emission. We developed ouranalysis for the case of R-DSR, where the curved momentum space (the κ-momentumspace) can be deduced from the properties of κ-Poincaré.

It is in Ch. 7, based on our work [44], that we report our results on the formulationof a DSR theory with three observer-invariant scales (c, `, H). We start discussingbriefly the setting of special relativity in De Sitter/expanding spacetime. We exhibitthen how the DSR formalism can be improved to include the scale of deformationH−1, keeping the analysis on a general level, without relying on a specific DSRformalism. Because of the phenomenological opportunities we have in sight we focuson cases such that in the H → 0 limit the (physical) velocity (1.3) is recovered.The most significant challenges for this analysis were met in implementing finitetranslations in the case where both ` and H are different from zero.

Finally, in Ch. 8, based on our works [43, 46], we will discuss the covariantquantum mechanics formulation of κ-Poincaré/κ-Minkowski. We will find that itis possible to introduce a “pregeometric representation of both the κ-Minkowskinon-commutative spacetime properties, and the action of the κ-Poincaré symmetrygenerators on the “kinematical Hilbert space”. A fundamental role will be playedby the κ-Poincaré differential calculus [19, 79, 80, 81, 82], which will allow us todescribe translation transformations and to study the role of relative locality inthe quantum theory. We conclude the chapter by discussing in greater detail thephenomenological consequences of this formulation related to the spacetime fuzzinessand distant quasars [46].

9

Chapter 2

Covariant formulation ofparticles motion in Galilean andspecial relativity

For the research work reported in this thesis a crucial role is played by how relativisticsymmetries govern the structure of physical laws. We can characterize the symmetriesof a physical system by the group of transformations that leave invariant its lawsof dynamics. This in turn means that all observers connected by that set oftransformations describe the laws of dynamics in the same form; they describe thesame physical laws. We can say (in a physical jargon) that the laws of motion arecovariant under the action of those transformations, i.e. they conserve the sameform when expressed in the coordinates of an observer connected by one of thosetransformations. This defines the class of inertial observers. For instance in specialrelativity the inertial observers are the class of observers connected by the Poincarétransformations which describe the same laws of dynamics.

Galilean relativity is the relativistic framework in which the Newtonian mechanicstakes place. Galilean transformations are the set of transformations which leaveinvariant Newtonian mechanics. This is the (Galilean) principle of relativity: thelaws of (Newtonian) dynamics are the same for all inertial observers (connected bythe Galilei transformations). In Galilei relativity there is no observer-independentscale, and in fact (for example) the dispersion relation is written as E = p2/(2m)(whose structure fulfills the requirements of dimensional analysis without the need fordimensionful coefficients). The dispersion relation E = p2/(2m), which characterizesNewtonian dynamics, is covariant under the Galilei group of transformation (seeSec. 2.2).

As experimental evidence in favor of Maxwell equations started to grow, thefact that those equations involve a fundamental velocity scale appeared to require(assuming the Galilei symmetry group should remain unaffected) the introductionof a preferred class of inertial observers (the “ether”). Einstein’s Special Relativityintroduced the first observer-independent relativistic scale (the velocity scale c), itsdispersion relation takes the form E2 = c2p2 + c4m2 (in which c plays a crucialrole for what concerns dimensional analysis), and the presence of c in Maxwell’sequations is now understood not as a manifestation of the existence of a preferred

10 2. Free particle in Galilean and special relativity

class of inertial observers but as a manifestation of the necessity to deform theGalilei transformations. The Galilei transformations would not leave invariant therelation E2 = c2p2 + c4m2 , which is instead covariant according to the Lorentztransformations (the Lorentz transformations are a dimensionful deformation ofthe Galilei transformations). Lorentz-Poincaré (in Einsteinian special relativity)transformations, enforce covariance of Maxwell equations of motion, so that thevelocity “c” of light is the same for all inertial observers (without the need for anaether).

It will be useful in what follows to devote this section to an approach to thedescription of the kinematics of a relativistic particle, both in Galilean and specialrelativity, starting from the algebra of symmetries of the relativistic theory. Thiswill be useful when generalizing to DSR theories, where, even if deformed withrespect to special relativistic transformations, there is still a full set of symmetrytransformations connecting inertial observers. The analysis will be developed in a“covariant” Hamiltonian formalism, in which the (mass) Casimir of the symmetryalgebra will play the role of Hamiltonian. The Casimir/Hamiltonian makes the systema constrained Hamiltonian system, where the motion emerges as the unfolding of agauge transformation [83]. Then the elements of the symmetry algebra generate thesymplectic transformations which map the coordinates between each inertial observer.In this set up we will stress how the transition from Galilean relativity to specialrelativity can be described as the introduction of an invariant velocity scale (thevelocity of light c) in the theory, and we will characterize such transition at the levelof a generalization (c-deformation) of the algebra of symmetries. This framework willallow us to recognize, in the following chapters, the essential features characterizingone of the striking concepts we will develop in the transition (through `-deformation)from special relativity to DSR theories: as the introduction of an invariant velocityscale enforces one to abandon the idealization of absolute simultaneity in favor of arelativity of simultaneity, the introduction, in DSR theories, of an observer invariantinverse-momentum scale, implies locality to become relative.

2.1 Some tools concerning Hamiltonian analysis in Pois-son brackets formalism1

We can describe the phase space of our Hamiltonian system, i.e. the space ofpositions and momenta (the cotangent bundle T ?Q of the configuration manifold Q),by defining the Poisson brackets between positions and momenta, which determinethe symplectic structure 2 of the Hamiltonian system. Recall that a Poisson bracketis defined as a bilinear map on the smooth functions C∞ (M) over a manifold M ,

1The material of this section is inspired by various classical mechanics textbooks. We cite asexamples [84, 85]. See also [86] and references therein.

2More generally one can define a Poisson manifold instead of a symplectic manifold, if one admitsdegeneration of the Poisson structure.

2.1 Poisson brackets 11

·, ·: C∞ (M)× C∞ (M)→ C∞ (M), satisfying the following properties

f, g = −g, f skew − symmetry , (2.1)f, g, h+ g, h, f+ h, f, g Jacobi identity , (2.2)

f, gh = g f, h+ f, gh Leibniz rule , (2.3)

for all f, g, h ∈ C∞ (M).Then, the Poisson structure can be described, in local coordinates k ≡ (p, x)

(the set of phase space variables, both positions and momenta), by the bilinear formΩab (k) (the Poisson bivector), characterized, in matrix representation, by a 2n× 2nskew-symmetric (Ωab (k) = −Ωba (k)) block matrix as

Ωab (k) = ka, kb , Ω =(pµ, pν pµ, xνxµ, pν xµ, xν

). (2.4)

In the case of canonical coordinates, the symplectic bivector takes the simplecanonical form

Ωcanonical =(

0 η

−η 0

), (2.5)

where we’ve taken into account the possibility for a Lorentzian metricη = diag (1,−1,−1,−1, · · ·). Once specified the symplectic form Ω (k), the Poissonbracket between any two functions of the phase space variables are defined as

f (k) , g (k) = Ωab (k) ∂f (k)∂ka

∂g (k)∂kb

. (2.6)

In the case of canonical bivector (2.5), it’s easy to show that the Poisson bracketsreduce to the canonical form

f, g = ∂f

∂p0

∂g

∂x0−∑j

∂f

∂pj

∂g

∂xj− ∂f

∂x0

∂g

∂p0+∑j

∂f

∂xj

∂g

∂pj, j = 1, 2, 3, · · · . (2.7)

Any (smooth) function f (k) on the phase space induces a (special) symplecticvector field Xf , called Hamiltonian vector field [86], defined by the relation

Xf = d

ds= f (k) , · , (2.8)

whose flow is a symplectic transformation, i.e. preserves the symplectic structure(or the Poisson structure). Then any f (k) = H can be used as Hamiltonian, andits flow, the Hamiltonian flow, determines the equations of motion, as evolution interms of the parameter τ :

d

dτ= H, · . (2.9)

It follows from (2.8) that an infinitesimal symplectic transformation (an infinitesimaltransformation preserving the symplectic structure of the theory (and hence theHamilton equations) generated by f (k), is such that the phase space coordinatesundergo the infinitesimal change

k′ = k + δk = k + ε f, k , (2.10)

where ε is the infinitesimal parameter of the transformation.


Suppose now that the generator G of an infinitesimal transformation has nullPoisson bracket with the Hamiltonian H:

H, G = 0 . (2.11)

It immediately follows from the equations of motions (2.9) that G is conserved alongthe motion of the particle:

dG

dτ= H, G = 0 . (2.12)

But from the antisymmetry property of the Poisson bracket (2.1), it follows also thatG,H = 0, and, using relation (2.10), this means that G generates an infinitesimalsymplectic transformation that does not change the value of the Hamiltonian [84]:

∂H = H (k + δk)−H (k) = ε G,H = 0 . (2.13)

This is related to the symmetry properties of the system: “if the system issymmetrical under the operation that produces a change of configuration, thenthe Hamiltonian will obviously remain unaffected under the corresponding trans-formation” [84]. So that Eq. (2.11) (with its consequences) can be understood asthe Noether theorem in Poisson bracket formalism: the constants of motion, i.e.the conserved quantities, are the generating functions of those infinitesimal sym-plectic transformations that leave the Hamiltonian invariant, i.e. of the symmetrytransformations.

Throughout this thesis we shall qualify an equation of motion as “covariant”under a given transformation if it has the same form when it is expressed in termsof the coordinates of two observers connected by that transformation. This pertainsto “passive” symmetry transformations, i.e. transformations that connect thedescriptions of the same physical system as given by different observers. (Converselyactive transformations act on a physical system being observed by a given observerand connect it to another physical system which might be observed by that sameobserver.)

We notice that one can codify the covariance of the equations of motion underan infinitesimal transformation (2.10), in the requirement that

dk′

dτ=(dk

dτ

)′. (2.14)

This relation ensures indeed that the equations of motion for the observer withcoordinates k′, take the same form of the equations of motion for the observer withcoordinates k.

We will proof now that the requirement (2.14) is satisfied if Eq. (2.11) holds.We notice first that for the last equation to hold, for an infinitesimal transformation,from Eq. (2.10), it suffices that

d

dτδk = δ

dk

dτ. (2.15)

2.1 Poisson brackets 13

From (2.9), (2.10), and the Jacobi identities (2.2), it follows that

d

dτδk=εH,G, k=G, H,k+H,G, k=εG, H, k+εH, G, k, (2.16)

where we used also the skew-symmetry property of the Poisson bracket (2.1). IfH, G = 0, it follows Eq.(2.15):

d

dτδk = ε G, H, k = δ

dk

dτ. (2.17)

We can thus characterize more precisely Noether theorem in Poisson bracketsformalism by stating that a function of the phase space variable G (k), that has van-ishing Poisson bracket with the Hamiltonian, generates infinitesimal transformationsunder which the equations of motion are covariant.

We conclude this section showing how one can construct finite transformationsby exponentiating the action of an infinitesimal transformation (2.10) (see forexample [84]). Considering Eq. (2.8), we can write k as a function of the parameterof transformation k (s). Then, starting from the initial system configuration denotedby s = 0, and assuming k0 = k (0), we can expand k (s) by Taylor series,

k (s) = k0 + sdk

ds

∣∣∣0

+ s2

2!d2k

ds2

∣∣∣0

+ s3

3!d3k

ds3

∣∣∣0

+ · · · . (2.18)

Using Eq. (2.8), we have that

dk

ds

∣∣∣0

= f, k∣∣∣0, (2.19)

where the zero subscript means that the Poisson bracket has to be taken at theinitial point, s = 0. Then, repeated application of Eq. (2.8), leads to the equation

k (s) = k0 + s f, k∣∣∣0

+ s2

2! f, f, k∣∣∣0

+ s3

3! f, f, f, k∣∣∣0

+ · · · , (2.20)

or, written in compact notation,

k (s) =∞∑n=0

sn

n! f, kn∣∣∣0, (2.21)

where f, kn is the n-nested Poisson bracket defined by the relation

f, kn =f, f, kn−1

, , f, k0 = k . (2.22)

In this notation, we can rewrite the infinitesimal symplectic transformation (2.10) as

k (s) = k0 + s f, k∣∣∣0. (2.23)


We close this section by mentioning the relativistic features of the Jacobi identities.It is evident from their definition (2.2), that when one of the elements in the Jacobiidentity is a symmetry generator G, the Jacobi identity ensures the covariance of thePoisson bracket between the remaining two elements f, g. Indeed, using Eq. (2.10),the (transformed) observer with coordinates k′, describes the Poisson brackets

f ′, g′

= f + ε G, f , g + ε G, g = f, g+ ε G, f , g+ ε f, G, g .(2.24)

Using now the antisymmetry property of the Poisson brackets (2.1) and the Jacobyidentities (2.2), we obtain

f ′, g′

= f, g+ ε G, f, g = (f, g)′ . (2.25)

We will consider in the following symplectic structures defined by the algebra of thesymmetry generators together with the phase space relations. The validity of theJacobi identities then ensures the covariance of the symplectic structure.

2.2 Galilean relativistic particle in covariant Hamilto-nian formalism

After this short review, we present in this section a discussion of the motion of aparticle in Galilean relativity, starting from the algebra of the symmetry generators.Consider then the Lie algebra relative to the 3+1D Galilean group of transformations(see e.g. [87] and [88] and references therein). This can be understood formally asthe algebra of generators of transformations which leave invariant both the metricgµν = diag (1, 0, 0, 0) and the (independent) dual metric gµν = diag (0, 1, 1, 1). Wecan represent the Galilean algebra 3 in terms of the following set of Poisson brackets4

pj , pk = 0 , p0, pj = 0 ,Rj , Rk = εjklRl , Rj , p0 = 0 , Rj , pk = εjklpl ,

Nj , Nk = 0 , Rj , Nk = εjklNl , Nj , p0 = pj , Nj , pk = δjkm ,(2.26)

C = mp0 −~p2

2 , (2.27)

where p0, ~p, ~R and ~N are respectively the time translation, space translation, rotationand Galilean boost generators, m is the mass of the particle (which coincides withthe first Casimir (central charge) of the Galilei algebra, while C is the (quadratic)Casimir of the algebra, having null Poisson bracket with all the generators.

One possible choice in order to describe the physical motion of a Galileanrelativistic particle from the above relations, is to enforce a constraint on the system

3More specifically, we are here considering the central extension of the Galilean algebra [88], bythe central charge m.

4The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra,so that Eqs. (2.26) reflect the properties of the Galilei algebra (the Lie algebra of the (centralextended) Galilei group).

2.2 Galilean relativistic particle 15

identifying the quadratic Casimir (2.27) with the “on-shell relation” for the particle:

C = mp0 −~p2

2 = mw , (2.28)

where w is the internal energy of the particle [88], and p0 and ~p have the physicalinterpretation of energy and momentum of the particle (p0 = E). We can take asHamiltonian of evolution the constraint

H = C −mw = mp0 −~p2

2 −mw . (2.29)

In this way the system becomes a covariant Hamiltonian system, where physicaltime and the dynamical variables of the system are treated more symmetrically [83].Although this setup might be redundant for the description of a Galilean relativisticparticle, it will prove useful for a comparison between Galilei relativity, specialrelativity and DSR. The on-shell relation is realized setting H = 0, giving thedispersion relation

p0 (pj) = ~p2

2m + w , (2.30)

We notice that with this choice of Casimir/Hamiltonian, the elements of the algebra(2.26) are automatically conserved along the motion, as shown in the previousparagraph, since they commute with the Hamiltonian.

We can define the symplectic structure of the system by introducing the spacetimecoordinates x0 and xj (j = 1, 2, 3), canonically conjugate to the momenta p0 andpj , where the Poisson bivector is the canonical one (2.5)5, i.e. the phase space isdescribed by the Poisson brackets

p0, x0 = 1 , p0, xj = 0 ,pj , x0 = 0 , pj , xk = −δjk . (2.31)

The rotation and boost generators can be represented in terms of the phase spacevariables x0, xj , p0, pj as

Rj = εjklxkpl , Nj = xjm− x0pj , j, k, l = 1, 2, 3 . (2.32)

One can then verify that the whole set of Poisson brackets (2.26) and (2.31) satisfythe Jacoby identities (2.2), so that the symplectic structure is well defined, and iscovariant, as shown in Eqs. (2.24), (2.25).

The evolution of the system, being generated by the constraint (2.29), is de-termined by the unfolding of a Gauge transformation [83], so that the evolution ischaracterized in terms of an “auxiliary” (not physically observable) parameter τ . Inthis specific case one finds the equations of motion

x0 = dx0dτ

= H, x0 = m ,

xj = dxjdτ

= H, xj = pj , (2.33)

5Notice again the Lorentzian metric.


which can be integrated to give

x0 (τ)− x0 (0) = mτ ,

xj (τ)− xj (0) = pjτ . (2.34)

Eliminating the auxiliary parameter τ from the last equations we find the worldlinesfor a Galilean relativistic particle, as given by expressing the space position xj interms of the time x0:

xj (x0)p,m = xj + pjm

(x0 − x0) , (2.35)

were we called x0 = x0 (τ = 0) and xj = xj (τ = 0).We can see that the velocity of the particle (which reflects the Newton law of

motion)

~vm (~p) = d~x

dx0= ~p

m, (2.36)

coincides with the velocity that one can derive directly from the dispersion relation(2.30) (the “wave velocity”) as6

dp0 (pj)dpj

= pjm

. (2.37)

This can be justified observing that the canonical phase space (2.31) induces, by Eq.(2.7), and using the chain rule of derivatives, the relation

vj = dxj (x0)dx0

≡ dxjdτ

(dx0dτ

)−1= H, xjH, x0

= −∂H∂pj

(∂H∂p0

)−1≡ ∂p0 (pj)

∂pj, (2.38)

where repeated use of the chain rule has been made 7, and where we used that fromthe canonical symplectic vector (2.5), it follows

H (p0, pj) , xj = pk, xj∂H (p0, pj)

∂pk+ p0, xj

∂H (p0, pj)∂p0

=− ∂H (p0, p1)∂pj

= ∂H (p0, p1)∂pj

,

H (p0, pj) , x0 = pk, x0∂H (p0, pj)

∂pk+ p0, x0

∂H (p0, pj)∂p0

=∂H (p0, pj)∂p0

.

(2.39)

6Notice that we have to derive respect to the momentum with upper index to get the velocityexpressed in terms of the momentum with lower index, which makes a difference of sign when themetric is Lorentzian.

7After enforcing on-shellness, by the chain rule applied on inverse functions (assuming they areinvertible, which is true if we restrict to p0 > 0)

1(∂H(p0)∂p0

)(pj)

= ∂p0

∂H (pj) ,

so that, using again the chain rule(∂H∂p0

)−1

(pj)∂H (pj)∂pj

= ∂p0

∂H∂H∂pj

(pj) = ∂p0

∂pj(pj) .


We mention here that in a covariant Hamiltonian formulation, with a pureconstraint (vanishing) Hamiltonian, all physical questions should be formulated interms of functions that have zero bracket with the constraint [83]. By doing so, onetreats on the same footing “ordinary gauge-invariant quantities” and constants ofmotion. We notice that within this approach, the space xj and time x0 coordinatesare not by themselves properly defined physical quantities, since they have non-zerobracket with the Hamiltonian. They are not classical “observables” (functions on theconstraint surface that are gauge invariant). However, one can consider functionsdefining the position in spacetime of a particle (more properly of an event), as“relational observables” between xj and x0. For example one can define an observable“spatial position” (in Galilean relativity) from the worldline (2.35) as

xj (x0 = 0)m,p = xj − x0vm (p0, ~p) . (2.40)

Conversely, one can define an observable “time position” (in Galilean relativity)from the worldline (2.35) as

x0 (xj = 0)p,m = x0 − v−1m (p0, ~p) xj . (2.41)

One can easily see that both the functions (2.40) and (2.41) have zero Poisson bracketwith the Hamiltonian constraint (2.29), where H, · xµ = H, xµ

∣∣∣τ=0

. The twofunctions (2.40) and (2.41) have vanoshong Poisson bracket with the Hamiltonian(2.29).

The physical interpretation of the functions (2.40), (2.41), as will be clarifiedin the following chapters, is that of the position of the event characterizing thecrossing of the particle worldline with, respectively, the spatial and time origin ofthe observer (the intercept). The function (2.41) will be (implicitly) the basis of ourresults on the particle time-of-flight in chapters 4, 6, 7. We will encounter again thefunction (2.40), generalized to its quantum mechanical form, in Ch. 8, where wewill recognize it to coincide with the Newton-Wigner operator [89].

We study now some properties of the symmetry transformations generated bythe algebra (2.26). We first focus on the translation transformations generatedby p0 and pj . We introduce a scheme of analysis which will prove to be fruitfulthroughout this thesis. We specialize for simplicity to the 1+1D case, so that thephase space is characterized by the variables p0, p1, x0, x1. Consider a first observerAlice, characterized by a set of phase space variables kA =

(pA, xA

). Alice will

describe particle’s motion in terms of the worldlines (2.35) as

xA1

(xA0

)p,m

= xA1 + pA1m

(xA0 − xA0

). (2.42)

Consider now a second observer, Bob, at rest relative to Alice, and such that Alicedescribes Bob’s position as being at the point

(xA0 , x

A1

)= (a0, a1), i.e. Bob is a “dot”

in Alice’s frame (characterized also by a relative velocity).If the theory is relativistic, in his coordinates kB =

(pB, xB

), the worldlines in

terms of which Bob will describe the motion of particles will have the same form(2.35) of Alice’s worldlines:

xB1

(xB0

)p,m

= xB1 + pB1m

(xB0 − xB0

). (2.43)


As shown in the previous paragraph, through equations (2.11-2.17), the covarianceof the worldlines under transformations generated by the elements of the algebra(2.26) is ensured by the vanishing of the Poisson bracket between the generators andthe Hamiltonian (2.29).

We now want to confront Alice’s and Bob’s description of the same worldline.Observing that for each observer (labeling the coordinates of a generic observerby the suffix I) a given worldline can be specified by the parameters m, pI1, xI0, xI1,one way to confront Alice’s and Bob’s description is to express Bob’s parameters interms of Alice’s. The expression of Bob’s coordinates in terms of Alice’s is obtainedthrough a passive translation, generated by p0 and p1. Recalling the expressions(2.21) and (2.23) for a symmetry transformation, by substituting Alice’s coordinateskA as the initial coordinates and Bob’s coordinates kB as final coordinates, we canwrite

kB = kA − ε0 p0, k∣∣∣A

+ ε1 p1, k∣∣∣A, (2.44)

where the higher orders in the nested Poisson brackets vanish, and the signs in frontof the parameters have been chosen so to have the correct translation. Since thehigher orders in the nested Poisson brackets vanish, we can obtain a finite translationto a point (a0, a1) by simply composing a sequence of N infinitesimal translationswith parameters(ε0, ε1) =

(a0N ,

a1N

), and then taking the limit N →∞. In this way

we obtain the finite translation

kB = kA − a0 p0, k∣∣∣A

+ a1 p1, k∣∣∣A. (2.45)

We can thus define a translation Ta0,a1 , connecting Alice and Bob’s coordinates,by its (passive) action on the phase space coordinates, as

kB = (Ta0,a1 . k)∣∣∣A, (2.46)

where

Ta0,a1 . k = (1− a0 p0, ·+ a1 p1, ·) k = k − a0 p0, k+ a1 p1, k . (2.47)

In general, if two observers (Alice and Camilla) are connected by the sequenceof transformations G1followed by G2, we can define their composition as (G2 G1)We can derive the law of composition by considering a third intermediate observerBob, so that

kC = (G2 . k)∣∣∣B

= ((G2 G1) . k)∣∣∣A. (2.48)

If G1 and G2 are generated respectively by G1 and G2 as in the exponential repre-sentation (2.21), one finds that the sequence of transformations is given by

kC = (G2 . k)∣∣∣B

= kB + ε2 G2, k∣∣∣B

+ ε222! G2, G2, k

∣∣∣B

+ · · ·

=kA + ε1 G1, k∣∣∣A

+ ε212! G1, G1, k

∣∣∣A

+ · · ·

+ ε2 G2, k∣∣∣A

+ ε1ε2 G1, G2, k∣∣∣A

+ · · ·

+ ε222! G2, G2, k

∣∣∣A

+ · · ·

= (G1 . (G2 . k))∣∣∣A.

(2.49)


The exponential representation induces then the law of composition

(G2 G1) . k = (G1 . (G2 . k)) . (2.50)

It is important to notice the reverse order in the action of the transformations: Aliceand Camilla are connected by the transformation G1 followed by the transformationG2, while from Eq. (2.50) we see that Camilla’s coordinates are obtained from Alice’sformally by the (exponential) action of G2 followed by G1.

Coming back to our specific case, we find, by (2.45) and (2.31), that Bob’sparameters are expressed in terms of Alice’s as

xB0 = xA0 − a0 , xB1 = xA1 − a1 , pB1 = pA1 . (2.51)

Substituting the last relations in Eq. (2.43) we find that Bob describes the motionof the particles in terms of the worldlines

xB1

(xB0

)p,m

= xA0 − a0 + pA1m

(xB0 − xA1 + a1

), (2.52)

so that defining the initial parameters in Alice’s frame, we can determine Bob’sdescription by the last formula.

We can repeat the same steps for the boost transformation, generated (in 1+1D)by N1, Considering now Bob purely boosted respect to Alice, i.e. Bob’s origincoincide with Alice’s, but he has velocity VB relative to Alice. We define theinfinitesimal 1+1D boost trasformation to be given by

Bξ . kA = (1− ξ N1, ·) kA . (2.53)

From Eq. (2.32) and (2.31), we find

xB0 = xA0 − ξ x1m− x0p1, x0∣∣∣A

= xA0 ,

xB1 = xA1 − ξ x1m− x0p1, x0∣∣∣A

= xA1 − ξ1xA0 ,

pB1 = pA1 − ξ x1m− x0p1, p1 = pA1 − ξ1m . (2.54)

As for translations, the higher order in the nested Poisson brackets for Galileanboosts vanish, and we can, with the same procedure adopted for translation, definethe action of a finite Galilean boost simply substituting ξ → VB in Eqs. (2.54).Then, substituting into Eq. (2.43), we find

xB1

(xB0

)p,m

= xA1 − VBxA0 +(pA1m− VB

)(xB0 − xB0

). (2.55)

For both T and B one can verify that the worldlines (2.35) are covariant, byapplying the generators to both side of the equation. For example for the boostsone finds

xB1 = B . xA1 = xA1 − VBxA0 (2.56)


for the left-hand side, while for the right-hand side

xB1 + pB1m

(xB0 − xB0

)= B .

(xA1 + pA1

m

(xA0 − xA0

))

= xA1 − VBxA0 +(pA1m− VB

)(xA0 − xA0

),

(2.57)

so that

xB1 = xB1 + pB1m

(xB0 − xB0

)⇔ xA1 = xA1 + pA1

m

(xA0 − xA0

). (2.58)

2.3 Special relativistic particle in covariant Hamilto-nian formalism

We now proceed, in analogy with the previous section, to describe the motion ofa particle in special relativity in a covariant Hamiltonian formalism, starting fromthe algebra of the spacetime symmetry generators. The generators of the symmetrytransformations (leaving invariant the metric η = diag(1,−1,−1,−1)) for Mankowskispacetime can be represented by the Poincaré Lie algebra given in terms of thePoisson brackets

pj , pk = 0 , p0, pj = 0 ,Rj , Rk = εjklRl , Rj , p0 = 0 , Rj , pk = εjklpl , (2.59)

Nj , Nk = −εjklRl, Rj , Nk = εjklNl, Nj , p0 = pj , Nj , pk = δjkp0,

C = p20 − ~p2 , (2.60)

where N now is the Lorentz boost generator. The special relativistic physical pictureis obtained identifying the Casimir (2.60) with the particles on-shell relation givenby

C = p20 − ~p2 = m2c2 , (2.61)

with c an invariant speed scale (the speed of light), while ~p is the particle momentum,and the physical interpretation of p0 is given by the relation p0 = E

c , with E theenergy of the particle.

The Hamiltonian of evolution is given by the constraint

H = C −m2c2 = p20 − p2

1 −m2c2 , (2.62)

and, setting H = 0, we find the dispersion relation

p0c = E = c√~p2 +m2c2 , (2.63)

where we restricted to positive energies p0 > 0.We assume again canonical symplectic structure (with implicit Lorentzian metric)

introducing the spacetime coordinates xµ (µ = 0, 1, 2, 3), where x0 = ct, t being

2.3 Special relativistic particle 21

the time. The Poisson bivector is the canonical one (2.5), the phase space beingdescribed by the Poisson brackets

p0, x0 = 1 , p0, xj = 0 ,pj , x0 = 0 , pj , xk = −δjk . (2.64)

The rotation and boost generators can be represented in terms of the phase spacevariables xµ, pµ as

Rj = εjklxkpl , Nj = xjp0 − x0pj , j, k, l = 1, 2, 3 . (2.65)

Following the same reasoning made in the previous section, since the symplecticstructure is canonical, the particles velocity can be obtained directly from (2.63) bythe relation

vj (~p)m = dE (pj)dpj

= cpj√~p2 +m2c2 , (2.66)

from which it is evident the interpretation of c as the speed of massless particles

~v (~p)m=0 = c~p

|~p|. (2.67)

The velocity (2.66) can be derived also from the Hamiltonian analysis, where theequations of motions are

x0 = dx0dτ

= H, x0 = p0 ,

xj = dxjdτ

= H, xj = pj , (2.68)

which can be integrated to give

x0 (τ)− x0 (0) = p0τ ,

xj (τ)− xj (0) = pjτ . (2.69)

and eliminating τ we find the worldlines

xj (x0)p = xj + pjp0

(x0 − x0) , (2.70)

were we called x0 = x0 (τ = 0) and xj = xj (τ = 0). Using the on-shell relation(2.63), we can eliminate p0 from the velocity

~v (p0, ~p) = ~p

p0, (2.71)

which then coincides with (2.66).As in the previous section, we analyze now some aspects of the symmetry

transformations. While the translation sector is similar to the Galilean one, we focusnow on the Lorentz boost sector, restricting again for simplicity to the 1+1D case.Using Eqs. (2.65) and (2.64), the Poisson brackets between the boost generator N1and the spacetime coordinates are given by

N1, x0 = x1 , N1, x1 = x0 , (2.72)


so that infinitesimal boost transformations are given by

Bξ . x0 = x0 − ξx1 , Bξ . x1 = x1 − ξx0 , (2.73)

with ξ, the boost parameter, called the “rapidity”. Differently from the Galileanboosts, the higher order in nested Poisson brackets here do not vanish, and to evaluatefinite boost transformations we have to sum the whole series (2.21). Considering alsothe Poisson bracket between N1 and p1 defined in (2.59) one finds the transformationrules between Alice’s and (purely boosted) Bob’s coordinates

xB0 = cosh (ξ1)xA0 − sinh (ξ1)xA1 , xB1 = cosh (ξ1)xA1 − sinh (ξ1)xA0 ,

pB0 = cosh (ξ1) pA0 − sinh (ξ1) pA1 , pB1 = cosh (ξ1) pA1 − sinh (ξ1) pA0 . (2.74)

Consider now the case of two observers, Alice and Bob, whose frames origincoincide, with relative speed V , i.e. Bob is purely boosted relative to Alice. Assumethat Bob is traveling inside a rocket, of mass M , of course with speed V too. Themotion of the rocket is described by Alice in terms of the worldline (2.70) (in 1+1D),as

xAj

(xA0

)p,m

= xAj + pAR√(pAR)2 +M2c2

(xA0 − xA0

), (2.75)

with pR the rocket momentum, so that, recalling that x0 = ct,

V

c= pAR√(

pAR)2 +M2c2

. (2.76)

Now, from the transformation laws (2.74) we find that if a particle’s velocity isdescribed by Alice as vA

(pA0 , p

A1

)= pA1

pA0, the same velocity is described by Bob as

vB(pB0 , p

B1

)= cosh (ξ1) pA1 − sinh (ξ1) pA0

cosh (ξ1) pA0 − sinh (ξ1) pA1. (2.77)

Since the rocket is at rest relative to Bob, we set V B = 0, where V B is the rocketvelocity described by Bob, so that, using Eq. (2.77), we find the relation

sinh (ξ1)cosh (ξ1) = tanh (ξ1) = pAR√(

pAR)2 +M2

= V

c. (2.78)

The last equation express the relation between the rapidity and the speed of themoving frame, and, substituting in Eqs. (2.74), calling γ = cosh (ξ), and β = V

c , werecover the Lorentz transformations for a collinear boost:

xB0 = γxA0 − γβxA1 , xB1 = γxA1 − γβxA0 ,

pB0 = γpA0 − γβpA1 , pB1 = γpA1 − γβpA0 . (2.79)

2.4 Inonu-Wigner contraction 23

2.4 Galilei algebra as Inonu-Wigner contraction ofcentral-extended Poincaré algebra

In this section we show how the Galilei algebra (2.26) can be obtained from Poincaréalgebra (2.59) by Inönü-Wigner contraction [90]. This justifies our assertion thatEinstein special relativity can be understood as a “c−1-deformation” of Galileirelativity, where we emphasize the characterization of special relativity as theintroduction of an invariant velocity scale (the velocity of light) in a relativistictheory. This will turn useful when generalizing to DSR theories, and even morewhen we will introduce a third invariant scale, the spacetime expansion rate, in thetheory in Ch. 7.

We first notice that, from a more formal point of view, the algebra (2.26) isnot properly the Lie algebra G associated to the Galilei group, but it is the Liealgebra G × 〈m〉 of the extended Galilei group, where m (the mass), is the centralcharge which commutes with all the generators of the Galilei group [88, 90]. Then,in order to obtain the Galilei algebra (2.26) from the Poincaré algebra, we first haveto perform a trivial central extension of the Poincaré algebra P (2.59) (the Poincarégroup does not admit non-trivial central extensions [88]), obtaining a new algebraP × 〈µ〉.

This is done by the change8

p0 → p0 + µ (2.80)

in (2.59). The only Poisson bracket that changes is the one between Nj and pk,which becomes

Nj , pk = δjk (p0 + µ) . (2.81)

The Casimir (2.60) changes also as

C = p20 + 2p0µ+ µ2 − ~p2 . (2.82)

We notice that the quadratic Casimir contains also the central charge µ, (whichrepresents the first Casimir). In order to obtain the correct limit to the Galileialgebra, we subtract first from the quadratic Casimir the unnecessary constant termµ2. So that

C → p20 + 2p0µ− ~p2 . (2.83)

The contraction is made by rescaling the generators and the charge as

p0 → εp0 , Nj →Nj

ε, µ→ m

ε. (2.84)

This rescaling is easily understood, once recalled that the physical interpretationwe adopted for the p0 generator is p0 = E

c for special relativity, while p0 = E forGalilean relativity, with E the particle’s (kinetic) energy, assuming ε = 1

c , and

8From a physical point of view this change is understandable. In Galilei algebra p0 ≡ E representsonly the kinetic energy, while in Poincaré algebra p0c ≡ E represents the total energy of the particle.


recalling the definition of the boost generators (2.32), (2.65) (the rescaling of themass follows by dimensional analysis). We thus obtain the algebra

pj , pk = 0 , p0, pj = 0 ,Rj , Rk = εjklRl , Rj , p0 = 0 , Rj , pk = εjklpl , (2.85)

Nj , Nk=−ε2εjklRl, Rj , Nk=εjklNl, Nj , p0=pj , Nj , pk=δjk(ε2p0 +m

),

C = ε2p20 + 2p0m− ~p2 , (2.86)

from which, taking the limit ε→ 0, we obtain the Galilei algebra (2.26), (2.27).

2.5 Loss of simultaneity and synchronization of clocksThe issue of locality will be of critical importance in the following chapters whenintroducing DSR theories. We here review some of the features of locality related toGalilean and special relativity.

Consider two observers, Alice and Bob, connected by a Galilean boost (2.56), i.e.Bob has velocity vx (in the direction of x-axis) respect to Alice. Consider now twodistinct points9 which, for Alice, have coordinates

(t1, x) , (t2, x) , (2.87)

with t2 > t1. I.e., the two points mark events that take place in the same spaceposition, but at different times; the events that they mark are at relative rest respectto Alice. By using the equations (2.56), we find that Bob describe those same twopoints as having coordinates

(t1, x− vxt1) , (t2, x− vxt2) . (2.88)

The situation is pictured in Fig. 2.1. While the two events are at relative rest inAlice’s description, they take place (at different times) in different positions in Bob’sdescription. The rest is relative in Galilean relativity10. We can also say that inGalilean relativity space locality is relative.

tA

xA

tB

xB

(t1, x) (t2, x)(t1, x− vxt1)

(t2, x− vxt2)

vx

vx|∆t|

Figure 2.1. Galilei relative rest

9With points here we indicate more precisely coordinates of an event.10We can use this scheme to describe the paradigmatic situation in which Bob is on a boat moving

at velocity vx respect to Alice, who is standing on the dock. Imagine that Alice is bouncing a ballon the dock, and that the two points mark the position and time of two of the ball’s bounces on theground. While Alice evidently observes the ball bouncing at the same point in space, Bob, who ismoving with velocity vx relative to Alice, observes, in his reference frame, the ball bouncing in twodifferent positions: if Bob is approaching the dock, for example (as in Fig. 2.1), Bob sees the secondbounce closer then the first.

2.5 Loss of simultaneity and synchronization of clocks 25

On the contrary, if we consider two points which in Alice’s frame have coordinates(t, x1

)and

(t, x2

)(with x2 > x1), we find by (2.56) that they are described by Bob

as (t, x1 − vxt

),

(t, x2 − vxt

). (2.89)

tA

xA

tB

xB

(t, x1)

(t, x2) vx(t, x2 − vxt)

(t, x1 − vxt)

Figure 2.2. Galilei absolute simultaneitySo that (see Fig. 2.2) two events which take place in different positions but at thesame time in Alice description, are still described by Bob as occurring at the sametime. Simultaneity (which we can also call “time locality”) is absolute11 in Galileanrelativity. As a limiting case it is worth noticing that if the two events are takingplace in the same spatial point and are also simultaneous in Alice’s frame, i.e. theyare marked by the same point

(t, x), they are still marked by a single point in Bob’s

frame:(t, x− vxt

). The rest is absolute if the two events happens simultaneously:

in Galilean relativity “space locality” is still absolute for simultaneous events.The picture changes when switching to special relativity. Consider the second

situation described before: two points marking two simultaneous events in Alice’sframe, but in different space positions. And consider Bob being boosted respect toAlice by an infinitesimal boost along the x direction with rapidity ξx. By using theLorentz (infinitesimal) boosts (2.73), and recalling that (see (2.78)) vx = ξc, we findthat Bob describes Alice’s points (2.87) as(

t− vxc2 x1, x1 − vxt

),

(t− vx

c2 x2, x2 − vct). (2.90)

(t, x1 − vxt

),

(t, x2 − vxt

). (2.91)

tA

xA

tB

xB

(t, x1)

(t, x2) ξx (t− vxx2/c2, x2 − vxt)

(t− vxx1/c2, x1 − vxt)

ξx|∆x|/cFigure 2.3. Einstein relative simultaneity

It thus follows, as shown in Fig. 2.3, that the two events, that were simultaneousin Alice’s description, are described by Bob as happening at different times. In thelimit in which c→∞ we recover the Galilean case of Fig. 2.2.

11Again we can imagine two balls which in Alice’s frame, bounce simultaneously. The situationremains the same in Bob’s frame. In particular if the balls bounce in sync in Alice’s frame, they arestill in sync in Bob’s frame.


We can state that the introduction of the invariant scale c in a relativistic theory,enforces the loss of absolute simultaneity. In special relativity simultaneity becomesrelative. We also stress that in the limit in which the two points coincide in Alice’sframe both in space and in time, simultaneity is still absolute. Indeed, we can say thatin special relativity, space and time locality, separately, are relative; but spacetimelocality is still absolute. We will see how, in a similar manner in which simultaneity(time locality) is affected by special relativity, in DSR, the introduction of anobserver-invariant inverse-momentum scale ` affects the absoluteness of spacetimelocality.

We can show as illustrative example of one of the consequences of relativesimultaneity, the situation depicted in Fig. 2.4: Alice and Bob, distant observersin relative motion (with constant speed), have stipulated a procedure of clocksynchronization and they have agreed to build emitters of blue photons (blueaccording to observers at rest with respect to the emitter). They also agreed tothen emit such blue photons in a regular sequence, with equal time spacing ∆t∗.We notice that Bob’s worldlines are obtained combining a translation and a boosttransformation (Bob = B . T .Alice), so that, using Eqs. (2.47) (2.74), (2.50), and(2.70), we find

xB1

(xB0

)p=γ

(xA−a1−β

(xA0−a0

))+ pA1 −βpA0pA0 −βpA1

(xB0 −γ

(xA0−a0−β

(xA−a1

))), (2.92)

where we made the same substitutions as in (2.79). We can use the last formula toobtain the worldlines in Fig. 2.4. The black worldlines are the worldlines of Alice’sand Bob’s emitter and detector.

We arranged the starting time of each sequence of emissions so that there wouldbe two coincidences between a detection and an emission, which are of coursemanifest in both coordinatizations, so to obtain a specular description. Relativesimultaneity is directly or indirectly responsible for several features that would appearto be paradoxical to a Galilean observer (observer assuming absolute simultaneity).In particular, while they stipulated to build blue-photon emitters they detect redphotons, and while the emissions are time-spaced by ∆t∗ the detections are separatedby a time greater than ∆t∗.

c tA

xA

c tB

xB

Figure 2.4. Relative simultaneity and synchronization of clocks.

27

Chapter 3

DSR theories andκ-Poincaré/κ-Minkowski

The principle of relativity, first stated by Galilei, affirms that the laws of physicstake the same form in all inertial frames (i.e. these laws are the same for all inertialobservers). Implicit in Galilei relativity, in which the laws Newtonian mechanicsare invariant, is that the laws of physics do not involve any fundamental scales ofvelocity and/or length. Then, one can think of special relativity as the introductionin a relativistic theory (based on the principle of relativity) of an invariant velocityscale c (the speed of light), so that the laws of physics involve c as a fundamentalvelocity scale. We have shown in the previous chapter that special relativity canbe thought as a c−1-deformation of Galilean relativity. Moreover Einstein specialrelativity requires that the value of the fundamental velocity scale c can be measuredby each inertial observer as the speed of light (speed of massless particles).

DSR theories where introduced [12] to investigate the possibility of introducing,beside c, a fundamental inverse-momentum1 scale ` (usually thought to be, in theQuantum Gravity literature, roughly at the Planck scale) as a relativistic invariant.The requirements of DSR then are that the laws of physics involve both a fundamentalvelocity scale c and a fundamental inverse-momentum scale `, and that each inertialobserver can establish the same measurement procedure to determine the value of` (besides the invariant measurement procedure to establish the value of c). Inparticular, since in DSR theories the speed of light could depend on its momentum,the value of c is assumed to be measured by each inertial observer as the value ofthe velocity of massless particles in the “soft” limit. This is the limit in which thephoton’s momentum is so small that the effects of the `-deformation are negligible.

An operative procedure for measuring the value of ` can be defined, for example,in DSR models in which the scale of deformation enters in the form of the dispersionrelation:

E2 − ~p2c2 + f (pµ; `)−m2c4 = 0 . (3.1)

Each inertial observer can establish the value of ` (the same value for all inertialobservers) by determining the dispersion relation (3.1) in its reference frame. Partic-

1In this thesis we focus on the point of view in which the relativistic invariant has the dimensionof an inverse-momentum (the inverse of the Planck scale). An equivalent definition could be for thedimension of the deformation scale to be a length scale (the Planck length).

28 3. DSR theories and κ-Poincaré/κ-Minkowski

ularly interesting (especially from the phenomenological point of view, due to thesmallness of the deformation scale), is the case where the leading order contribute of` in the dispersion relation (3.1) does not vanish, in which case, neglecting higherorder in `, one can assume, for the dispersion relation, the generic form (uponenforcing analyticity of the deformation and invariance under classical space-rotationtransformations)

E2 − ~p2c2 + αÈ3c−1 + βÈ~p2c−m2c4 = 0 , (3.2)

where α and β are numerical parameters.It is worth noticing here a technical difference between DSR theories and theories

in which Lorentz symmetries are broken (LSB theories), which, as we will show inmore detail in chapter 5, could imply significant phenomenological consequences.As already manifest from the previous discussion, the laws of physics, in presenceof a fundamental inverse-momentum scale `, cannot be invariant under ordinaryLorentz transformations. For example, supposing the scale to be introduced in thedispersion relation as in Eq. (3.2), it is evident (consider equations (2.79)) thatunder a Lorentz transformation Eq. (3.2) changes its form.

Take for simplicity the α = 0, β = 1 case2 of Eq. (3.2). Suppose that thisrepresents the particles dispersion relation for an observer Alice, so that in hercoordinatization she describes the dispersion relation E2

A−~p2Ac

2+ÈA~p2Ac−m2c4 = 0.

For an observer Bob, connected by infinitesimal Lorentz boost with rapidity ~ξ, thedispersion law takes the form E2

B−~p2Bc

2+ÈB~p2Bc−`~ξ · ~pB~p2

Bc2−2`~ξ · ~pE2

B−m2c4 =0.If Bob adopted the same operative procedure used by Alice to measure the valueof the fundamental scale `, he would obtain a different value of the scale (differentfrom `).

This is essentially the approach adopted in LSB theories, in which the funda-mental scale `LSB, is introduced as a scale at which the covariance under Lorentztransformations of the laws of physics involving `LSB is lost. In the LSB frameworkthe physical definition of the fundamental scale changes under (Lorentz) boosttransformations. Observers connected by (Lorentz) boost transformations woulddescribe different laws of physics, which implies the existence of a sort of “quantumgravity aether”, i.e. a preferred reference frame in which the laws of physics take aspecific form, for example, the specific form (3.2).

The assumption of DSR theories is that the principle of relativity is not violated,even in presence of an observer-invariant inverse-momentum scale `, so that forexample the dispersion relation (3.2) must have the same form for all inertialobservers. It is clear from the above discussion that in DSR theories the laws oftransformation between inertial observers must be modified respect to the specialrelativistic ones. This is achieved, in analogy with the c−1-deformation that bringsfrom the Galilei relativity description to Einstein relativity (as shown in the previouschapter), through a `-deformation of special relativity transformation laws.

In this chapter we characterize the κ-Poincaré/κ-Minkowski model of DSR (whichwe will consistently call R-DSR) to which we will refer as illustrative example. We will

2Notice that Eq. (3.2) represents in principle a dispersion relation with 2 independent deformationscales `′ = α`, `′′ = β`. We restrict to the case α = 0 to consider the simpler illustrative case withonly one deformation scale.

3.1 κ-Minkowski/κ-Poincaré properties description in terms of plane waves 29

not give a self contained review of the mathematical structures of Hopf algebras, andin particular we will skip rigorous demonstrations of the properties of κ-Poincaré/κ-Minkowski, for which we refer to the wide literature on the subject (see [91] andreference therein). But we shall be satisfied with pointing out (furnishing in somecases a reasonable justification) the properties of κ-Poincaré/κ-Minkowski that willbe useful in the following for the analysis we propose. We will define the relevantproperties relying on a description based on plane waves in κ-Minkowski [19, 91].

From now on we work in units such that the speed-of-light scale (speed of masslessparticles in the infrared limit) and the Planck constant are 1 (c = 1, ~ = 1).

3.1 κ-Minkowski/κ-Poincaré properties description interms of plane waves

In this section we shall review the most studied formulation of theories in κ-Minkowski spacetime, often labeled as “bicrossproduct basis” [55, 58] or “time-to-the-right basis” [55, 61], characterizing its properties in terms of plane waves inκ-Minkowski [19, 91]. From the commutation relations (3.3)κ-Minkowski spacetimecan be defined as a Hopf algebra Xκ with the properties

[xj , x0] = i`xj , [xj , xk] = 0 , (3.3)

∆ (xµ) = xµ ⊗ 1 + 1⊗ xµ , (3.4)

S (xµ) = −xµ , ε (xµ) = 0 , (3.5)

with the symbols ∆, S, ε, meaning respectively the coproduct, the antipode, andthe counit of the Hopf algebra.

κ-Minkowski spacetime can be connected by nonAbelian Fourier theory preciselyto functions on a classical but nonAbelian momentum group [19]. This suggests [19]a natural definition of (time-to-the-right) plane waves associated to a point pµin momentum space, provided by the generalization of the same notion in thecommutative case:

ψp = eipj xje−ip0x0 . (3.6)

Directly by use of Eq. (3.3), one finds that these respect the group law on momentumspace in the sense that their product gives

ψpψq = ψp⊕q , (3.7)

with the symbol ⊕ defining the composition law of momenta3

(p⊕ q)0 = p0 + q0 ,

(p⊕ q)j = pj + e−`p0qj , (3.8)

The plane waves (3.6) admit inverse (plane wave in the reverse direction in momentumspace) as

ψ−1p = ψp , (3.9)

3Notice that the composition law is associative due to the fact that the group multiplication isalso associative.


with the symbol defining the inversion

(p)0 = −p0 , (p)j = e`p0pj . (3.10)

We can use the time-to-the-right basis of plane waves to define the action of theκ-Poincaré generators of translations, space-rotations and boosts. The generators ofκ-Poincaré translations are conveniently characterized through the following rule

Pµ . ψp = pµψp . (3.11)

Similarly one has that the generators of space rotations are given by

Rj . ψp = εjklxkplψp , (3.12)

and the generators of boosts are given by

Nj . ψp =[−x0pj + xj

(1− e−2`p0

2` + `

2~p2)]

ψp . (3.13)

Notice these actions can be expressed in momentum space as

Rj . ψp = iεjklpk∂

∂plψp , (3.14)

Nj . ψp = i

[pj

∂

∂p0+(

1− e−2`p0

2` + `

2~p2)

∂

∂pj− `pjpk

∂

∂pk

]ψp . (3.15)

The fact that the translation sector of the κ-Poincaré Hopf algebra is itself aHopf algebra is essentially manifest in the observation that the rule of action (3.11),together with the noncommutativty property of coordinates (3.3) and the consequentproduct law (3.7), (3.8), implies that for these translation generators the action onproducts of functions is governed by a deformed Leibniz rule

Pµ . ψpψq = Pµ . ψp⊕q = (p⊕ q)µ . (3.16)

from this follow

P0 . ψpψq = p0ψpψq + ψpq0ψq = (P0 . ψp)ψq + ψp (P0 . ψq) , (3.17)

Pj . ψpψq = pjψpψq + e−`p0ψpqjψq = (Pj . ψp)ψq +(e−`P0 . ψp

)(Pj . ψq) , (3.18)

i.e. the spatial translation generators have “non-primitive coproduct”

∆Pj = Pj ⊗ 1 + e−`P0 ⊗ Pj . (3.19)

With this coproduct one can show that the commutator (3.3) is conserved in thefollowing sense

Pµ . [x1, x0] = i`Pµ . x1 . (3.20)

We notice also that while the composition law ⊕ of momentum space reflects thecoproduct properties of κ-Poincaré, the inversion reflects the “antipode” of Pµ(see also [92]): (p)µ ↔ S (Pµ), where S (P0) = −P0, S (Pj) = −e`P0Pj .

3.1 κ-Minkowski/κ-Poincaré properties description in terms of plane waves 31

We can reconstruct the whole bicrossproduct (time-to-the-right) κ-Poincaréalgebra Pκ, defined in [55], by relations (3.11), (3.12) and (3.13), finding

[Pµ, Pν ] = 0 , [Rj , Rk] = iεjklRl , [Nj ,Nk] = −iεjklRl ,[Rj ,Nk] = iεjklNl , [Rj , P0] = 0 , [Rj , Pk] = iεjklPl ,

[Nj , P0] = iPj , [Nj , Pk] = iδjk

(1− e−2`P0

2` + `

2~P 2)− i`PjPk , (3.21)

with non primitive coproducts (the other coproducts are all primitive) (3.19) and

∆Nj = Nj ⊗ 1 + e−`P0 ⊗Nj + `εjklPk ⊗Rl , (3.22)

with antipodes

S (P0) = −P0 , S (Pj) = −e`P0Pj ,

S (Rj) = −Rj , S (Nj) = −e`P0Nj + `εjkle`P0PkR` , (3.23)

and with all zero counits. One also finds the quadratic Casimir to be

` =(2`

)2sinh2

(`

2P0

)− e`P0 ~P 2 , (3.24)

which, keeping in mind the review of the relativistic particle in covariant formalismof the previous chapter, can be linked to the possibility of having a `-deformeddispersion relation of the kind (3.1).

We also note that, from the rules of action of the translation generators Pµ (3.11),(3.17) and (3.18), one finds the relations

P0 . x0ψp − x0P0 . ψp = iψp

P0 . xjψp − xjP0 . ψp = 0 ,Pj . x0ψp − x0Pj . ψp = −i`pjψp ,Pj . xkψp − xkPj . ψp = −iδjkψp .

(3.25)

Inspired by these relations, one can argue “deformed commutation relations” betweenpositions and translation generators

[P0, x0] = i , [P0, xj ] = 0 ,[Pj , x0] = −i`Pj , [Pj , xk] = −iδjk . (3.26)

We notice that these commutation relations coincide with the ones constructed by“Heisenberg double” [93, 94], where a product law, inherited by the action rules ofPµ on Xκ, is defined in the space Xκ ⊗Pκ. We will use these relations directly, inthe following section, to construct our `-deformed (not quantum) phase space, whilewe will see in Ch. 8 how a more precise definition of a quantum phase space inκ-Minkowski/κ-Poincaré can be made in terms of a “pregeometric” approach.


3.2 Time-to-the-right DSR symplectic structureWe now define the symplectic structure that we will use in the next chapter asstarting point for the analysis of the motion of a free particle in DSR. This setup isinspired by the bicrossproduct (time-to-the-right) version of κ-Poincaré/κ-Minkowskiintroduced in the previous section, and we will refer briefly to it as R(Time-to-the-Right)-DSR. We start by characterizing the algebra of symmetry generatorsin terms of Poisson brackets, by analogy with the algebraic sector (3.21) of Pκ,substituting4 [·, ·]→ i ·, ·, and substituting the generators Pµ with the momentumspace coordinates pµ:

pµ, pν = 0 , Rj , Rk = εjklRl , Nj ,Nk = −εjklRl ,Rj ,Nk = εjklNl , Rj , p0 = 0 , Rj , pk = εjklpl ,

Nj , p0 = pj , Nj , pk = δjk

(1− e−2`p0

2` + `

2~p2)− `pjpk , (3.27)

C` =(2`

)2sinh2

(`

2p0

)− e`p0~p2 , (3.28)

In the same way, we define the phase space in analogy with the relations (3.26)as

p0, x0 = 1 , p0, xj = 0 ,pj , x0 = −`pj , pj , xk = −δjk , (3.29)

where, from (3.3), we have also the Poisson bracket

xj , x0 = `xj . (3.30)

The equations (3.29), (3.30), together with the first of (3.27), defines the Poissonbivector in R-DSR as

ΩR−DSR =

0 0 0 0 1 0 0 00 0 0 0 −`p1 −1 0 00 0 0 0 −`p2 0 −1 00 0 0 0 −`p3 0 0 −1−1 `p1 `p2 `p3 0 −`x1 −`x2 −`x30 1 0 0 `x1 0 0 00 0 1 0 `x2 0 0 00 0 0 1 `x3 0 0 0

. (3.31)

In terms of the phase space variables pµ, xµ, the rotation and boost generators haverepresentations

Rj = εjklxkpl , Nj = −x0pj + xj

(1− e−2`p0

2` + `

2~p2). (3.32)

analogous to (3.12) and (3.13).4Remember that we are assuming ~ = 1. it would be [·, ·]→ i

~ ·, ·.

3.2 Time-to-the-right DSR symplectic structure 33

One can verify, using the representations (3.32), that given the Poisson bracketsdefined by Eqs. (3.29), (3.30) and (3.27), the Jacobi identities for the whole set ofphase space functions pµ, xµ, Nj , Rj , are satisfied, and it is worth noticing that inorder for this to be true, if the cross relations of phase space are defined by (3.29), onehas to consider the nontrivial Poisson bracket (3.30) between spacetime coordinates,i.e. the Jacoby identities would not be satisfied by the relations (3.29) if xj , x0where zero. This ensures, as shown in Sec. 2.1, that the symplectic transformationsgenerated by the elements of the algebra (3.27) preserve the Poisson structureitself, in such a way that every inertial observer connected by such transformationsdescribes the same symplectic structure in his coordinates.

The relations from (3.27) to (3.32), will define the `-deformation of the Poincaréalgebra (2.59) that we will use in the next session to describe the motion of a freeparticle in an example of DSR theory. They will be also the starting point tointroduce interactions in DSR in Ch. 6, through the Principle of Relative localityapproach. To complete this section we write again the composition law of momenta(3.8) and the inverse (3.10) that will be needed to the latter purpose:

(p⊕ q)0 = p0 + q0 ,

(p⊕ q)j = pj + e−`p0qj , (3.33)

(p)0 = −p0 , (p)j = e`p0pj . (3.34)

Since these relations derive from the κ-Poincaré/κ-Minkowski description firstlydefined in [55], we will call them in the following the “Majid-Ruegg compositionlaw”.

We close this section noticing that one can define a map on the spacetimecoordinates xµ,

x0 → x0 = x0 − `xjpj , xj → xj = xj , (3.35)

that trivialize the symplectic structure, in the sense that

xj , x0 = 0 (3.36)

p0, x0 = 1 , p0, xj = 0 ,pj , x0 = 0 , pj , xk = −δjk , (3.37)

so that the Poisson bivector is canonical. One can show that the Jacobi identitiesare satisfied also for the set of functions pµ, xµ, Nj , Rj .

35

Chapter 4

Free particle in DSR: relativityof locality

We have seen in chapter 2 (see Eq. (2.43), (2.71), and (2.66)), that when thesymplectic structure is canonical, the velocity derived as dxj(t)/dt, coincides with thevelocity derived as dE(pj)/dpj . In the κ-Poincaré/κ-Minkowski inspired model forDSR theories we defined in the previous chapter (R-DSR), the symplectic structureis not canonical, and, as we will see, dxj(t)/dt 6= dE(pj)/dpj . We will call the velocitydxj(t)/dt the “coordinate velocity”, and we will show that it is not the physical velocity,which instead still coincides with dE(pj)/dpj . This is the chapter where the notion ofrelative locality plays a role. We shall see that awareness of the possibility of relativelocality is an important aspect for DSR relativistic theories. It is through relativelocality that a long dispute [19, 56, 62, 63, 64, 65, 67, 66, 68, 69, 70, 71, 72, 73])between advocates of dxj(t)/dt [62] and advocates of dE(pj)/dpj is solved. We willdiscuss this topic in Sec., were we analyze the motion of free particles in the R-DSRframework defined in Sec. 3.2. In Sec. 4.1 we show how Relative locality emergeswhen the non-trivial role of deformed (R-DSR) translations is taken into account.

The preparation of the previous chapters, allows us to stress the analogy (Sec. 4.2)between the loss of of absolute locality due to the introduction of an observer-invariantinverse-momentum scale ` in DSR theories and the loss of absolute simultaneity dueto the introduction of an observer-invariant velocity scale c in special relativity. Asstressed at the beginning of the previous chapter, the introduction of the invariantmomentum scale `−1 requires a deformation of Poincaré transformations. And forthe logical balance of the relativistic theory one finds that having one more thinginvariant (the scale `) requires rendering one more thing relative, which is the roleplayed by relative locality. As shown in chapter 2, the same logics applies to thetransition from Galilean Relativity to Einstein’s Special Relativity. Taking as startingpoint Galilean Relativity, the introduction of the invariant velocity scale c requires adeformation of Galilean boosts. And for the logical balance of the relativistic theoryone finds that having one more thing invariant (the scale c) requires rendering onemore thing relative, which is the role played by relative simultaneity.

In Sec. 4.3 we discuss how the same analysis of Sec. 4.1 can be performed incanonical spacetime coordinates, defined in Sec. 3.2. In canonical coordinates one hasthat the translations are undeformed, so that in these coordinates relative locality

36 4. Free particle in DSR: relativity of locality

for translations is removed. We show however that the physical results coincide withthe ones derived in the case of R-DSR (non-canonical) spacetime coordinates.

It could seem then that relative locality can be removed by a suitable choiceof spacetime coordinates. We show in the remainder of the chapter that this isnot the case by considering the whole set of spacetime transformations (other thantranslations). We show first (Sec. 4.4) that relative locality is still present for thecase of observers connected by R-DSR boost transformations, even in canonicalcoordinates. We then exhibit (Sec. 4.5) a more general result. Without relying inany specific DSR model, we show how relative locality is an unavoidable feature ofany DSR theory predicting a (leading-order in `) momentum dependent velocity ofphotons. Moreover we show in general how locality is still absolute for events takingplace locally to the origin of the observer.

We are working in units such that c = 1, so that the speed of a “soft” photon(massless particle), i.e. a photon for which we can neglect the effects of deformationon its motion, will be v = 1. For simplicity, unless otherwise specified, we willdevelop the analysis in 2D dimensions (1+1D: one time and one space dimension), sothat the spacetime metric will be defined by the tensor ηµν = (1,−1). Moreover, dueto the smallness of the deformation scale we want to contemplate for our physicalinterpretation (close to the inverse of the Planck scale, 1/Ep ∼ 10−19GeV) of thetheory 1 we will limit our analysis to the first order in the deformation parameter `,neglecting higher orders contributes.

With these assumptions, we take as starting point of our analysis the Poissonstructure R-DSR defined2 in Sec. 3.2 by the equations (3.27), (3.28),(3.29),(3.30).So that, in 1+1D, the Poisson structure to which we will refer as R-DSR will begiven by

pµ, pν = 0 , xj , xk = 0x1 , x1, x0 = −`x1 ,

p0, x0 = 1 , p0, x1 = 0 ,p1, x0 = `p1 , p1, x1 = −1 , (4.1)

N , p0 = p1 , N , p1 = p0 + `p20 + `

2p21 , (4.2)

where N is the boost generator (in the x1 direction), represented by

N = −x0p1 + x1p0 + `x1

(p2

0 −p2

12

). (4.3)

The quadratic Casimir becomes

C` = p20 − p2

1 + p0p21 . (4.4)

The results of this chapter are reported in our works [11] and [38].1Notice that in principle the DSR theory we defined does not depend on the interpretation of `

as the inverse of the Planck scale, and can describe a generic invariant-observer inverse-momentumscale.

2We take the opposite sign of ` respect to the formulas of Sec. 3.2, in order to describe,consistently throughout this thesis, subluminal (superluminal) deviations from speed-of-light forpositive (negative) `.

4.1 Relative locality for translations 37

4.1 Relative locality for translations: physical and co-ordinate speed of particles

4.1.1 R-DSR particle in covariant Hamiltonian formalism

We now proceed in analogy with the analysis developed in Sec. (2.2) and (2.3) tostudy the motion of a particle in R-DSR model. We start by defining, from theCasimir (4.4), the Hamiltonian constraint

H = p20 − p2

1 + `p0p21 −m2 . (4.5)

By setting the constraint to zero, H = 0, we find the on-shell dispersion relation

p0 (p1) =√p2

1 +m2 − `

2p21 . (4.6)

From the dispersion relation, one finds the “wave velocity”

v (p1)wavem = dp0 (p1)dp1

= p1√p2

1 +m2− `p1 . (4.7)

We see in particular that for a photon,

|v (p1)wavem=0 | = 1− ` |p1| , (4.8)

the wave velocity depends is momentum dependent.The Hamiltonian flow generates the equations of motion

x1 = H, x1 = 2 (p1 − `p0p1) ,

x0 = H, x0 = 2(p0 −

12`p

21

),

(4.9)

After integration, we have

x1 (τ)− x1 (τ) = 2 (p1 − `p0p1) τ ,

x0 (τ)− x (τ) = 2(p0 −

12`p

21

)τ ,

(4.10)

we can thus factor out the affine parameter τ and find the worldlines

x1 (x0; p0, p1; x0, x1) = x1 +(p1p0− `

(p1 −

p31

2p20

))(x0 − x0) , (4.11)

where xµ = xµ (τ). We can identify the (coordinate) velocity as given by

v (p0, p1) = p1p0− `

(p1 −

p31

2p20

). (4.12)

Notice that after enforcing the onshell relation H = 0, substituting the dispersionrelation (4.6), the (coordinate) velocity is given by

v (p1)m = p1√p2

1 +m2− ` m2p1

m2 + p21, (4.13)


so that the worldline is

x1 (x0; p0, p1; x0, x1) = x1 +

p1√p2

1 +m2− ` m2p1

m2 + p21

(x0 − x0) , (4.14)

which, for a massless particle, becomes

x1 (x0; p1; x0, x1)m=0 = x1 + p1|p1|

(x0 − x0) , (4.15)

from which one can see that for massless particles the (first order) ` contribution tothe coordinate velocity vanishes:

|v (p1)m=0| = 1 . (4.16)

The coordinate velocity (4.13), for which the speed of photons is momentumindependent, is in conflict with the wave velocity (4.7), for which the speed ofphotons depends on momentum.

Formally this discrepancy can be understood once taken into account of the noncanonicity of the phase space (4.1). Indeed, by the chain rule

v = dx1 (x0)dx0

≡ dx1dτ

(dx0dτ

)−1= H, x1H, x0

, (4.17)

but, differently from (2.39), now

H, x1H, x0

=(p1, x1

∂H∂p1

)(p0, x0

∂H∂p0

+ p1, x0∂H∂p1

)−1

= −∂H∂p1

(∂H∂p0

+ `p1∂H∂p1

)−1' −∂H

∂p1

(∂H∂p0

)−1+ `p1

(∂H∂p1

(∂H∂p0

)−1)2

,

(4.18)

which, using again the chain rule, can be rewritten as

− ∂p0 (p1)∂p1

+ `p1

(∂p0 (p1)∂p1

)2, (4.19)

and, raising the momentum index in the derivative with the metric ηµν = (1,−1),the coordinate velocity finally writes

v (p1) = ∂p0 (p1)∂p1 + `p1

(∂p0 (p1)∂p1

)2, (4.20)

which can be confronted with (4.7): the deformed symplectic structure is at theorigin of the mismatch between coordinate and wave velocity.

We will see that the key to understand this discrepancy relies on the role ofthe symmetry transformations generated by Eqs. (4.1) and (4.2) in characterizingthe description of physical events made by different inertial observers. Therefore,we introduce now, as made in Sec. (2.2) and (2.3) for the Galilean and specialrelativistic case, the action of the symmetry transformations on the phase spacecoordinates.


4.1.2 Covariance of the worldlines

Before going through the main analysis, we show explicitly that the worldlines (4.14)are covariant under the transformations generated by pµ and N by Poisson brackets(see Sec. 2.1), as it should be guaranteed by the arguments of Sec. 2.1 (see inparticular Eq. (2.14) and the following). We characterize the worldline described bya generic inertial observer I by labeling the coordinates in (4.14), (4.13), with thesuffix I, so that he describes the motion of a particle in terms of the worldlines

xI1

(xI0

)m,p1

= xI1 + v(pI1

)m

(xI0 − xI0

). (4.21)

We consider two observers, Alice and Bob, connected by infinitesimal symmetrytransformations. What we want to show is that, if the worldlines described by Aliceare given by the relation

xA1 − xA1 − v(pA1

)m

(xA0 − xA0

)= 0 , (4.22)

then the worldlines described by Bob, whose coordinates are connected to Alice’s,as introduced in Sec. 2.1, by a translation or infinitesimal boost generated by pµ orN as in (4.24), (2.53), have the same form of Alice’s worldlines, and are given bythe relation

xB1 − xB1 − v(pB1

)m

(xB0 − xB0

)= 0 . (4.23)

Consider first a translation, whose action on the set of phase space variablesk = (pµ, xµ) is given by (see (2.47))

kB = Ta0,a1 . kA = kA − a0

p0, k

A

+ a1p1, k

A, (4.24)

then, by (4.1), Bob’s coordinates will be related to Alice’s by

xB0 = xA0 − a0 + à1pA1 , xB0 = xA0 − a0 + à1p

A1 ,

xB1 = xA1 − a1 , xB1 = xA1 − a1 ,

pB1 = pA1 .

(4.25)

Substituting these relations in (4.23), we find straightforwardly that the equation(4.23) is true if (4.22) holds.

For the infinitesimal boosts, defined as in (2.53) by

Bξ . kA = kA − ξN , kA

, (4.26)

one finds, using (4.2),

xB0 = xA0 − ξxA1 − `ξ(√

m2 + p21xA1 − pA1 xA0

),

xB0 = xA0 − ξxA1 − `ξ(√

m2 + p21xA1 − pA1 xA0

),

xB1 = xA1 − ξxA0 , xB1 = xA1 − ξxA0 ,

pB1 = p1 − ξ√m2 + p2

1 − `ξ(m2 + p2

1

).

(4.27)


where we also enforced on-shellness to eliminate p0 through (4.6). We first noticethat under these substitutions, the velocity (4.13) in Bob’s coordinates is related tothe velocity in Alice’s coordinates by (remember that we working at leading orderboth in ` and in ξ)

v(pB1

)m

= pB1√(pB1)2 +m2

− ` m2pB1

m2 +(pB1)2

= pA1√m2+

(pA1)2 − ξm2

m2+(pA1)2 − `

m2pA1

m2+(pA1)2 +

2ξm2(pA1

)2

(m2+

(pA1)2)3/2

.(4.28)

Substituting the last expression and Eqs. (4.27) in Bob’s worldline (4.23), one finds,after some cumbersome calculations,

xB1 − xB1 − v(pB1

)m

(xB0 − xB0

)=(xA1 −xA1 −v

(pA1

)m

(xA0 −xA0

))1+ξ pA1√m2+

(pA1)2 + `ξ

(pA1

)3

m2+(pA1)3, (4.29)

so that if Eq. (4.22) is true then (4.23) holds.

4.1.3 The role of deformed translations: relative locality

We have shown in Sec. 4.1.1 that the coordinate velocity, i.e. the velocity that anobserver, in its coordinatization of events, attributes to a massless particle worldline,does not present any `-deformed term, so that a single observer would say that thevelocity of a massless particle does not depend on the particle momentum. But wehave to keep in mind the role of inferences in coordinatization effects. We have seenbefore that in relativistic theories coordinate artifacts can arise in different ways(see Sec. 2.5).

Thus, in order to obtain a correct, physical derivation of the velocity of a particlein our deformed framework, we rely on an operative definition of velocity based onthe exchange of signals between two different inertial observers, guided by the ideathat all physical measurements are local measurements (see also the discussion ofclassical physical observables in Sec. 2.2). We study the case of a simultaneousemission of two photons (massless particles): a low energy (soft) one, such that allthe effects of deformation to its motion can be neglected, i.e. with momentum pssuch that the `-deformed terms in the formulas describing its motion fall below theexperimental sensitivity available, and a high energy (hard) one, for which the effectsof the `-deformation are tangible, i.e. with momentum ph big enough that at leastthe leading `-deformed terms in formulas fall within the experimental sensitivityavailable3. The two photons are emitted from an ideally pointwise emitter, and aredetected by a detector placed at a certain distance L.

3To be more precise, for the specific case here of interest, we are assuming that the detectorresolution δtdet for measuring the times of arrival is much greater then the momentum dependentcontribute to the time of arrival of the soft particle: δtdet

∣∣tsarr(`ps)− Lc

∣∣, where c is the velocityof light, and L the propagation distance; while it is comparable to the momentum dependentcontribute of the hard particle: δtdet .

∣∣tharr(`ph)− Lc

∣∣.


We consider then a first observer, Alice, local to the emitter, and a secondobserver, Bob, at rest respect to Alice, local to the detector. Alice and Bob’scoordinates are connected by a (passive) translation (4.24), as defined in Sec. 2.1.Since the worldlines, as shown in the previous section, are covariant under thetransformations (4.24), i.e. they take the same form for all “inertial” observers, Aliceand Bob respectively describe the motion of the particles in their coordinates interms of the worldlines (see (4.21))

xA1

(xA0

)p1,m

= xA1 + v(pA1

)m

(xA0 − xA0

),

xB1

(xB0

)p1,m

= xB1 + v(pB1

)m

(xB0 − xB0

).

(4.30)

Consider first the worldlines of the two photons described by Alice. Enforcingboth photons to be emitted simultaneously in Alice origin (xA0 = xA1 = 0), the twoparticles are described by Alice in terms of the worldlines

xA1

(xA0

)ps,m=0

= xA0 ,

xA1

(xA0

)ph,m=0

= xA0 ,(4.31)

where we also set ps > 0 and ph > 0 for the spatial component of the particlesmomenta, so that they propagate towards the positive xA1 axis direction.

In Alice’s description, the two worldline would reach the detector simultaneously,at xA0 = L. But Alice is distant from the detector, so she is not a reliable observer tomeasure their arrival time . We consider then the second observer Bob, local to thedetector. To explicitly write Bob’s description of the photons worldlines, we considerthat Bob’s coordinates are related to Alice’s by (4.25), and assume that Bob detectsthe soft particle in his spacetime origin, so that the translation parameters are4

a1 = a0 = L, so thatxB0 = xA0 − a0 + à1p

A1 = −L+ `LpA1 ,

xB1 = xA1 − a1 = −L ,

pB1 = pA1 ,

(4.32)

and the two particles are described by Bob in terms of the worldlines

xB1

(xB0

)ps,m=0

= xB0 ,

xB1

(xB0

)ph,m=0

= xB0 − `LpBh .(4.33)

Setting to zero the last two equations, we find that while Bob detects the soft particlein his spacetime origin, he detects the hard particle with a delay

∆tB = `LpBh . (4.34)

4Notice that we implicitly assumed that the relative rest of Alice and Bob is established byexchanges of infrared massless particles, so that indeed it can borrow from the special-relativisticoperative definition of inertial observers in relative rest. By construction the distance a betweenAlice and Bob is also defined operatively just like in special relativity, with the only peculiaritythat Alice and Bob should determine it by exchanging infrared massless particles. This setupguarantees that infrared massless particles are timed and observed in κ-Minkowski exactly as inclassical Minkowski spacetime.


xA0

xA1

detector

emitter

(L,L)

xB0

xB1

detector

emitter

∆t = ℓ|ph|L

Figure 4.1. On the left (right) panel is shown Alice’s (Bob’s) description.

This is the delay that one would expect if the velocity of massless particles was(4.7)

|v| = 1− ` |p1| , (4.35)

instead of the coordinate velocity (4.13). Since this is true independent from thedistance between Alice and Bob, we can interpret the velocity (4.35) as the “physical”velocity, unlike the coordinate velocity (4.13), which can be understood indeed as aneffect of the coordinatization of an event by a given observer.

The physical meaning of Bob’s observation of the photons arrival time is connectedwith the discussion in Sec. 2.2 (see formulas (2.40) and (2.41) and the followingdiscussion). Bob is the observer local to the event of detection, so that he is theone that witnesses the physical detection of the particles arrival time. In the sameway the time of arrival defined by setting to zero Eqs. (4.33) and solving for x0, isjust the (DSR analogous of) classical observable (2.41), which commutes with theHamiltonian (4.5).

The non triviality of translation transformations thus, on one side solves thediscrepancy between coordinate and wave velocity, restoring the validity of the latteras “physical velocity”. On the other side introduces a novel effect of “inference”:Alice, distant from the detector, describes the soft and hard photon’s worldlinesto be coincident, and the arrival time at the detector to be simultaneous. Bob,local to the detector, describes distinct worldlines for the soft and hard photons and“witnesses” different arrival times. Conversely Bob, distant to the event of emission,describes the two particles to be emitted at different times, while Alice, local to theemission event, witnesses it to be simultaneous.

We thus understand that in this κ-Minkowski inspired DSR theory, what appearsas a distantly local event to an observer may not be local for an observer witness tothe event: one can say that distant coincidence is a coordinate artifact. To be moreprecise, we established that what actually happens to locality is that it remainsobjective to observers local to the coincidence of events, but observers who aredistant in their coordinatization of spacetime see those same pairs of events as notcoincident: locality becomes relative [11, 38].

4.2 Relative locality: analogy with relativity of simultaneity 43

We will see in Ch. 8, that relativity of locality, which we here derived in anon quantum approach for a model of κ-Poincaré/κ-Minkowski DSR theory, is stillpresent in a full quantum description of κ-Poincaré/κ-Minkowski, enriched of anotherimportant aspect: the fuzzyness of spacetime.

4.2 Relative locality: analogy with relativity of simul-taneity

In section 2.5 we have discussed some aspects of how the transition from Galileanrelativity to special relativity, and in particular the introduction of an observer-invariant velocity scale c, enforces one to abandon the idealization of absolutesimultaneity. We also stressed how the logical balance of the relativistic theorysuggests that having one more thing invariant requires rendering one more thingrelative. The discussion of Sec. 2.5 permits us to draw an analogy between the theemerging of relative locality in DSR theories and the emerging of relative simultaneityin special relativity.

Recalling the considerations of Sec. we can trace the following scheme:

• In Galilean relativity, we can say that one has relativity of spatial locality

tA

xA

tB

xB

(t1, x) (t2, x)(t1, x− vxt1)

(t2, x− vxt2)

vx

vx|∆t|

the rest is relative

tA

xA

tB

xB

(t, x1)

(t, x2) vx(t, x2 − vxt)

(t, x1 − vxt)

while time simultaneity is still absolute

We can say then that one describes space as a whole (while time as a separatequantity). Space is Euclidean, and we can associate every point of it toan observer, such that all the inertial observers are connected by Galileantransformations.


• In Einstein special relativity:invariant scale”c“⇒ absolute (time) simultaneity → relative (time) simultaneity.

tA

xA

tB

xB

(t, x1)

(t, x2) ξx (t− vxx2/c2, x2 − vxt)

(t− vxx1/c2, x1 − vxt)

ξx|∆x|/cThus one has relative space locality and relative simultaneity (time locality),but still absolute spacetime locality.There is no observer-independent projection from spacetime to separately spaceand time. We can say that one describes spacetime as a whole.Spacetime is Minkowski, and we can associate every point of it to an observer,such that all the inertial observers are connected by Poincaré transformations.

• In DSR theories:invariant (inverse-momentum) scale ` ⇒ absolute spacetime locality→ relative spacetime locality

To stress the analogy with the preceding cases, we have to consider not only thespace and time characterization of a point, but the whole phase space. Considertwo observers, Alice and Bob, connected by a a translation (4.25) with parametersa0 = 0, a1 = bx, i.e. Bob is at the spatial distance bx respect to Alice. Take twopoints, characterizing the position of two particles in Alice’s frame, with differentmomenta p1 and p2 (with p2 > p1), but with the same space and time coordinates,for Alice, x,t. If we consider only Alice’s space and time description of the two points,clearly they coincide. But if we picture also the momenta axis, we find that the twopoints are distinct, while their projection on Alice’s spacetime “plane” coincides. IfBob is connected by the translation (4.25) (with parameters a0 = 0, a1 = bx), hedescribes the two points as (see Fig. 4.2)(

t+ `bxp1, x− bxt, p1),

(t+ `bxp2, x− bxt, p2

). (4.36)

t

x(t, x, p1)

(t, x, p2)

p

bx

t

x

ℓbx|∆p|

p

(t+ ℓbx|p2|, x− bx, p2)

(t+ ℓbx|p1|, x− bx, p1)

Figure 4.2. Relative locality in DSR

For Bob, the two points, distinct in phase space, are projected into two dif-ferent points of its spacetime “plane”. Spacetime locality has become relative.

4.3 Canonical coordinates 45

The limiting case in which the momenta of the particles coincide, suggests that“phase space locality” is still absolute in DSR. We thus can say that one describesphase space as a whole, and that there is no observer-independent projection from aone-particle phase space to a description of the particle separately in spacetime andin momentum space. To complete the analogy with the previous cases we noticethat in a full quantum description of R-DSR, spacetime is κ-Minkowski, and all theinertial observers are connected by κ-Poincaré transformations (see also Ch. 8).

4.3 Canonical coordinatesWe have shown in Sec. 3.2, that the phase space can be reduced to the canonicalphase space by the map

x0 → x0 = x0 + `x1p1 , x1 → x1 = x1 , (4.37)

such that the coordinates x0, x1have vanishing Poisson brackets:

x0, x1 = 0 . (4.38)

It is obvious that the translations become the ordinary special relativity transla-tions in these coordinates:

xB0 = xA0 − a0 ,

xB1 = xA1 − a1 ,

pB1 = pA1 ,

(4.39)

and we can see already from relations (4.39) that in these coordinates one doesnot have relative locality for translations, since there are no momentum dependentterms in Eqs. (4.39). Then, thinking of the derivation from Eq. (4.17) to (4.20),and of Eq. (2.39), one can also deduce that, since the phase space is canonical, thecoordinate velocity coincides with the physical, wave velocity in canonical coordinates.Let’s verify it explicitly repeating the analysis of Sec. 4.1 for the exchange of signalsbetween two distant observers at rest, but now assuming that both observers describethe particle motion in the canonical coordinates (4.38).

We start again from the Hamiltonian constraint (4.5)

H = p20 − p2

1 + `p0p21 −m2 , (4.40)

but this time with the canonical phase space

p0, x0 = 1 , p0, x1 = 0 ,p1, x0 = 0 , p1, x1 = −1 ,

(4.41)

x0, x1 = 0 , p0, p1 = 0 . (4.42)

The Hamiltonian flow is given by

˙x1 = H, x1 = 2λ (p1 − `p0p1) ,

˙x0 = H, x0 = 2λ(p0 + 1

2`p21

).

(4.43)


Again we can factor out the affine parameter τ and find the worldlines

x1 (x0; p0, p1; x0, x1) = x1 +(p1p0− `

(p1 + p3

12p2

0

))(x0 − x0) , (4.44)

where we can identify the (coordinate) velocity as given by

v (p0, p1) = p1p0− `

(p1 + p3

12p2

0

)(4.45)

After enforcing the onshell relation H = 0, and the dispersion relation (4.6), the(coordinate) velocity is given by

v (p1)m = p1√p2

1 +m2− `p1 , (4.46)

from which one can see that, differently from the R-DSR coordinates (4.1) case, the` contribution does not vanish even for massless particles,

x1 (x0; p1; x0, x1)m=0 = x1 +(p1|p1|− `p1

)(x0 − x0) , (4.47)

and the velocity coincides with the wave velocity dp0 (p1) /dp1.Again one can verify that the worldlines (4.44) are covariant under the set of

translations generated by (4.41). If we repeat the analysis of Sec. 4.1.3, now, fromEq. (4.47), we find that Alice describes the motion of the two particles in terms ofthe worldlines

xA1

(xA0

)ps,m=0

= xA0 ,

xA1

(xA0

)ph,m=0

=(1− `pAh

)xA0 ,

(4.48)

where again we set ps > 0 and ph > 0 for the spatial component of the particlesmomenta, so that they propagate towards the positive xA1 axis direction.

Thus, in canonical coordinates, we find (substituting xA1 = L and solving for xA0 )that Alice describes the soft and hard particles arriving at the detector respectivelyat xA0 = L and xA0 = L+ `LpAh . Assuming again that Bob detects the soft particlein his spacetime origin, so that the translation parameters are a1 = a0 = L, we findthat Bob’s coordinates are related to Alice’s by the trivial relations (4.39)

xB0 = xA0 − a0 = −L ,

xB1 = xA1 − a1 = −L ,

pB1 = pA1 ,

(4.49)

and the two particles are described by Bob in terms of the worldlines

xB1

(xB0

)ps,m=0

= xB0 ,

xB1

(xB0

)ph,m=0

= xB0 − `LpBh =(1− `pBh

)xB0 − `pBh L .

(4.50)

4.3 Canonical coordinates 47

xA0

xA1

detector

emitter

(L,L)

(L + ℓL|ph|, L)

xB0

xB1

detector

emitter

∆t = ℓ|ph|L

Figure 4.3. On the left (right) panel is shown Alice’s (Bob’s) description.

Setting to zero the last two equations, we find that while Bob detects the soft particlein his spacetime origin, he detects the hard particle with a delay

∆tB = `LpBh . (4.51)

The “physical velocity” of the particles, determined by the delay measured byBob, is the same in both R-DSR (4.1) and canonical coordinates (4.38), and it is inagreement with what expected from a wave analysis approach (4.7), but in canonicalcoordinates, which trivialize translations (4.39) and phase space (4.41), Alice andBob, connected by a pure translation, agree on the arrival time of the two photons,and then on the physical velocity. This is due to the fact that the canonical phasespace (4.41) does not allow for relativity of locality.

We notice that the observable “time-of-arrival” on which we are basing ouranalysis, which we have shown to be a “good” physical observable (see Sec. 2.2and the discussion at the end of sub. 4.1.1), since it is defined to be a “local”measurement, i.e. it takes place in the origin of the observer frame, is not affectedby the change of coordinates defined by (4.37), which reduces to identity in theorigin of the coordinate system. One could argue that relativity of locality can beremoved by a suitable choice of coordinates, which one could call “absolute locality”coordinates: with the map (4.37), one can define canonical coordinates, whichtrivialize phase space, and thus trivialize translations, so that observers connectedby a pure translation agree on the locality of distant events, and they describeobjectively particles motion in their coordinatization of worldlines. While this logicalchain is true in the case of an undeformed special relativistic theory5, this is not thecase for a DSR theory. Indeed we considered so far only inertial observers connectedby a pure translation, but when we take the whole set of R-DSR transformations,including boost transformations, we cannot find a map to coordinates which are

5See [38] for the case where special relativity is described in R-DSR coordinates. In thatcase, the use of coordinates with non-zero Poisson brackets between them turns out to be just acumbersome description ordinary special relativity. Since one can always map back to “absolutelocality coordinates” valid for all the set of transformations.


“absolute locality” coordinates for all inertial observers. Thus relativity of localityseems to be a necessary (unavoidable) feature of DSR theories. One could then saythat the for R-DSR, canonical coordinates are “absolute locality coordinates fortranslations”, i.e. only for observers at relative rest. To show how this arises, weconsider now the boost sector of transformations.

4.4 Relative locality for boosts

4.4.1 canonical coordinates

We start by analyzing the boost sector in canonical coordinates defined by (4.37),(4.38), to show how, even if in those coordinates there is no relativity of localityfor observers connected by a pure translations, relative locality arises for boostedobservers. Using the map (4.37) on the boost representation (4.3), we find the boostrepresentation in canonical coordinates

N = −x0p1 − x1p0 + `x1

(p2

0 + p21

2

), (4.52)

so that it has Poisson brackets with xµ

N , x0 = −x1 + 2`x1p0 , N , x1 = x0 − `x1p1 . (4.53)

One can show that the whole set of variables xµ, pµ,N , together with the Poissonbrackets (4.2), (4.41), and (4.53), satisfy the Jacobi identities.

We consider again two observers, Alice and Bob, but this time Bob is connectedto Alice by a pure boost. We take also Alice and Bob to be at rest respect to their“own” detectors. Bob’s coordinates are related to Alice’s by the relations defined bythe action of N (4.53) as

xB0 = xA0 − ξxA1 − 2`ξpA0 xA1 ,

xB1 = xA1 − ξxA0 + `ξpA1 xA1 ,

pB1 = pA1 − ξpA0 − `ξ((pA0

)2+ 1

2(pA1

)2),

(4.54)

where ξ is the infinitesimal Boost parameter (the rapidity), so that we restrict ouranalysis to first order in ξ (and in `ξ). Then, if we focus on massless particles,substituting relations (4.54) in (4.47), we find that their motion is described by Bobin terms of the worldlines (we also assume pI1 > 0, the particles moving towards thepositive xI1 direction, for both observers, so that for massless particles pI0 = pI1 afterimposing the on-shell dispersion relation (4.6))

xB1

(xB0

)p1,m=0

=xB1 +(1− `pB1

) (xB0 − xB0

)=(1− `pB1

) (xB0 − xA0

)+ xA1 − ξ

(xA0 − xA1

)+ 2`ξpB1 xA1 ,

(4.55)

Consider first the case of a soft and a hard photon, such that in Alice’s descriptionthey are emitted simultaneously from a source (which we assume to have mass

4.4 Relative locality for boosts 49

M) at Alice’s spatial distance L (assumed to be at rest respect to Alice, so thatits worldline is simply xA1

(xA0

)pA1 =0,M

= −L), i.e. at the same spacetime point(xA0 , x

A1

)= (−L,−L). Alice describes the emission (crossing of the two photon

worldlines) as a distant local event, and, consequently, since the coordinate velocityis vm=0 = 1− `p1, the hard particle arrives at Alice’s spatial origin (to her detector)with a delay ∆t = `Lph (see Sec. 4.3). Alice’s worldlines are

xA1

(xA0

)ps,m=0

= xA0 ,

xA1

(xA0

)ph,m=0

=(1− `pAh

)xA0 − `LpAh ,

(4.56)

Substituting(xA0 , x

A1

)= (−L,−L) in (4.55) we find that Bob’s worldlines are

xB1

(xB0

)ps,m=0

= xB0 , (4.57)

xB1

(xB0

)ph,m=0

=(1− `pBh

)xB0 − `LpBh (1 + 2ξ)

=(1− `pAh (1− ξ)

)xB0 − `LpAh (1 + ξ) ,

(4.58)

The source is is described by Bob in terms of the worldline

xB1

(xB0

)pA1 =0,M

= −L− ξxB0 . (4.59)

Alice’s detector is described by Bob by the worldline

xB1

(xB0

)= −ξxB0 . (4.60)

From Eqs. (4.56) and (4.58) we can point out various features. One is that thedelay measured respectively by Alice and Bob, if the particles reach respectivelytheir detectors (so that in this case we are considering two different physical events),(solving for xA,B0 the equations xA,B1

(xA,B0

)ph,m=0

= 0) is given by

∆tA = `LpAh ,

∆tB = `LpBh (1 + 2ξ) = `LpAh (1 + ξ) = ∆tA (1 + ξ) .(4.61)

Bob’s description of Alice’s detection of the hard particle (i.e. considering Alice’sand Bob’s description of the same physical event) is obtained by the intercept of thehard photon worldline (4.58) with Alice’s detector worldline (4.60). We find thatBob describes the detection to take place in the point(

`LpB(1 + ξ),−`ξLpB)

=(`LpA,−`ξLpA

). (4.62)

The most relevant feature for the role of locality is that Bob describes the eventof emission, i.e. the crossing of the photons worldlines with the worldline of thesource, as non local: evaluating the intercept of the worldlines (4.57) and (4.58) with


the worldline of the source (4.59) one finds that in Bob’s description, while the softphoton and the source cross at the point(

xB0 , xB1

)pS ,M

= (−L (1− ξ) ,−L (1− ξ)) , (4.63)

the hard photon and the source cross at the point(xB0 , x

B1

)ph,M

=(−L (1− ξ) + 3`ξLpBh ,−L (1− ξ)

)=(−L (1− ξ) + 2`ξLpAh ,−L (1− ξ)

).

(4.64)

Thus, even in canonical coordinates (4.38), for which locality is absolute for translatedobservers, one has relativity of locality for a distant event described by purely boostedobservers.

Consider now the following situation: a soft and a hard photon, emitted from adistant source, arrive simultaneously at Alice’s origin (i.e. to Alice’s detector). Theirmotion then, is described by Alice in terms of the worldlines (

(xA0 , x

A1

)= (0, 0))

xA1

(xA0

)ps,m=0

= xA0 ,

xA1

(xA0

)ph,m=0

=(1− `pAh

)xA0 .

(4.65)


A1

)= (0, 0) in (4.55), we find that the same particles are described

by Bob in terms of the worldlines

xB1

(xB0

)ps,m=0

= xB0 ,

xB1

(xB0

)ph,m=0

=(1− `pBh

)xB0 =

(1− `pAh (1− ξ)

)xB0 .

(4.66)

As it is evident from the last equations, the particles still pass through Bob’s origin:the event of detection then, for two purely boosted observers local to the detector,remains local.

xA0

xA1

detAlice

emitter

∆tA = ℓ|pAh |L

xB0

xB1

detBob

emitter

∆tB = ℓ|pBh |L(1 + 2ξ)

(−L(1−ξ),−L(1−ξ))

detAlice

(−L(1−ξ)+3ℓξL|pBh |,−L(1−ξ))

Figure 4.4. The first (second) case is represented by the Blue (Violet) photon.

4.4 Relative locality for boosts 51

Thus we have seen that the relativity of locality, for an event described byobservers connected by a pure boost (which of course have coincident origin), thoughpresent also in canonical coordinates, vanishes as the distance of the event approacheszero, i.e. as the event is local to the observers. Locality of an event is still objectiveto observers local to the event, in the same way in which simultaneity in specialrelativity is still absolute for spatially local events.

This could also be seen directly by the fact that the transformations (4.54) arehomogeneous in the coordinates, (as in ordinary special relativity), so that the effectof boost transformations is null in the origin of the reference frame.

Holding on the conclusions of the last two sections, we argue that relativity oflocality is encoded in the non linearity of the Poisson structure of the theory, whichdefines the action of the generators on the coordinates: if one could find a map tospacetime coordinates such that all the Poisson brackets of the generators with thecoordinates were not momentum dependent, then relative locality would be removed.But we argue that if the generator algebra is not linear as for (4.2), such map doesnot exist. Indeed we attribute to such non linearity the origin of the shift fromspacetime to phase space as invariant arena advocated in Sec. 4.2.

4.4.2 Boosts in R-DSR coordinates

For completeness, we carry out the analysis of the previous section for boostedobservers, in R-DSR coordinates (4.1). In these coordinates, we have seen in Sec. 4.1that one has relativity of locality for purely translated observers, and the coordinatevelocity for massless particles is 1 (4.16). From (4.3) and (4.1), we find that thePoisson brackets between N and xµ are

N , x0 = x1 + ` (x1p0 − x0p1) ,

N , x1 = x0 . (4.67)

From (4.26) and (4.67), Bob’s coordinates are related to Alice’s by the relations

xB0 = xA0 − ξxA1 − `ξ(pA0 x

A1 − pA1 xA0

),

xB1 = xA1 − ξxA0 ,

pB1 = pA1 − ξpA0 − `ξ((pA0

)2+ 1

2(pA1

)2),

(4.68)

Then, for massless particles, Bob’s worldlines are

xB1

(xB0

)p1,m=0

= xB0 + xA1 − xA0 − ξ(xA0 − xA1

)+ `ξpB1

(xA1 − xA0

). (4.69)

We consider the first case of the previous section: the simultaneous emissionof a soft and a hard photon from a distant source. In that case, in canonicalcoordinates, Alice detects a soft photon in her origin and a hard photon with a delay∆tA = `Lph, and, the coordinate velocity being vm=0 = 1 − `p1, they cross at apoint

(xA0 , x

A1

)= (−L,−L). In R-DSR coordinates, since the coordinate velocity

for massless particles is vm=0 = 1, if the (massless) particles arrive at Alice’s spatialorigin with a certain delay, their worldlines must be parallel to each other, so they


cannot cross. Thus Alice does not describe the emission as a distant local event,which is obviously a consequence of the fact that in R-DSR coordinates there isrelativity of locality for translations. Indeed, the R-DSR analogous of the first casestudied in the previous section, is the one in which Alice’s worldlines are obtainedby substituting

(xA0 , x

A1

)=(`LpAh , 0

)in (4.30):

xA1

(xA0

)ps,m=0

= xA0 ,

xA1

(xA0

)ph,m=0

= xA0 − `LpAh .(4.70)

while, substituting(xA0 , x

A1

)=(`LpAh , 0

)in (4.69) we find that Bob’s worldlines are

xB1

(xB0

)ps,m=0

= xB0 ,

xB1

(xB0

)ph,m=0

= xB0 − `LpAh (1 + ξ) .(4.71)

The source is described by Bob to be in relative motion, with worldline

xB1

(xB0

)pA1 =0,M

= −L− ξxB0 , (4.72)

as well as Alice’s detector

xB1

(xB0

)pA1 =0,M

= −ξxB0 . (4.73)

Again we have, in agreement with the previous section, that Alice and Bobmeasure the delay (at their respective detector)

∆tA = `LpAh ,

∆tB = `LpBh (1 + 2ξ) = `LpAh (1 + ξ) = ∆tA (1 + ξ) ,(4.74)

We consider now the R-DSR coordinate version of the second case treated in theprevious section: the two photons, one soft and one hard, arrive simultaneous atAlice’s spacetime. In this case, Alice’s worldlines are simply (substituting

(xA0 , x

A1

)=

(0, 0) in (4.30))

xA1

(xA0

)ps,m=0

= xA0 ,

xA1

(xA0

)ph,m=0

= xA0 .(4.75)


A1

)= (0, 0) in (4.55), we find that the same particles are described

by Bob in terms of the worldlines

xB1

(xB0

)ps,m=0

= xB0 ,

xB1

(xB0

)ph,m=0

= xB0 .(4.76)

4.5 A general result: “Taming non locality” 53

xA0

xA1

detAlice

emitter

∆tA = ℓ|pAh |L

xB0

xB1

detBob

emitter

∆tB = ℓ|pBh |L(1 + 2ξ)detAlice

Figure 4.5. The first (second) case is represented by the Blue (Violet) photon.

4.5 A general result: “Taming non locality”

We are going to show now a general result, reported in our work [11]: any DSRtheory predicting a momentum-dependent velocity of massless particles (appreciableat first order in deformation parameter), presents relative locality effects whichvanish as the distance from the origin of the reference frame of the observer goes tozero.

In order to derive this result we will rely on some of the results of the previoussections:

1. We have seen in Sec. 4.3 that in canonical coordinates, the translations areundeformed, i.e. we have absolute locality for translations, so that coordinatevelocity and physical velocity coincide in these coordinates.

2. We studied then in Sec. 4.4, for the R-DSR case, how, even in canonicalcoordinates, relative locality is still present for relatively boosted observers.

3. We noticed also how relativity of locality can be understood to be encoded inthe deformed, momentum dependent, terms in the Poisson brackets defining theaction of symmetry transformation generators on the spacetime coordinates.

We choose to work in canonical coordinates, and we present this result in 3+1D, sothat the canonical phase space is defined by the Poisson brackets (4.41)

p0, x0 = 1 , p0, xj = 0 ,pj , x0 = 0 , pj , xk = −δjk ,

(4.77)

x0, xj = xj , xk = p0, pj = pj , pk = 0 , (4.78)

and we want to investigate the class of DSR theories predicting a momentum-dependent (physical) speed of massless particles of the kind (at first order in thedeformation parameter `)

v = 1− ` |~p| . (4.79)

We start our analysis by considering a general form for the deformed Hamilto-nian constraint, reproducing the speed (4.79). Upon enforcing analyticity of the


deformation and invariance under classical space-rotation transformations, we find atwo-parameter family of O(`) possibilities

H = p20 − ~p2 + `

(γ1p

30 + γ2p0~p

2)−m2. (4.80)

The Hamiltonian flow gives the equations of motion

˙xj = H, xj = 2pj − 2`γ2p0pj

˙x0 = H, x0 = 2p0 + `(γ2p

2j + 3γ1p

20

),

(4.81)

from which, after enforcing the on-shell relation H = 0, giving the dispersion relation(for p0 > 0)

p0 (~p) =√~p2 +m2 − `

2(γ1(~p2 +m2

)+ γ2~p

2), (4.82)

one finds the worldlines

xj (x0)~p,m = xj + pj√p2 +m2 (x0 − x0)− `pj(x0 − x0) , (4.83)

which for massless particles become

xj (x0)~p,m=0 = x1 + pj|~p|

(x0 − x0)− ` (γ1 + γ2) pj (x0 − x0) , (4.84)

and reproduce (4.79) for γ1 + γ2 = 1 (we restrict for simplicity to the choice (4.79)γ1 + γ2 = 1 since in the case v′ = 1 − ` (γ1 + γ2) |~p| we can always include theconstant factor (γ1 + γ2) in a redefinition of `, and thus it does not represent asignificative generalization of (4.79)).

We assume the Hamiltonian constraint (4.80) to correspond with the quadraticCasimir (C = H+m2) of our algebra of spacetime symmetries (including translationspµ, rotations Rj and boosts Nj), so that

C = p20 − ~p2 + `

(γ1p

30 + γ2p0~p

2), (4.85)

and the relationsC, pµ = C,Nj = C, Rj = 0 , (4.86)

hold.We look now for the suitable representation of boosts Nj compatible with C.

The form of the correction terms introduced in (4.85) suggests the following four-parameter family of O(`) deformed boosts, which enforces also compatibility withundeformed space rotations 6:

Nj = −x0pj + xjp0 + `

(α1x0p0pj + α2~p

2xj + α3p20xj +

∑k

α4xkpkpj

). (4.87)

6Note that this four-parameter family of deformed boosts includes, as different particularcases, all the proposals for deformed boosts that were put forward in this first decade of DSRresearch[12, 13, 14, 15, 16, 17].


The compatibility between boost transformations and form of the Casimir isencoded in the vanishing of the Poisson brackets between them, which also meansthat the boost charge is conserved:

0 = C,Nj =H+m2,Nj

= H,Nj = Nj , (4.88)

and straightforwardly leads to the following constraints on the parametersγ1, γ2, α1, α2, α3, α4:

2α2 + 2α4 = γ2 , 2α1 + 2α3 − 3γ1 − 2γ2 = 0 . (4.89)

Combining these with the requirement γ1 + γ2 = 1 derived above, we finally arriveat a three-parameter family of Hamiltonian/boost pairs

C = p20 − ~p2 + `

(2γp3

0 + (1− 2γ) p0 ~p2),

Nj = −x0pj + xjp0 + ` αx0p0pj

− `∑k

(γ + β − 1/2) xkpkpj + `xj(β~p2 + (1 + γ − α) p2

0

), (4.90)

where γ = γ1/2, α = α1, β = α2. The compatibility of γ, α, β-deformed boostgenerators with C then ensures also covariance of the worldlines under DSR-deformedboosts, as one can also verify by computing explicitly the action of an infinitesimaldeformed boost with rapidity vector ξj

p′j = pj − ξjp0 − ` ξj(β~p2 + (1 + γ − α) p2

0

)− `

∑k

(1/2− γ − β) ξkpkpj ,

x′0 = x0 −ξj xj −`(αx0ξjp

j + 2 (1 + γ − α) p0ξj xj), (4.91)

x′j = xj − x0ξj + `(αx0p0ξj + 2βξkxkpj

)− ` (γ + β − 1/2)

(ξkp

kxj + xkpkξj).

Using these one easily verifies that when a first observer describes the particle interms of the worldline (4.83), a second observer, purely boosted by rapidity ξj , willdescribe the particle in terms of the worldlines

x′j = x′j +p′j√

m2 + p′2(x′0 − x′0)− ` p′j(x′0 − x′0) ,

consistently with the relativistic nature of our framework, so that for any givenchoice of γ, α, β relativistic covariance is ensured and we have a rigorous Hamiltonianderivation of worldlines for which the speed law (4.79) is satisfied. We notice also,remembering the first point recalled at the beginning of this section, that, since weare working in canonical coordinates, the coordinate speed (4.79) coincide with the“physical” speed. More precisely, as shown in Sec. 4.3, the delay between particleswith different momenta, inferred through the equations of motion (4.83) by anobserver distant from the detector, coincides with the delay measured by an observerlocal to the detector. Thus we have defined a three parameter (γ, α, β) class of DSRtheories for which massless particles have momentum dependent “physical” speed(4.79).


For completeness we show the Poisson brackets between boost and translationgenerators, once we have fixed the parameters to γ, α, β.

Nj , p0 = pj − ` αp0pj ,

Nj , pk = δjkp0 + `

(12 − γ − β

)pjpk + `δjk

(β~p2 + (1 + γ − α) p2

0

). (4.92)

We are now ready to exploit our technical results for a “physical” characterizationof the relative locality produced by DSR boosts. The observations we shall makeon relativity of locality apply equally well to all choices of γ, α, β. We noticehowever that by enforcing the condition α−β−γ =1/2 one has the welcome [16, 95]simplification of undeformed Poisson brackets among boosts and rotations (“theLorentz sector is classical” [16, 95]). And in particular for the case γ=1/2, α=1,β=0, on which we focus for our graphical illustrations, the laws of transformationtake a noticeably simple form:

p′j = pj − ξj(p0 + `p2

0/2)

(4.93)

x′0 = x0 −ξj xj −`(x0ξjp

j + p0ξj xj)

(4.94)

x′j = xj − (1− `p0) x0ξj (4.95)

This case preserves much of the simplicity of classical boosts for what concernsboosts acting transversely to the direction of motion. In what follows we shall notoffer any additional comments on transverse boosts (and our figures focus on boostsalong the direction of motion). But it is easy to verify using (4.91) (and even easierusing (4.94)-(4.95)) that boosts acting transversely to the direction of motion leadto features of nonlocality that are of the same magnitude and qualitative type asthe ones we visualize for boosts along the direction of motion.

We first notice that the deformed boosts still act, like ordinary Lorentz boosts,in a way that is homogeneous in the coordinates. Then, since a boost connects twoobservers with the same origin of their reference frames and, it follows, as shown inFig. 4.6, that the differences between DSR-deformed boosts and classical boosts areminute for points that are close to the common origin of the two relevant referenceframes, but gradually grow with distance from that origin. Eventually, as shown bytwo of the worldlines in Fig. 4.7, when an observer Alice is local to a coincidenceof events (the violet and a red photon simultaneously crossing Alice’s worldline)all observers that are purely boosted with respect to Alice, and therefore share herorigin, also describe those two events as coincident: we have found that, at least atleading order in ` and ξ, in the DSR framework “locality”, a coincidence of events,preserves its objectivity if assessed by local observers.

The element of “nonlocality” that is actually produced by DSR-deformed boostsis seen by focusing on the “burst” of three photon worldlines also shown in Fig. 4.6,whose crossings establish a coincidence of events for Alice far from her origin, anaspect of locality encoded in a “distant coincidence of events”. We argue thena generalization of points 2 and 3 at the beginning of this section: relativity oflocality emerges as an unavoidable feature of any DSR theory predicting momentumdependent speed of light of the kind (4.79). And is encoded in the deformed,momentum dependent action of the symmetry transformations (boosts in this case)between inertial observers.


Figure 4.6. We here show a hard-photon worldline as seen by Alice (solid blue), byDSR-boosted Bob (dashed blue) and by classically-boosted Bob (dashed-black). In spiteof assuming (for visibility) the unrealistically huge p= 0.05/`, ξ= 0.15, the differencebetween DSR boosts and undeformed boosts is minute near the origin. But accordingto Bob’s coordinates the emission of the hard particle appears to occur slightly off the(thick) worldline of the source.

Figure 4.7. A case with two hard (violet) worldlines, with momentum pv = 0.13/`, a“semi-hard” (blue) worldline with momentum pb = pv/2, and a ultrasoft worldline (red,with pr 1/`). According to Alice (whose lines are solid, while boosted Bob hasdashed lines) three of the worldlines give a distant coincidence of events, while two ofthe worldlines cross in the origin.

We stress again that actually, as explained in Sec. 2.2, and encountered through-out this chapter, in a DSR framework two relatively boosted observers should notdwell about distant coincidences, but rather express all observables in terms of localmeasurements. For example, for the burst of three photons shown in Fig. 7.9 themomentum dependence of the speed of photons is objectively manifest (manifest bothfor Alice and Bob) in the linear correlation between arrival times and momentum ofthe photons.

59

Chapter 5

Opportunities for DSRphenomenology

In this chapter, we introduce some of the phenomenological opportunities availableto the DSR framework. We focus on two topics, which we we will discuss in greaterdetail on later parts of this thesis:

• The test of Planck scale deformations of the dispersion relation through ultra-energetic astronomical sources.

• The test of Planck scale contributions to the fuzziness of spacetime throughan associated effect blurring the images of distant quasars.

During this thesis we also contributed to clarify some of the differences betweenDSR and LSB (Lorentz symmetry breaking) scenarios, and in parts of this chapterwe present a discussion on the different perspective of the two approaches towardsPlanck scale phenomenology.

Effects like in-vacuo dispersion and spacetime fuzziness can be contemplatedeven within LSB scenarios. But a crucial difference between the LSB and DSRapproaches is due to the full relativistic framework of DSR. Effects of departure fromLorentz symmetry which are not compatible with a relativistic scenario, are indeedforbidden in DSR theories. On the contrary, this kind of effects are allowed in LSBscenarios, and tend to produce much more virulent phenomenological consequencesthen the effects allowed in both DSR and LSB approaches.

Moreover, even for the case of effects of departure from Lorentz symmetry allowedin both DSR and LSB scenarios, the quantitative predictions derived within a LSBscenario, can be in general much greater (even several orders of magnitude) thenthe ones derived within a DSR scenario. This can be ascribed to the softer mannerin which DSR theories modify Lorentz symmetries: deforming them rather thenbreaking.

For these reasons the LSB and DSR approaches have focused respectively ondifferent aspects of Planck scale phenomenology. LSB research has focused mainlyon those effects which are present only in scenarios allowing for a preferred frame.These effects are generally stronger then the effects allowed in both DSR and LSBscenarios, and as a result they tend to prevail over the latter, making them ineffectivefor LSB phenomenology.

60 5. Opportunities for DSR phenomenology

Threshold anomalies effects fall into this category. These are particle processeswhich are forbidden by kinematics in ordinary special relativity, but are allowed,in a preferred frame scenario, above a certain threshold energy. We discuss in Sec.5.3 the absence of threshold anomalies for DSR theories, and show explicitly, asillustrative example, the different predictions for the stability of photons in the LSBand DSR approaches.

We close the chapter showing an example of the case, above mentioned, in whichan effect allowed in both DSR and LSB approaches, is quantitatively much greaterin the LSB approach than in the DSR approach. We discuss, for an illustrative caseof LSB and DSR scenarios, a modification from the standard special relativisticformula of the synchrotron radiated power. One finds that the effects predictedby DSR are much softer than the one predicted by LSB. As a side effect, we showthat the stronger (indirect) bounds presently available on the superluminality ofelectrons, coming from the analysis of synchrotron radiation at LEP, are much lessconstraining in the DSR case respect to the LSB case. This discussion is reported inour work [41].

5.1 In vacuo dispersionWe have seen in the previous sections how DSR theories allow us to consider deformeddispersion relations of the kind (3.1),

E2 − ~p2c2 + f (pµ; `)−m2c4 = 0 . (5.1)

We also shown in detail in Ch. 4 that, in some cases, deformed dispersion relationsof this kind may lead to theories where the (physical) speed of massless particlesis momentum dependent. We focused on the case in which, at leading order, thevelocity of massless particles takes the form

vphys = 1− ` |~p| . (5.2)

We mentioned that a deformed dispersion relation of the kind (5.1) can be con-templated also in scenarios where the Lorentz symmetries are broken at the `scale.

The possibility of momentum dependence of speed of light is considerably relevantfor Planck scale phenomenology, as it has been shown to lead, in some cases, totestable predictions. These effects could be within the reach of present experimentsdue mainly to an effective amplification, as for instance in the observations of burstsof particles from cosmological distances [23, 24]. The amplification would be providedthen by the (cosmological) distance traveled by the particles before reaching thetelescope. This would be the case for photons produced by the most energeticgamma ray bursts [19, 10, 74]. If we consider for example the typical observationof a gamma-ray burst, with GeV particles that travel for, say, T ∼ 1017s beforereaching our telescopes, assuming the deformation scale ` to be close to the Planckscale (` ' `p ∼ 10−19GeV) the delay accumulated by a hard (ph ∼ O (GeV)) photonrelative to a soft photon (e.g. ∼ O(KeV)), due to the velocity (5.2), would be (seeprevious chapter, and in particular formula (4.34))

∆t ∼ `pT |phard| ∼ O(10−2s

), (5.3)

5.2 Spacetime fuzziness and blurring of images from distant sources 61

within the reach of the sensibility of presently available astronomical telescopes[10, 97, 96].

While (5.3) can be considered to be an approximate estimate of the magnitudeof the effect, it is clear however that with the huge, cosmological distances traveledby the GRB photons, one cannot neglect the effects of spacetime curvature in theestimate of the photons time of arrival. For the LSB scenario, due to the simplerconceptual framework (one is spared the need of enforcing logical compatibility withthe relativity of frames of reference), after a few steps [2, 98, 99, 100] of gradualimprovement, a consensus [99, 100] was reached on a formulation of the curvature(redshift) contribute to the time-delay (5.3).

On the contrary, until our work “Deformed Lorentz symmetry and relative localityin a curved/expanding spacetime” [44], DSR-deformed relativistic frameworks havebeen satisfactory investigated only for flat(Minkowskian) spacetimes There had beenprevious attempts of investigating the interplay between DSR-type deformationscales and spacetime expansion (see, e.g., Refs. [51, 52]), but without ever producinga fully satisfactory picture of how the worldlines of particles should be formalizedand interpreted. In retrospect we can now see that these previous difficulties weredue to the fact that the notion of relative locality had not yet been understood, andwithout that notion the interplay between DSR-deformation scale and expansion-ratescale remains unintelligible.

We will come back in detail to this topic in Ch. 7, where we will discuss theresults reported in our work [44]. We will show how it is possible to obtain adescription of particles worldlines within a DSR framework in a spacetime expandingat a constant rate. This will allow us to discuss a preliminary 1 comparison betweenthe DSR and LSB predictions for the photons time delays.

5.2 Spacetime fuzziness and blurring of images fromdistant sources

There has been growing interest [28, 29, 30, 31, 32] in the possibility of testing thehypothesis of spacetime fuzziness at the Planck scale (EP ∼ 1028eV ) exploitingan associated effect blurring the images of distant astrophysical sources, such asquasars. The arguments providing encouragement for these phenomenologicalstudies are merely heuristic, but this could be a rare opportunity [101, 102, 9] fortesting experimentally an aspect of the interplay between gravitational and quantum-mechanical phenomena. The scenario considered in Refs. [28, 29, 30, 31, 32] (buildingon earlier analogous pictures, such as those in Refs. [5, 25, 26, 27]) is centered onthe possibility that the quantum-gravity contribution to the fuzziness of a particle’sworldline might grow with propagation distance, in such a way that, in spite of itsultra-microscopic characteristic scale (which we assume [101, 102, 9] to be of theorder of the Planck length ` ' 10−35m), it could turn into a macroscopic effect forpropagation over suitably large (cosmological) distances.

The most crucial aspect of the phenomenological proposals is the assumption[28, 29, 30, 31, 32, 26, 27] that in a fuzzy/quantum spacetime there should be an

1A more satisfactory DSR phenomenological proposal would have to be formulated in a spacetimewith a non-constant expansion rate (FRW).


irreducible Planck-scale contribution to the uncertainty of energies, governed by alaw of the form

δE[`] ∼ `α E1+α , (5.4)

with α, assumed [28, 29, 30, 31, 32, 26, 27] to take values between 1/2 and 1, beingthe single parameter which should discriminate in this respect among differentproposals for a quantum spacetime. [We place here an index “[`]” when reportingan estimate of the Planck-scale contribution to a given quantity.]

As first observed in Ref. [26], Eq. (5.4) would in turn produce uncertaintiesin the phase velocity vp = E/p and in the group velocity vg = dE/dp, both oforder `αEα and uncorrelated. This would imply that [26] as a wave propagates itwill experience essentially random mismatches, of order `αEα, between its phasevelocity and its group velocity. In turn one can then notice [26] that during thepropagation time tprop = D/vg the phase advances from its initial value by an amount2πvptprop/λ, i.e. 2π(vp/vg)D/λ (denoting with λ the wavelength), and thereforethe random mismatches of order `αEα between phase velocity and group velocityend up producing a contribution to the phase uncertainty that can be written (alsoconverting wavelengths into energies) as

δφ[`] = δ

[2π vp

vgD

λ

][`]∼ 2πD

λδE[`] ∼ 2πD`αE1+α , (5.5)

resulting indeed in blurred images, with blurring amplified by the propagationdistance D.

It has also been motivated in Ref. [28] that the actual law of accrual of the phaseuncertainty might have to accommodate a further factor of (D/λ)−α attributable toa lack of coherence of the fluctuations:

δφ[`] ∼ 2πD1−α

λ1−α δE[`] ∼ 2πD1−α`αE . (5.6)

Within the framework that motivated Eq. (5.5), the authors of [26], haveconstrained the space of the parameter α so that models of spacetime foam withα < 1 are ruled out. Within the framework of Eq. (5.6), the authors of [28] haveruled out models with α . 0.6.

It is important for us important to notice that both (5.5) and (5.6) rely cruciallyon the assumed presence of an irreducible Planck-scale contribution to the energyuncertainty δE[`] of the form (5.4). This assumption is motivated in the relevantheuristic arguments by essentially noticing [28, 29, 30, 31, 26, 27] that operativelyenergy is most primitively derived from spacetime observations, and with spacetimebeing “fuzzy” at the Planck scale one should, according to Ref. [28, 29, 30, 31, 26, 27],also inevitably get “fuzzy energies”.

This link from spacetime fuzziness to irreducible contributions to energy un-certainty could be a striking characterization of the quantum-gravity realm, and,as we stressed, is the key ingredient of the phenomenology on which we are herefocusing. But its only basis are indeed heuristic arguments. Even in the moststudied formalizations of quantum properties of spacetime at the Planck scale, suchas Loop Quantum Gravity [22] and spacetime noncommutativity [103, 55], spacetimefuzziness has been so far characterized only at a rather formal level, unsuitable

5.3 Absence of threshold anomalies 63

for comparison with observations and inconclusive for what concerns Planck-scalecontributions to energy uncertainties.

We will present in Ch. 8, based on our works [43, 46], a first ever example of acharacterization of spacetime fuzziness derived constructively within a quantum de-scription of spacetime. We will show however, that our formalization predicts effectsthat are quantitatively different from those previously contemplated heuristically.

5.3 Absence of threshold anomalies

For the case of Lorentz symmetry breaking scenarios, it has been shown that thresholdanomalies at Planck scale can lead to observables effects [104]. A major effort inthe LSB Planck scale phenomenology literature has been directed in the analysis ofthese effects [104, 105, 106, 107].

An anomalous threshold occurs when a full breakdown of Lorentz symmetry(with emergence of a preferred “ether” frame) produces the effect that a processwhich is not allowed (respectively allowed) in Special Relativity is still not allowed(resp allowed) at sufficiently low energies in the new Lorentz-breaking theory, butabove a certain threshold energy the process is instead allowed (resp not allowed).

As an example, we refer to how certain observations in astrophysics, which allowus to establish that photons of energies up to ∼ 1014 eV are not unstable, canbe particularly useful [108, 109, 110, 111] in setting limits on some schemes fordepartures from Lorentz symmetry.

The situation is different for DSR theories. We have shown that DSR can bethought as a `-deformation of special relativity. In particular it must be such thatlow-energy particles are not sensitive to the effects of deformation. Then, if a processis forbidden in standard special relativity, it must be forbidden also for low-energyparticles in DSR. Since in a relativistic theory a particle which is low-energy forone (local) observer Alice has different higher energy for another relatively-boostedobserver Bob (also local to the particle) it must then be the case that the process isforbidden for particles of any energy, since then evidently the laws that establishwhether or not a process can happen must be observer independent.

We are therefore assured that there cannot be any anomalous thresholds inDSR theories, since they do not admit preferred frames. This was noticed alreadyseveral years ago [75], and has been more recently verified explicitly in severalstudies [75, 76, 77, 78] by taking different DSR-compatible examples of combinationsof deformed boosts N and deformed laws of composition of momenta ⊕N , findingthat indeed when this mutual DSR-compatibility is enforced one has that themodifications of the on-shell relation combine with the corresponding modificationsof the law of conservation of total momentum (total momentum computed with ⊕Ncomposition law) to produce no anomalous thresholds.

We are going to exhibit, in the following, as an example of this “absence ofthreshold anomalies in DSR”, the case, forbidden in standard special relativity (andthus also in DSR), of a photon decay into a positron and an electron (γ → e+ + e−).


5.3.1 DSR theories and deformed conservation laws

We have already mentioned that in a LSB scenarios one can have deformed dispersionrelations, but it is assumed that the transformation laws between observers in relativemotion (with constant velocity) are still given by ordinary Lorentz transformations(that’s how a preferred frame arises (see the introduction to Ch. 4). It is alsoassumed in LSB that the conservation laws of momenta in particle processes aregiven by the ordinary undeformed linear summation of momenta. In this subsectionwe are going to show how in DSR theories, to enforce the deformed dispersionrelation to be relativistically invariant, one necessarily has to consider deformedconservation laws of momenta.

We consider the illustrative case of R-DSR defined in Sec. 3.2 (in its first orderin ` approximation). As in the previous chapter, we enforce the dispersion law to beencoded in the on-shell constraint C −m2, C being the Casimir (3.28):

p20 − ~p2 − `p0~p

2 −m2 = 0 . (5.7)

Evidently this law is not Lorentz invariant (see the introduction to Ch. 4). If weinsist on this law and on the validity of classical (undeformed) Lorentz transformationsbetween inertial observers we clearly end up with a preferred-frame picture, and thePrinciple of Relativity of inertial frames must be abandoned: the scale ` cannot beobserver independent, and actually the whole form of (5.7) is subject to vary fromone class of inertial observers to another.

If one enforces (5.7) to be valid in a DSR framework, the action of the boosts onmomenta must indeed be deformed. We have shown that a compatible choice is theone encoded in the Poisson brackets (3.27)

Nj , p0 = p1 , Nj , pk = δjk

(p0 − `p2

0 + `

2~p2)− `pjpk , (5.8)

so that the dispersion relation (5.7) is covariant under the action of boosts (allobservers connected by boosts describe the same dispersion relation in their coordi-nates):

Nj , p20 − ~p2 − `p0~p

2 −m2

=Nj , C −m2

= 0 . (5.9)

Equally evident is the fact that these deformed boosts are relativistically incom-patible with the standard linear law of composition of momenta. Let us considerfor example the case of a process with two incoming and two outgoing particlesa+ b→ c+ d. For this case one easily finds that

Nj ,(p(a) + p(b)

)µ−(p(c) + p(d)

)µ

6= 0 , (5.10)

even when(p(a) + p(b)

)µ

=(p(c) + p(d)

)µis enforced. So that observers connected

by boosts would describe different conservation laws in their coordinates.One can verify that a possible momentum composition law relativistically com-

patible with the boosts defined by (5.8), is the law defined by the relations (5.11)(we will come back in detail to this argument in Ch. 6)

(p⊕ q)0 = p0 + q0 ,

(p⊕ q)j = pj + qj − `p0qj , (5.11)

5.3 Absence of threshold anomalies 65

so that Nj ,

(p(a) ⊕ p(b)

)µ−(p(c) ⊕ p(d)

)µ

= 0 (5.12)

when the conservation law(p(a) ⊕ p(b)

)µ

=(p(c) ⊕ p(d)

)µ

(5.13)

holds.

5.3.2 Photon stability

We choose now two illustrative scenarios of LSB and DSR, for both of which weassume the dispersion relation for all particles to be given by (5.7), so that they aresuitable for comparison. But for the LSB scenario we indicate with λ the scale atwhich the Lorentz symmetries break, while for the DSR scenario we use ` to indicatethe deformation scale2:

E2 − ~p2 − λE~p2 −m2 = 0 (LSB) E2 − ~p2 − È~p2 −m2 = 0 (DSR) . (5.14)

We study now the different predictions on the stability of the photon for the twoscenarios. We study the process of photon decaying in a positron and an electron:

γ → e+ + e− . (5.15)

As already mentioned, in the LSB scenario, the boost transformations are thestandard Lorentz transformations, and the law of composition of momenta is also thestandard linear summation law. For the DSR case we have the deformed momentumcomposition law (5.11). So that the conservation laws of momenta for the process(5.15) are given by

Eγ = E+ + E− ,

~pγ = ~p+ + ~p− ,(LSB)

Eγ = E+ + E− ,

~pγ = ~p+ + ~p− − È+~p− .(DSR) (5.16)

Consider first the LSB case. Let’s take Eq. (5.14) for the electrons, which canbe written as:

m2e = 1

2(E2

+ + E2− − p2

+ − p2− − λE+p

2+ − λE−p2

−

). (5.17)

Squaring the second (LSB) equation of (5.16), using the dispersion relation (5.14)for the photon, and substituting the first (LSB) equation of (5.16), we can write

p2+ + p2

− = − 2p+p− cos θ + E2+ + E2

− + 2E+E−

− λ (E+ + E−)(p2

+ + p2−

)− 2λ (E+ + E−) p+p− cos θ .

(5.18)

Substituting the last equation in (5.17), and considering the ultrarelativistic limit forthe outgoing electrons only for the terms proportional to ` (È2 6= 0 but `m2 ≈ 0),we find

cos θ 'm2 + E+E− − 3λ

2(E+E

2− + E−E

2+)

p+p−. (5.19)

2In the following we call the energy and momentum of the particles E and ~p.


We then substitute in the last expression the dispersion relations (5.14) for theelectrons, so that it becomes

cos θ 'm2 + E+E− − λ

(E+E

2− + E−E

2+)√(

E2+ −m2) (E2

− −m2) . (5.20)

For λ = 0, one has the standard special relativity result: one finds that theprocess would require cos θ > 1 for any E+, E− combination, so that photon decayis forbidden by kinematics. In the limit of ultrarelativistic outgoing particles thistakes the form

cos θ ' E+E− +m2

E+E− − m2

2

(E+E−

+ E−E+

) . (5.21)

For λ 6= 0, there would be some high values of E+, E− (and correspondingly Eγ )for which a real θ could solve (5.20). It must be noticed that the energies needed forthis are ultra-high but below-Planckian: photon decay starts to be allowed alreadyat scales roughly of order

(m2 |`|−1

)1/3(which indeed, for m the electron mass and

|`|−1 roughly of order the Planck scale, is |`|−1 ).Let’s perform the same analysis for the DSR case. Consider Eq. (5.14) for the

electrons, which can be written as

m2 = 12(E2

+ + E2− − p2

+ − p2− − È+p

2+ − È−p2

−

). (5.22)

Squaring the second (DSR) equation of (5.16), using the dispersion relation (5.14)for the photon, and substituting the first (DSR) equation of (5.16), we can write

p2+ + p2

− = − 2p+p− cos θ + E2+ + E2

− + 2E+E−

− `(E+

(E2

+ − E2−

)+ E−

(E2

+ + E2−

))− 2È+E

2− cos θ ,

(5.23)

where we considered the ultrarelativistic limit for the outgoing electrons only for theterms proportional to ` (È2 6= 0 but `m2 ≈ 0). Substituting the last equation in(5.22), we find

cos θ 'm2 + E+E−

(1− `

2 (E+ − E−))

p+p− (1 + È−) . (5.24)

We then substitute in the last expression the dispersion relations (5.14) for theelectrons, so that it becomes

cos θ 'm2 + E+E−

(1− `

2 (E+ + E−))

√(E2

+ −m2) (E2− −m2) (1− ` (E− + E+))

' m2 + E+E−√(E2

+ −m2) (E2− −m2) ,

(5.25)

which coincides with the standard special relativity relation, cos θ > 1 for any choiceof E+, E−, thus not allowing for the possibility of photon decay.

5.4 Synchrotron radiated power 67

Amusingly during the course of this thesis work we had a brief “OPERA season”,and some of the issues highlighted in this chapter came to the attention of a broadercommunity. We are referring of course to the much discussed announcement by theOPERA collaboration [112]. It is now well established that the results reportedby OPERA were affected by a very significant systematic error. Nonetheless it isnoteworthy the central role played by some of the topics discussed in this chapterduring the debate on the OPERA anomaly, as the different behavior of DSR and LSBrespect threshold anomalies [77]. While a possibility of a LSB interpretation of theOPERA anomaly had been ruled out by arguments relying on in-vacuo Cerenkov-likeprocesses [113, 114], which are a manifestation of threshold anomalies effects, therewas no evidence of incompatibility of the OPERA anomaly with a DSR scenario[77] (also see our analysis in [40]), aside from the unexpectedly large scale of theeffect (several orders of magnitude larger then Planck scale), which could not havebeen accommodated within a DSR/Quantum Gravity point of view.

5.4 Synchrotron radiated power

In the previous section we discussed how, the relativistic framework of DSR theoriesprevents from virulent effects such as the emerging of threshold anomalies, whichare instead allowed in preferred frame scenarios, where Lorentz symmetries arebroken. We mentioned at the beginning of this chapter how in cases where thePlanck scale effects are present as a modification of some effects which are alreadypresent in ordinary special relativity, i.e. of effects which are not in conflict with therelativity of inertial frames, the softer departure from ordinary special relativity dueto DSR, is such that the DSR new effects are usually softer the LSB ones. As a sideeffects indirect bounds for departures from Lorentz symmetry, which of course arein general model dependent, may be weaker if established within DSR models thenLSB models.

In this section, based on our work [41], we exhibit as example an analysis of thecomparison between the LSB and the DSR modifications to the ordinary specialrelativistic synchrotron radiated power formula. We apply our analysis to DSR/LSBcomparison of the reference laboratory bounds on departures from Lorentz symmetryfor the electron, of which the most frequently quoted are based on electron studiesat LEP [115, 116]. It is usually assumed that these analyses establish the validity ofa standard special-relativistic description of the electron with accuracy of at least afew parts in 1014, and in particular this is used to exclude electron superluminalitywith such an accuracy.

We first notice that a particularly stringent limit on superluminality of theelectron in scenarios with a preferred frame is credited [115, 116, 114] to be providedby the “in-vacuo-Cerenkov threshold” : above a certain threshold energy for the(presumedly) superluminal electron the Cerenkov process e− → e− + γ is allowed forelectrons propagating in vacuum. The fact that such in-vacuo-Cerenkov processes(if at all present) did not produce noticeable energy losses for LEP electrons impliesthat the threshold is higher than ∼ 100GeV , which in turn allows us to conclude thatthe parameters responsible for breaking Lorentz symmetry must be correspondinglysmall [115, 116, 114].


As explained in the previous section, these bounds on departures from Lorentzsymmetry based on anomalous thresholds are completely inapplicable to scenarioswhere the departures from Lorentz symmetry do not produce a preferred-framepicture. We will focus then only in the bounds provided by synchrotron radiation.

For scenarios with a full breakdown of Lorentz symmetry, and therefore a preferredframe, the determinations of synchrotron radiated power by LEP electrons providea very stringent constraint on electron superluminality, as emphasized recently byAltschul [115]. These determinations agree with the corresponding special-relativisticprediction to better than 0.1% accuracy. More precisely the available data can beused to establish that [115] ∣∣∣Pexp − PSR

PSR

∣∣∣ < 6× 10−4 (5.26)

where of particular interest is the dependence of the special-relativistic predictionPSR on the rapidity ξSR which is needed to connect the instantaneous rest frame ofa LEP electron to its lab frame: PSR = P0 cosh4(ξSR) (where P0 is obtained fromcomputing the radiated power in the instantaneous rest frame of a LEP electron).

One can then argue [115] that in the rest frame of the electron the implicationsof electron superluminality, and therefore the departures from special relativity,should be negligibly small, so that Larmor formula for the radiated power in theinstantaneous rest frame of the electron remains unmodified:

P0 = 23e

2a20 (5.27)

where e is the electron charge and a0 is the acceleration in the instantaneous restframe of the electron (and we keep working with units such that c = 1).

This line of reasoning allows one to focus [115] exclusively on the factor cosh4(ξSR)as the key for establishing bounds: as in the standard special-relativistic case thecentral role is played by the way a boost in direction orthogonal to the rest-frameacceleration changes that acceleration. The factor cosh4(ξSR) in PSR = P0 cosh4(ξSR)indeed comes from this boost of the acceleration

a[lab]SR = a0 cosh2(ξSR) .

Theories with departures from Lorentz symmetry would require a different valueof rapidity for connecting the lab frame to the instantaneous rest frame. And thisis indeed the line of analysis adopted by Altschul [115], who properly specified asreference framework a framework with a full breakdown of Lorentz symmetry, theso-called Standard Model Extension (see, e.g., Refs. [117, 118]). For our purposeshere it suffices to consider Altschul’s argument for the case of superluminal electronswithin the Coleman-Glashow broken-Lorentz-symmetry picture of Ref. [114], whichis the part of the Standard Model Extension most intuitively connected to thestructure of the argument. This amounts to attributing to superluminal electronsan on-shell relation

m2 = E2 − (1 + 2δ)p2 , (5.28)

and then looking [115] for the rapidity ξLIV that connects such a superluminal LEPelectron of 91 GeV to its rest frame, assuming (as standard for scenarios with broken


Lorentz symmetry and a preferred frame) that the boost transformations are stillgoverned by classical Lorentz boosts. Of course, classical Lorentz boosts are a veryawkward match for (5.28) (this is after all why a preferred frame arises). As a resultAltschul correctly finds [115] that the relationship between ξLIV and ξSR manifestsa large mismatch even for small values of δ:

cosh4(ξLIV ) ' cosh4(ξSR)[1 + 4δ cosh2(ξSR)] . (5.29)

We see here very vividly how the awkward mismatch between the superluminality ofthe electron and classical Lorentz boosts amplifies the effects: in the correction termthe inevitable factor of δ is amplified by cosh2(ξSR), which for a 91 GeV electron iscosh2(ξ91GeV

SR ) ' 3.2× 1010. It should be clear that this huge amplification sets thestage for when the boost transformation from the lab frame to the rest frame mustgo totally pathological, which is when the electron is actually superluminal (of coursethe bound of Ref. [115] is derived assuming the electron would turn “superluminal”only at some higher energies, with the parameter δ small enough that at 91 GeV theelectron still actually has speed smaller than the speed-of-light scale).The striking result is that one can use (5.29), in light of (5.26), to conclude thatδ . 5× 10−15.

It should be clear that in the case of deformation rather than breakdown ofLorentz symmetry the pathological amplification that produced this terrific boundon δ is not to be expected. But in general even in a DSR framework there willbe modifications to the synchrotron radiated power: unlike the case of anomalousthresholds, a modification of the synchrotron radiated power is not in conflict withthe relativity of inertial frames, so in general such a modification should be expectedin a DSR framework. And different DSR setups will produce different modificationsof the synchrotron radiated power. Without the aim to motivate some sort of generalbound applicable to DSR scenarios, we consider explicitly two examples of suchscenarios, just to show that the line of analysis which Altschul correctly appliedto the Lorentz-breaking case fails to produce “amplified bounds” when Lorentzsymmetry is deformed.

Also in the DSR case it appears safe to assume, as assumed by Altschul for theLIV case, that in the instantaneous rest frame of the electron (where speeds andenergies are all small, so that the deformations are mute) Larmor’s formula for theradiated power still holds3.

So once again, also in the DSR case, the key issues concern(i) the way instantaneous-rest-frame acceleration is changed into a “lab-frame acceler-ation” by a boost, of given rapidity ξ, in direction orthogonal to the that acceleration,and(ii) the computation of the rapidity that connects the instantaneous rest frame andthe laboratory frame.For these two features we can in general expect different predictions in differentDSR scenarios.

3One can easily verify however that if we allowed for modifications of the Larmor formulagenerically of the type

P0 = 23e2a2

(1 + α1 m

M∗ + α2 a

M∗

)(which are admissible on dimensional ground) our conclusions would not change.


Let us start contemplating these issues within the DSR setup introduced in Sec.3.2, for which the action of boosts on momenta can be described as

dE

dξ= p , (5.30)

dp

dξ= E − È2 − `

2p2 (5.31)

(here given for simplicity at leading order in ` and focusing on the case of collinearboost and spatial momentum).This is such that in particular the relationship between the rest energy m and theenergy and momentum in a frame boosted with respect to the rest frame by arapidity ξ is

E(ξ) = m cosh(ξ)− m2`

2 sinh2(ξ) (5.32)

p(ξ) = m sinh(ξ)− m2`

2 sinh(2ξ) (5.33)

and in general ensures a description compatible with the relativity of inertial framesfor the on-shell relation

m2 = E2 − ~p2 − È~p2 . (5.34)

In particular for point (i) above (the way a boost in direction orthogonal to theinstantaneous-rest-frame acceleration changes that acceleration) it is valuable forus to use the results for boosts acting on spacetime coordinates given in Ref. [11](Sec. 4.5). From those results one easily infers that in this specific DSR setup therelationship between acceleration in the instantaneous rest frame and the accelerationin a frame obtained by boosting in direction orthogonal to the acceleration remainsundeformed4:

a[lab]DSR = a0 cosh2(ξ) .

So in this DSR setup the differences for the derivation of the lab-frame synchrotronradiated power with respect to the LIV case analyzed by Altschul, obtaining hisresult (5.29), are all contained in the different predictions for the rapidity needed toconnect the lab frame to the instantaneous rest frame of the electron. In our DSRcase we easily deduce from (5.32) and (5.33) that

cosh4(ξ) ' cosh4(ξSR) (1 + 2`m cosh(ξSR)) . (5.35)4The case of DSR setup we are here considering is the case of parameters α = γ = 0 using

the notation of Ref. [11] (Sec. 4.5). And one sees from Ref. [11] that (even specifically for thiscase α = γ = 0) the action of boosts on generic coordinates is deformed, but just in the casehere of interest the deformation vanishes: for boosts connecting to the rest frame and acting indirection orthogonal to the rest-frame acceleration there is indeed no deformation [11]. Also therelativity of locality is mute, according to Ref. [11] (Sec. 4.5), in the case here of interest sincewe are contemplating LEP electrons and the radiation they emit as observed by detectors placedvery close to the electrons (at the LEP site the electron is crossing at a given time). So we shouldformalize the situation in the DSR framework as the map from the reference frame O where theelectron is instantaneously at rest and in the origin of the reference frame to a reference frame O′purely boosted with respect to O in direction orthogonal to the acceleration of the electron in O.


Unsurprisingly this formula shows no peculiar amplification of the sort we high-lighted in relation to Eq. (5.29). The correction term is just of the order of thesuperluminality this picture endows to the electron, which is È ' `m cosh(ξ). Thisshould be contrasted to the broken-Lorentz case, where the superluminality of theelectron was codified in δ, and determinations of synchrotron radiated power lead to4δ cosh2(ξSR) ≤ 6× 10−4 (i.e. δ . 5× 10−15). In our DSR picture, in light of (5.35),the line of analysis developed by Altschul only affords us a much weaker bound onsuperluminality: È . 3× 10−4.

While, as stressed above, different DSR setups will give different predictionsfor the synchrotron radiated power, let us offer some evidence that the qualitativeaspects of our findings might have applicability that goes beyond the single case ofDSR setup on which we so far focused. For this purpose we consider, borrowing fromthe Hopf-algebra literature [58, 54], a case where the superluminality is governed by`2E2, in which the boosts

dE

dξ= p ,

dp

dξ= E − `2

6 E3 (5.36)

provide a description compatible with the relativity of inertial frames for the on-shellrelation

m2 = E2 − ~p2 − `2

12E4 . (5.37)

From (5.36) it follows that the rapidity ξ that connects a frame where the electronhas energy E to its rest frame is such that

E(ξ) = m cosh(ξ)− `2m3 cosh3(ξ)− cosh(ξ) + 3ξ sinh(ξ)48

For the large values of rapidity which are here of interest this formula can be comparedto its special-relativistic limit (of course obtained again for ` → 0) producing theestimate

cosh4(ξ) ' cosh4(ξSR)(

1 + `2m2

12 cosh2(ξSR)), (5.38)

Therefore, also this other DSR setup produced a prediction for the rapidity connectingthe lab frame to instantaneous rest frame which is free from the peculiar amplificationwe highlighted in relation to Eq. (5.29). The correction term in (5.38) is just ofthe order of the superluminality this picture endows to the electron, which is`2E2/8 ' `2m2 cosh2(ξ)/8.

73

Chapter 6

Interactions and curvedmomentum space

So far we have studied the effects of relative locality in DSR scenarios only for thecase of free propagating particles. But it is clear that our strategy of analysis relieson the identification of points in spacetime as events, or crossing of worldlines, asfor example in the measurement of time delays in particles detection: even in ournon-quantum approach of the last sections, the description of the time of arrival ofa given particle, gains a physical interpretation only if thought as the (pointwise)interaction of the particle with a detector local to the observer.

In order to make room in our non-quantum DSR model for the description ofinteractions, we rely on an approach based on the recently proposed [48, 49] “principleof relative locality”. The framework of Refs. [48, 49] is centered on the possibility ofa non-trivial geometry for momentum space, and links to those geometric propertiessome effects of relative locality. Several approaches to the study of the quantum-gravity problem have led to speculations about nonlinearities in momentum spacethat may admit geometric description (see, e.g., Refs. [58, 60, 12, 15, 119, 120]and references therein), and some related studies had hinted at possibly strikingimplications of such nonlinearities for the fate of locality at the Planck scale [33, 34,121, 122, 123, 35].

We will discuss the aspects of relative locality focusing again on the role playedby deformed translations in characterizing the description of events made by distantobservers, and we will introduce interactions relying on a Lagrangian formalismfirst introduced in [48]. According to Ref. [48], one could describe the process inFigure 6.1, which is the idealized case with 3 particles of energy-momenta kµ, pµ, qµall incoming, on the basis of the following action:

74 6. Interactions and curved momentum space

Sexample=∫ s0

−∞ds(xµkµ+yµpµ + zµqµ+NkC [k]+NpC [p]+NqC [q]

)−ξµKµ(s0). (6.1)

p, y

k, x

q, z

K[0] = k ⊕ p⊕ q

Figure 6.1. A process with 3 particles of energy-momenta kµ, pµ, qµ, all incoming. Thelines in the graph are not intended as representatives of worldlines of particles. Theyshould rather be looked at, going from left to right, as a schematic portrait of the discretesteps in the redistribution of momentum among particles, changing at every subsequentinteraction (but the case in this figure is a single-interaction process). A similar graphicalcharacterization of processes is often adopted for quantum-field-theory Feynman diagrams(but our entire analysis is confined to the context of classical particles).

As we shall here discuss in greater detail in Section 6.2, the bulk part of Sexampleends up characterizing [48] the propagation of the 3 particles, with the Lagrangemultiplier Nk (and similarly Np and Nq) enforcing the on-shell relation C[k] =D2(k)−m2, with D2(k) in turn derived from the metric on momentum space as thedistance of kµ from the origin of momentum space. And the form of the boundaryterm ξµKµ(s0) is such that [48] the Lagrange multipliers ξµ enforce the conditionKµ(s0) = 0, so that by taking for Kµ a suitable composition of the momenta kµ, pµ,qµ the boundary terms enforce a law of conservation of momentum at the interaction.The form of the law of composition of momenta used for the conservation lawKµ(s0) = 0 is governed by the affine connection on momentum space [48], andmay involve nonlinear terms which are ultimately responsible for the relativity ofspacetime locality. This is indeed seen by studying the invariance of the actionSexample under translations of the coordinates of worldline points xµ(s), yµ(s), zµ(s),which each observer introduces as variables that are canonically conjugate to thecoordinates on momentum space kµ(s), pµ(s), qµ(s).

The observations used in Ref. [48] in the derivations that established the presenceof this translational invariance appeared to rely crucially on some simplifications af-forded only by the case of a single-interaction process. As here stressed in Section 6.2,for a single interaction several alternative choices of boundary terms enforcing thesame conservation law are consistent with the presence of relativistic translation sym-metries within the relative-locality framework of Ref. [48]. However, the demands oftranslational invariance become much more constraining when 2 or more interactionsare causally connected, i.e. when a particle outgoing from one interaction is incominginto another interaction. In Section 6.3 we provide an explicit example of formulationof the boundary terms which ensures translational invariance when several causally-linked interactions are analyzed. And the logical structure of our proposal is easilydescribed: translation transformations are generated by the total-momentum charge(obtained from individual particle momenta via the connection-induced composition

75

law) and the boundary terms are written as differences between the total momentumbefore the interaction and after the interaction (so that the associated constraintsautomatically ensure conservation of the total momentum).

Before getting to that main part of our analysis it will be useful to do somepreparatory work. In the next section we motivate our focus on results that areobtained only at leading order in the deformation scale, by observing that, ifindeed the deformation scale is roughly given by the Planck scale, the experimentalsensitivities foreseeable at least for the near future will not afford us investigationsgoing much beyond the leading-order structure of the geometry of momentum space.And we also show that working at leading order not only simplifies matters in theway that is commonly encountered in physics, by shortening some computations, butin this case also provides some qualitative simplifications, including most notably thefact that at leading order the momentum-composition laws of the relative-localityframework are automatically associative.

Then in Section 6.1 we characterize the specific example of relative-localitymomentum space on which we shall test our proposal for relativistic translationalsymmetries, based on the R-DSR κ-Minkowski/κ-Poincaré-inspired momentum space(defined in Secs. 3.1 and 3.2), which we find to have de Sitter metric and paralleltransport governed by a non-metric and torsionful connection.

Equipped with these preliminary observations we then discuss, in Section 6.2, thechallenges that must be dealt with in seeking a translationally-invariant descriptionof chains of causally connected interactions within the relative-locality frameworkproposed in Ref. [48]. The main insight gained from the analysis reported inSection 6.2 is that in order to achieve translational invariance it is not sufficient toensure that the boundary terms at endpoints of worldlines enforce some suitablemomentum-conservation laws, since in general two such choices of boundary termsat the endpoints of a finite worldline (a worldline going from one interaction toanother) will spoil translational invariance.

It is in Section 6.3 that the reader finds our main results concerning the proposaland analysis of a relativistic formulation of processes involving several interactions,within the general framework of Ref. [48], with translational invariance assured by acorresponding specification of the boundary conditions that implement momentumconservation. We test the robustness of our proposal mainly by applying it to theillustrative example of the κ-Poincaré-inspired momentum space.

Section 6.4 contains our results that are of particular significance from theperspective of phenomenology. We show that, within our setup for κ-Poincaréinteracting particles, simultaneously-emitted massless particles do not necessarilyreach the same detector at the same time. Since our κ-Poincaré momentum space hasnonmetricity, this is consistent with the thesis put forward in Ref. [124], accordingto which these time delays at detection for simultaneously-emitted massless particlesare to be expected when nonmetricity is present. In addition we also investigatehow the torsion of our κ-Poincaré momentum space affects these time-of-detectiondelays, an issue for which no previous result is applicable. And we find that thetorsion does affect the time delays, by essentially rendering the effect non-systematic:the time-of-detection difference for two simultaneously emitted massless particlesdepends non only on the momenta of the two particles involved but also on someproperties of the events that emitted the two particles. We also discuss the first


elements of a phenomenology that could exploit this striking feature.In Section ?? we keep the R-DSR momentum space while switching to a trivial

symplectic structure, and we reproduce again the results of Section 6.4. This allowsus to establish that the predictions derived in Section 6.4 are purely manifestationsof momentum space geometry.

The issues studied in this chapter are of exactly the same nature in the case of a4D momentum space and in the case of a 2D momentum space, and we shall often(but not always) focus for definiteness and simplicity on the 2D case. When nototherwise specified we shall switch between 4D and 2D formulas by simply denotingwith p0, pj the momentum in the 4D case and with p0, p1 the momentum in the 2Dcase.

The results of this chapter are reported in our work [39].

6.0.1 Leading-order anatomy of relative-locality momentum spaces

Refs. [48, 49] (also see Refs. [124, 125]) raised the issue of determining experimentallythe geometry of momentum space, much like it is traditional in physics to studyexperimentally the geometry of spacetime.It is however important to notice a crucial difference: while we do have experimentalaccess to distance scales larger than the scales of curvature of spacetime, it isvery unlikely that in the foreseeable future we could have experimental access tomomentum scales even just comparable to the Planck scale, which is the naturalcandidate for the scale of curvature of the relative-locality momentum space [48].It should be appreciated that this disappointing limitation of our horizons on thegeometry of momentum space can also be turned in some sense into a powerfulweapon for the phenomenology of momentum-space geometry: evidently all we needis a characterization of the geometry of momentum space near the origin, where|p| |`|−1 'Mp. And at least at first this will essentially be focused on the searchof leading-order evidence of a nontrivial geometry of momentum space.

For what concerns the affine connection on momentum space, responsible forthe nontrivial properties of the law of composition of momenta [48], all we need forthe purposes of this phenomenology are the (`-rescaled) connection coefficients onmomentum space evaluated at pµ = 0, which we denote by Γαβµ :

(p⊕ q)µ ' pµ + qµ − `Γαβµ pα qβ + · · ·

And evidently the fact that the phenomenology only needs leading-order resultsimplies (also considering that we already rescaled the connection coefficients by thePlanck scale) that we can treat the Γαβµ as pure numbers.

Analogous considerations lead us to focus on momentum-space metrics that areat most linear in the momenta:

gµν = ηµν + `hµνρpρ (6.2)

and, just like the Γαβµ , we should handle the coefficients hµνρ as pure numbers inour leading-order phenomenology.

In this chapter we shall mainly work only at leading order in the deformationscale, and it will be evident that this provides with significant advantages. In

77

particular, at leading order in the deformation scale the momentum-compositionlaw is always associative. This can be established by writing a general leading-ordercomposition law as follows:

(k ⊕ p)µ = kµ + pµ − `Γαβµ kαpβ , (6.3)

and then noticing that indeed (of course to leading-order accuracy)

[(k ⊕ p)⊕ q]µ = kµ + pµ + qµ − `Γαβµ (kαpβ + kαqβ + pαqβ) = [k ⊕ (p⊕ q)]µ . (6.4)

Beyond leading order the composition law could be nonassociative, and in thatcase one could appreciate the curvature of the momentum-space connection, withinteresting but technically challenging consequences which we shall not encounter inthis chapter, and will never be encountered when working at leading order in thedeformation scale.

The fact that our horizons on the geometry of momentum space probably areconfined to leading order may be viewed as an unpleasant philosophical limitation,but pragmatically can be turned into a powerful asset for phenomenology workon relative-locality momentum spaces, since the task of phenomenologists then isvery clearly and simply specified: the target should be to determine experimentally(as accurately as possible) a few dimensionless numbers for the leading-order (andpossibly the next-to-leading order) geometry of momentum space.

To make this point fully explicit let us for simplicity imagine a 2D relative-locality momentum space. In the 2D case a full leading-order characterization of themomentum-space geometry requires establishing experimentally (in hypothetical 2Dexperiments) the 8 dimensionless parameters of the affine connection on momentumspace,

Γ000 , Γ01

0 , Γ100 , Γ11

0 ,

Γ001 , Γ01

1 , Γ101 , Γ11

1

and the 6 dimensionless parameters of the leading-order description of the metric,which one can conveniently encode1 into the 6 free parameters of the associatedChristoffel symbols Cµνα ,

C000 , C01

0 = C100 , C11

0 ,

C001 , C01

1 = C101 , C11

1 .

6.0.2 curved κ- momentum space

We introduced κ-momentum space in Secs. 3.1 and 3.2. In this subsection we reviewits properties from a slightly different point of view, focusing on the properties of the

1We are here implicitly using the fact that our “leading-order momentum-space metrics” can beparametrized, in the example of the 2D case, equivalently in terms of the 6 independent numbersthat specify hµνσ or in terms of the 6 independent Christoffel symbols. Indeed one finds that

Cµνρ = 12 gρσ (gσµ,ν + gνσ,µ − gµν,σ) = `

2 ηρσ (hσµν + hνσµ − hµνσ) (6.5)


κ-momentum space as a curved momentum space. We follow Ref. [59] (see also [126]for a discussion of the relation between κ-Poincaré and curved κ-momentum space),so we describe κ-momentum space as a manifold of the group AN(3) (dubbed alsothe Borel group), which is, as a manifold, essentially a half of de Sitter space. TheAN(3) group is a subgroup of the de Sitter SO(4, 1) group, defined by its Lie algebraan(3) which has the following form 2:

[X 0,X i] = −i`X i , [X i,X j ] = 0 . (6.6)

This algebra is a subalgebra of so(4, 1) and one can represent it as an algebra of5× 5 real matrices, with the matrices representing X i being nilpotent. Knowing theform of the Lie algebra, one can readily write down a group element. It is convenientto split it into the product of two elements, one generated by nilpotent elements X iand the second generated by the abelian one X 0,

AN(3) 3 g(p) = exp(ipiX i) exp(ip0X 0) . (6.7)

Clearly pµ can be thought of as the coordinates on the group manifold.Since AN(3) is a subgroup of de Sitter group SO(4, 1), g(p) defined by (6.7) acts

naturally on points of the five dimensional Minkowski space M5. Therefore, if wetake a point O, the group AN(3) as a manifold is just a set of all points of the formgO. If O has coordinates (0, . . . , 0, 1/`) than the point g(p)O, with g(p) given by(6.7) and represented as a 5× 5 matrix has Minkowski coordinates

P0(p0, pi) = 1`

sinh `p0 −`p2i

2 e−`p0 ,

Pi(p0, pi) = pi e−`p0 , (6.8)

P4(p0, pi) = −1`

cosh `p0 + `p2i

2 e−`p0 .

One can easily check by direct computation that the coordinates PI = (Pµ, P4),µ = 0, . . . , 3 of these points satisfy the conditions

P 20 − P 2

1 − P 22 − P 2

3 − P 24 = − 1

`2, (6.9)

and (assuming that ` is negative)

P0 + P4 > 0 . (6.10)

Thus, as a manifold, the AN(3) group is an open subset of the four dimensional deSitter space (6.9) defined by the condition (6.10), and the points in this manifoldcan be parametrized by coordinates pµ. It is worth noticing in passing that the unitelement of the group AN(3), g(0) naturally corresponds to the zero momentum pointpµ = 0 of the momentum space, whose existence is required for the relative localityconstruction [48].

Since our momentum space, AN(3), is defined as a hyper-surface embedded inthe five dimensional Minkowski space it possesses a natural induced metric, which

2Again, as in Ch. 4 we choose sign conventions so that the speed of massless particles insubluminal for ` > 0.

79

can be obtained by inserting the relations (6.8) into the five dimensional Minkowskimetric

ds2 = dP 20 − dP 2

1 − dP 22 − dP 2

3 − dP 24 .

Using (6.9) one finds that this metric is nothing but the de Sitter metric in flatcoordinates

ds2 = dp20 − e−2`p0

(dp2

1 + dp22 + dp2

3

). (6.11)

We shall use this form of the metric in the next section, in the derivation of theon-shell relation of a particle on the κ-momentum space.

Since our momentum space is a group manifold it is natural to assume that themomentum composition and is defined by the group multiplication law. If we havetwo group elements g(p) and g(q) then their product is a group element itself sothat we can define the momentum composition ⊕ as follows:

g(p) g(q) = g(p⊕ q) . (6.12)

It is worth stressing that since the group multiplication is associative, the composition⊕ is associative as well.

In the case of the AN(3) group elements defined by (6.7) we find

g(p) g(q) = exp(iX i (pi + e`p0 qi)

)exp

(iX 0 (p0 + q0)

), (6.13)

so that we find again Eqs.(3.8) or (3.33),

(p⊕ q)i = pi + e`p0 qi , (p⊕ q)0 = p0 + q0 , (6.14)

which to the leading order in ` reads

(p⊕ q)i = pi + qi + ` p0 qi +O(`2) , (p⊕ q)0 = p0 + q0 . (6.15)

We can then introduce p, the “antipode” of p (see (3.10),(3.34)), using the factthat the inverse of a group element is a group element itself:

g−1(p) = g(p) , g−1(p)g(p) = 1⇔ p⊕ (p) = 0 (6.16)

and in the case of the AN(3) group we find

(p)i = −e−`p0 pi , (p)0 = −p0 (6.17)

and in the leading order we have

(p)i = − (1− `p0) pi +O(`2) , (p)0 = −p0 . (6.18)

Since this composition law can be derived within the time-to-the-right formulation ofthe κ-Poincaré/κ-Minkowski framework, which first appeared in Ref. [55] by Majidand Ruegg, we shall refer to this composition law as the “Majid-Ruegg compositionlaw” and to the associated affine connection on momentum space as the “Majid-Rueggconnection”.

In closing this subsection let us also observe that when the momentum spaceis a Lie group there is a natural way to construct a free particle action. The idea


is to identify the position space with a linear space dual to the Lie algebra (as avector space) and to make use of the canonical pairing between these dual spaces.Concretely let us define the basis of the vector space Yµ dual to the Lie algebraan(3) as follows:

〈Yµ,X ν〉 = δνµ . (6.19)

And let us take the space dual to the Lie algebra of AN(3) to be the space of positionsso that

x = xµ Yµ . (6.20)

Then the kinetic term of the action of a particle with AN(3) momentum space is3

Lkin ≡ −⟨x, g−1 d

dτg

⟩. (6.21)

Substituting (6.7), (6.19), and (6.20) into (6.21) one easily finds that

Lkin = xµpµ − ` pi xi p0 . (6.22)

It is worth noticing that the same procedure can be applied to the standard casewith flat momentum space, when the group associated with momentum compositionis just an abelian group R4 (in our case we get the abelian limit when `→∞.)

It follows from (6.22) that positions variables xµ have a nontrivial Poisson bracket.To see this most easily, notice that with the help of the transformation

xµ → xµ , x0 = x0 − ` pi xi , xi = xi , (6.23)

one can diagonalize the kinetic Lagrangian (6.22), Lkin = xµpµ, so that the Poissonbrackets in these new variables (see Sec. 3.2 Eqs. (3.35)-(3.37)) read

xµ, xν = 0 , xµ, pν = δµν .

Using (6.23) one easily finds the phase space4 (4.1)

x0, xi = −` xi , xi, xj = 0 , (6.24)

x0, p0 = 1 , xi, pj = δij , x0, pj = ` pj . (6.25)

6.1 The example of κ-Poincaré-inspired momentumspace with Majid-Ruegg connection

Ref. [48] introduced the idea of using the geodesic distance from the origin to ageneric point pµ in momentum space P as the mass of a particle. We shall here argue

3In the mathematical literature the symplectic form associated with this kinetic term is calledKirillov symplectic form.

4Throughout this chapter we adopt conventions consistent with the ones in [48, 49], so thatspacetime coordinates are characterized by upper indexes, while momenta are characterized bylower endexes. The R-DSR equations of the previous chapters are recovered once the indexes forthe spacetime coordinates are lowered with the metric ηµν = diag(1,−1).

6.1 Majid-Ruegg connection 81

that according to this proposal one should view the κ-Poincaré/κ-Minkowski-inspiredR-DSR framework as a case in which the metric on momentum space is de-Sitterlike (see Eq. (6.11)),

gµν =

1 0 0 00 −e−2`p0 0 00 0 −e−2`p0 00 0 0 −e−2`p0

, (6.26)

and, as already anticipated, parallel transport is given in terms of the Majid-Rueggconnection.

The other objective of this section is to establish the torsion and nonmetricity ofthis κ-Poincaré-inspired setup.

6.1.1 Distance from the origin in a de Sitter momentum space

In order to calculate the geodesic distance from the origin to a generic pointpµ = (p0, pj) in momentum space P we must find

D(0, pµ) =∫ 1

0ds√gµν pµpν , (6.27)

where pµ is the solution of the geodesic equation

pρ + Cµνρ pµpν = 0 , (6.28)

gµν is the metric of P and Cµνρ are the Christoffel symbols for the metric gµν .To find an approximate solution consider the metric slightly away from zero,

which has the formgµν = ηµν + ` hµνρ pρ + . . . (6.29)

A simple calculation of the Christoffel symbols to the leading order,

Cµνρ = 12 gρσ (gσµ,ν + gνσ,µ − gµν,σ) = `

2 ηρσ (hσµν + hνσµ − hµνσ) , (6.30)

shows that the only non vanishing components are:

Cij0 = −è−2`p0δij ,

Cj0i = C0ji = −`δij ,

(6.31)

so that the geodesic equation (6.28) can be easily solved perturbatively with theboundary conditions

pµ(0) = 0 , pµ(1) = Pµ . (6.32)

The solution at leading order is

pρ(s) = Pρ s+ 12 Cρ

µν Pµ Pν(s− s2) (6.33)

andpρ(s) = Pρ + 1

2 Cρµν Pµ Pν(1− 2s) . (6.34)


To compute the distance one must find√gµν pµ(s)pν(s) =

√ηµν Pµ Pν + Cρµν Pρ Pµ Pν(1− 2s) + `hµνρ Pρ Pµ Pν s . (6.35)

To do that we use the identity that results from eq. (6.30)

Cρµν Pρ Pµ Pν = `

2 hρµν Pρ Pµ Pν . (6.36)

So that finally we find√gµν pµ(s)pν(s) =

√P 2 + Cρµν Pρ Pµ Pν . (6.37)

Integrating this from 0 to 1 and taking the square we get the final result

D(0, Pµ) = m2 = P 2 + Cρµν Pρ Pµ Pν . (6.38)

Substituting the values of the connections found in eq. (6.31) we have

m2 = P 20 − P 2

i + `P0P2i , (6.39)

consistently with the leading-order form of the κ-Poincaré inspired on-shell relation.

6.1.2 Momentum space with de Sitter metric and Majid-Rueggconnection: torsion and (non)metricity

To further investigate the geometrical properties of momentum space we take theMajid-Ruegg composition law:

(p⊕ q)0 = p0 + q0 ,

(k ⊕ p)j = pj + e`p0qj .

Using the Majid-Ruegg composition law, we can define a parallel transport on themomentum space P as

(p⊕ dq)µ = pµ + dqµ − Γαβµ pαdqβ + . . . (6.40)

In particular for the (leading order) composition law

(p⊕ q)0 = p0 + q0 ,

(p⊕ q)i = pi + qi + `p0qi ,(6.41)

we find that the only non-vanishing components of the connection are:

Γ0ji = −`δji . (6.42)

Given the components of the connection we can easily find the other geometricproperties as torsion, nonmetricity and curvature.

For the torsion, we use [48]

Tαβµ = − ∂

∂pα

∂

∂qβ(p⊕ q − q ⊕ p)µ = 2Γ[αβ]

µ = Γαβµ − Γβαµ (6.43)

6.2 Partial anatomy of distant relative-locality observers 83

to find that at leading order the only non vanishing components of the torsion tensorare

T 0ji = −T j0i = Γ0j

i = −`δji . (6.44)

And for the nonmetricity tensor,

Nαµν = ∇αgµν = gµν,α + Γµαβ gβν + Γναβ gµβ , (6.45)

the only non-vanishing components to the leading order are

N0ij = 2`δij ,N i0j = `δij ,

N ij0 = `δij .

(6.46)

For what concerns the curvature of the connection, determined by [48]

Rαβγµ = 2 ∂

∂p[α

∂

∂qβ]

∂

∂rγ

((p⊕ q)⊕ r − p⊕ (q ⊕ r)

)µ

, (6.47)

it is evident that it vanishes by construction in any leading-order analysis (in a powerseries in ` the first contribution to the curvature of the connection is of order `2).It is worth noticing that in the case of the Majid-Ruegg connection this curvaturevanish exactly (to all orders) as a result of the fact that the Majid-Ruegg compositionlaw is associative.

6.2 Partial anatomy of distant relative-locality observers

6.2.1 A starting point for the description of distant relative-localityobservers

Let us now return to the preliminary results on translation invariance reported inRef. [48], which we already briefly summarized in the first section, but we shall nowanalyze in greater detail. In Ref. [48] translation invariance was explicitly checkedonly for the idealized case of the process we already showed in Figure 6.1, with 3particles of energy-momenta kµ, pµ, qµ all incoming into the interaction.

Let us note down again here the action Sexample which, according to Ref. [48],could describe the process in Figure 6.1:

Sexample=∫ s0


)−ξµKµ(s0). (6.48)

where C[k] = D2(k)−m2 is the distance of kµ from the origin of momentum space,and the on-shell condition is C[k] = 0, while the deformed law of energy-momentumconservation has been enforced by first introducing a connection-induced compositionof the momenta,

Kµ(s) ≡ [k(s)⊕ p(s)⊕ q(s)]µ ,

and then adding to the action a boundary term (in this case, at the s = s0 boundary)with this Kµ. The Lagrange multipliers enforcing Kµ = 0 are denoted by ξµ andplay the role of “interaction coordinates” in the sense of Ref. [48]. This is a theory


on momentum space in the sense that the “particle coordinates” xµ, yµ,zµ areintroduced as “conjugate momenta of the momenta”, and for the action Sexampleone evidently has that

xµ, kν = δµν , yµ, pν = δµν , zµ, qν = δµν . (6.49)

Following again Ref. [48] we vary the action Sexample keeping the momenta fixedat s = ±∞ (so that, for the case we are here considering, one has that δkµ

∣∣∣s=−∞

= 0,

δpµ∣∣∣s=−∞

= 0, δqµ∣∣∣s=−∞

= 0) and we find the equations of motion

kµ = 0 , pµ = 0 , qµ = 0 ,C[k] = 0 , C[p] = 0 , C[q] = 0 ,

Kµ = 0,

xµ = NkδC[k]δkµ

, yµ = NpδC[p]δpµ

, zµ = NqδC[q]δqµ

,

and the boundary conditions at the endpoints of the 3 semi-infinite worldlines

xµ(s0) = ξνδKνδkµ

, yµ(s0) = ξνδKνδpµ

, zµ(s0) = ξνδKνδqµ

. (6.50)

The relative locality is codified in the fact that for configurations with ξµ = 0the endpoints of the worldlines must coincide and be located in the origin of theobserver (xµ(s0) = yµ(s0) = zµ(s0) = 0), but for configurations such that ξµ 6= 0 theendpoints of the worldlines do not coincide, since in general

δKνδkµ

6= δKνδpµ6= δKν

δqµ, (6.51)

so that in the coordinatization of the (in that case, distant) observer the interactionappears to be nonlocal.

As noticed in Ref. [48], taking as starting point of the analysis some observerAlice for whom ξµA 6= 0, i.e. an observer distant from the interaction who sees theinteraction as nonlocal, one can obtain from Alice an observer Bob for whom ξµB = 0if the transformation from Alice to Bob for endpoints of coordinates has the form

xµB(s0) = xµA(s0)− ξνAδKν(s)δkµ(s)

∣∣∣s=s0

,

yµB(s0) = yµA(s0)− ξνAδKν(s)δpµ(s)

∣∣∣s=s0

,

zµB(s0) = zµA(s0)− ξνAδKν(s)δqµ(s)

∣∣∣s=s0

.

(6.52)

Such a property for the endpoint is produced of course, for the choice bν = ξνA, by


the following corresponding prescription for the translation transformations:

xµB(s) = xµA(s)− bν δKν(s)δkµ(s) ,

yµB(s) = yµA(s)− bν δKν(s)δpµ(s) ,

zµB(s) = zµA(s)− bν δKν(s)δqµ(s) ,

ξµB = ξµA − bµ .

(6.53)

Indeed one finds by direct substitution that these transformations leave the equationsof motion and the boundary conditions unchanged. And also the action is invariant;indeed

Sexample=∫ s0


)−ξµKµ(s0)

=∫ s0

−∞ds

((xµA − b

ν δKνδkµ

)kµ +

(yµA − b

ν δKνδpµ

)pµ +

(zµA − b

ν δKνδqµ

)qµ

+NkC [k] +NpC [p] +NqC [q])− ξµBKµ(s0)

= Sexample,bulkA −∫ s0

−∞ds

d

ds(bνKν)− ξµBKµ(s0)

= Sexample,bulkA − (ξµB + bµ)Kµ(s0)

= Sexample,bulkA − ξµAKµ(s0) = SexampleA ,

(6.54)

where Sexample,bulkA coincides with SexampleA with the exception of boundary terms.This also shows that all interactions are local according to nearby observers

(observers themselves local to the interaction): if ξµA 6= 0 for observer Alice, so thatin Alice’s coordinates the interaction is distant and nonlocal, one easily finds aobserver Bob for whom ξµB = 0, an observer local to the interaction who witnessesthe interaction as a sharply local interaction in its origin.

For the purposes of the proposal we shall put forward in the following sections, itis important to notice here that these observations reported in Ref. [48] actually canbe viewed as a prescription for translations generated by the “total momentum” Kµ(in which however individual momenta are summed with a nonlinear compositionlaw). In fact, in light of (6.49) the description of translation transformations givenin (6.53) simply gives

δxµB(s) = xµB(s)− xµA(s) = bν(k ⊕ p⊕ q)ν , xµ = bνK ν , xµ = −bν δK ν(s)

δkµ(s) ,

δyµB(s) = yµB(s)− yµA(s) = bν(k ⊕ p⊕ q)ν , yµ = bνK ν , yµ = −bν δK ν(s)

δpµ(s) ,

δzµB(s) = zµB(s)− zµA(s) = bν(k ⊕ p⊕ q)ν , zµ = bνK ν , zµ = −bν δK ν(s)

δqµ(s) .

(6.55)


6.2.2 Some properties of our conservation laws

Our next task is to focus on another issue which also needs to be fully appreciated inorder to work with relative locality: the issue of ordering momenta in the nonlinearcomposition law.

That ordering might be an issue is evident from the fact that relative-localitymomentum spaces can in general allow [48] for interactions characterized by con-servation laws which are possibly noncommutative (torsion) and/or non-associative(curvature of the connection). For leading-order analyses, of the type we are heremotivating, only noncommutativity is possible, but that is enough to introducequite some novelty with respect to standard absolute-locality theories. It shouldbe noticed however that the number of truly different conservation laws is muchsmaller than one might naively imagine, as we shall now show for our illustrativeκ-Poincaré-inspired example.

Let us first notice that while for arbitrary choices of k and p our compositionlaw is evidently such that k ⊕ p 6= p⊕ k (noncommutativity), in the cases of interestwhen discussing interactions, cases in which the composition of momenta is used towrite a conservation law, we actually do have

k ⊕ p = 0⇐⇒ p⊕ k = 0 .

This is easily checked in the case which is of primary interest for us here:

0 = k1 + p1 + `k0p1 = k1 + p1 + `p0k1 ,

where on the right-hand-side we used in the leading-order correction the propertiesk0 = −p0 and k1 = −p1 which follow (at zero-th order) from k ⊕ p = 0.

And actually k ⊕ p = 0⇐⇒ p⊕ k = 0 holds for any choice of affine connectionon momentum space, as shown by the following chain of properties:

k ⊕ p = 0 =⇒ p = k =⇒ p⊕ k = k ⊕ k = 0 .

This observation also simplifies the description of 3-particle interactions. In fact,since we have established that k ⊕ p = 0 ⇐⇒ p ⊕ k = 0 it then evidently followsthat5

k ⊕ p⊕ q = 0⇐⇒ q ⊕ k ⊕ p = 0 .

So, while there is no cyclicity property of the rule of composition of generic momenta,when the rule of composition is used for a conservation law it produces a conservationlaw with cyclicity.

6.2.3 Boundary terms and conservation of momenta

So we have seen that the number of truly independent conservation laws that can bepostulated using the deformed composition law “⊕” is smaller than one might have

5Note that from k ⊕ p = 0 ⇐⇒ p ⊕ k = 0, which holds for any choice of momentum-spaceaffine connection (and associated composition law), it evidently follows that (k ⊕ p)⊕ q = 0⇐⇒q ⊕ (k ⊕ p) = 0 but unless the composition law is associative this will not amount to a cyclicityproperty. When, as in the case which is here of our primary interest, the composition law isassociative we then have q⊕ (k⊕p) = (q⊕k)⊕p = q⊕k⊕p and a genuine cyclicity property arises.


naively imagined, because of cyclicity. For some of the observations we report later onin this manuscript it is however important to appreciate that different compositionsof momenta that (when set to zero) would produce the same conservation law stillcan lead to tangibly different choices of boundary terms enforcing the conservationlaws.

Let us first illustrate the issue within the specific example of an interaction withtwo incoming and one outgoing particle, with conservation law

p⊕ k ⊕ (q) = 0 .

This conservation law can be enforced by adding to the action a term of the formξµKµ, withKµ = [p⊕ k⊕ (q)]µ and ξµ are Lagrange multipliers. But this evidentlyis not the only choice of constraint term that enforces the chosen conservation law.For example let us observe that6

p⊕ k ⊕ (q) = 0 ⇐⇒ p⊕ k = q ⇐⇒ (p⊕ k)− q = 0 ,

and also that

p⊕ k ⊕ (q) = 0 ⇐⇒ p⊕ k = q ⇐⇒ q ⊕ p⊕ k = q ⊕ q = 0 .

So we see that the same conservation law7 can be enforced by adding a boundaryterm of the form ξµKµ with Kµ given by any choice among Kµ = [p⊕ k ⊕ (q)]µ,Kµ = [(p⊕ k)− q]µ, and Kµ = [(q)⊕ p⊕ k]µ, However, it is easy to verify (and thiswill play a role in the analysis reported in the following section) that these differentpossible choices of boundary terms enforcing the same momentum-conservation lawactually produce boundary conditions that are physically different.

In the case of our interest, which is the case of the Majid-Ruegg connection, weshall be confronted with the observation that

(p⊕ k ⊕ (q))0 = ((p⊕ k)− q)0 = ((q)⊕ p⊕ k)0

but

((p⊕k)−q)1 = (p⊕k⊕(q))1 +`(q0−k0−q0)q1 = ((q)⊕p⊕k)1 +`q0(q1−k1−q1) .

However, when ((p⊕ k)− q)µ = 0 one evidently also has8. (neglecting O(`2)) that`(q0 − k0 − q0)q1 = 0 = `q0(q1 − k1 − q1), so also for the specific case of the Majid-Ruegg connection one has this possibility of different boundary terms enforcing thesame conservations laws, but producing physically-different boundary conditions.

6This elementary chain of equivalence may at first appear striking since evidently in generalp⊕ k ⊕ (q) 6= (p⊕ k)− q, or, more precisely, if p⊕ k ⊕ (q) 6= 0 then p⊕ k ⊕ (q) 6= (p⊕ k)− q.However the chain of equivalences immediately follows upon observing that in the special caseswhere p⊕ k ⊕ (q) = 0 (conservation laws) one then has that also (p⊕ k)− q = 0.

7It should be noticed that in Ref. [12], where nonlinear conservation laws were analyzed from theperspective of the doubly-special-relativity research program, a possible role of such conservationlaws of the type p[in]− (p[out,1]⊕p[out,2]) = 0 or (p[in,1]⊕p[in,2])− (p[out,1]⊕p[out,2]) = 0 was alreadymotivated on different grounds.

8We stress again that the 3 conservation laws in question are exactly equivalent, equivalent to allorders in `. We are however working here to leading order in `, and for example the antipode forthe Majid-Ruegg connection was here determined only to leading order. So the equivalence of the3 conservation laws in question is of course verified within our computations only upon droppingsubleading, O(`2), contributions.


6.2.4 A challenge for spacetime-translation invariance in theorieson a relative-locality momentum space

We shall now characterize preliminarily the nature of some consistency conditionsthat should be enforced in order to produce a relativistic formulation with relativelocality for interacting particles. As emphasized at the beginning of this section,in the relative-locality frameworks here of interest essentially what happens is thatthe correct notion of translation to distant observers must act on the endpoints ofworldlines in a way that reflects the form of the boundary terms used to implementthe conservation laws. As also shown at the beginning of this section this notion isnever problematic for semi-infinite worldlines, with a single endpoint. But we mustnow highlight a challenge which materializes in all instances where two interactionsare causally connected, i.e. there is a particle “exchanged” between the interactions,described by a finite worldline with two endpoints. In those instances we are going tohave that the conservation laws essentially impose two conditions on the “exchangedworldline”, for the translation of the two endpoints. But we must request, for arelativistic description, that the worldline of distant observer Bob is solution ofthe same equations of motion that the initial observer Alice determines, and theserelativistic demands are not automatically satisfied.

In order to render our concerns more explicit let us consider a specific examplewhich does not admit the sort of relativistic description we are here interested in.For simplicity we consider a case in which the on-shell relation is undeformed and thesymplectic structure is trivial. And we consider the situation shown in Figure 6.2,in which the two outgoing particles of a first decay themselves eventually decay.

k, z

p, x

p′, x′

p′′, x′′

q, y

K[1] = p⊕ (⊖p′′)⊕ (⊖p′)

K[0] = k ⊕ (⊖q)⊕ (⊖p)

q′, y′

q′′, y′′

K[1] = q ⊕ (⊖q′′)⊕ (⊖q′)

Figure 6.2. The action given and analyzed in this subsection would be intended for thedescription of the three causally-connected interactions shown here. But it appears thatsuch a description is incompatible with a relativistic description of distant observers

A suitable description of the relevant conservation laws is the following:

0 = (k ⊕ (q)⊕ (p))µ = kµ − pµ − qµ − `δjµ (k0qj − q0qj + k0pj − p0pj − q0pj) ,

0=(p⊕ (p′′)⊕ (p′)

)µ=pµ−p′µ−p′′µ−`δjµ

(p0p′′j − p′′0p′′j + p0p

′j − p′0p′j − p′′0p′j

),

0=(q ⊕ (q′′)⊕ (q′)

)µ=qµ−q′µ−q′′µ−`δjµ

(q0q′′j − q′′0q′′j +q0q

′j−q′0q′j−q′′0q′j

).

(6.56)

where for definiteness (it is easy to check that none of the points made in thissubsection depend crucially on this choice) we specified as composition law the


one coming from the “Majid-Ruegg connection”. Evidently the conservation lawsconcern a first interaction where a particle of momentum k decays into a particle ofmomentum p plus some other particle of momentum q, followed by two more decays,one where the particle of momentum p decays into particles of momentum p′ and p′′and one where the particle of momentum q decays into particles of momentum q′

and q′′.

The main observation we here want to convey is that the following choice of K’sto be used in writing up constraints implementing the conservation laws

K[0](s0) = k ⊕ (q)⊕ (p) ,K[1](s1) = p⊕ (p′′)⊕ (p′) ,K[2](s2) = q ⊕ (q′′)⊕ (q′) .

which appears to be a very natural way to implement the conservation laws asconstraints, does not lead to a relativistic description of distant observers.

To see this let us first write the action which would implement all this:

SA =∫ 0

−∞ds(zµkµ +Nk[k2 −m2

a])

+∫ s1

s0ds(xµpµ +Np[p2 −m2

b ])

+∫ s2

s0ds(yµqµ +Nq[q2 −m2

c ])

+∫ +∞

s1ds(x′µp′µ +Np′ [p′2 −m2

d])

+∫ +∞

s1ds(x′′µp′′µ +Np′′ [p′′2 −m2

e])

+∫ +∞

s2ds(y′µq′µ +Nq′ [q′2 −m2

f ])

+∫ +∞

s2ds(y′′µq′′µ+Nq′′ [q′′2−m2

g])−ξµ[0]AK

[0]µ (s0)−ξµ[1]AK

[1]µ (s1)−ξµ[2]AK

[2]µ (s2),

(6.57)

where we restricted our focus on the undeformed on-shell condition k2−m2 = 0 andwe allowed for the presence of particles of different mass. The equations of motionthat follow from varying this action evidently are:

kµ = pµ = qµ = p′µ = p′′µ = q′µ = q′′ = 0 ,k2−m2

a = p2−m2b = q2−m2

c = p′2−m2d = p′′2−m2

e = q′2−m2f = q′′2−m2

g = 0,K[0]µ = 0 , K[1]

µ = 0 , K[2]µ = 0 ,

zµ = 2Nkδµ0 k0 − 2Nkδµ1 k1 , xµ = 2Npδµ0 p0 − 2Npδµ1 p1 ,

x′µ = 2Np′δµ0 p′0 − 2Np′δµ1 p′1 , x′′µ = 2Np′′δµ0 p′′0 − 2Np′′δµ1 p′′1 ,yµ=2Nqδµ0 q0−2Nqδµ1 q1, y′µ=2Nq′δµ0 q′0−2Nq′δµ1 q′1, y′′µ=2Nq′′δµ0 q′′0−2Nq′′δµ1 q′′1 .


And for the boundary conditions at endpoints of worldlines one finds:

zµA(s0) = ξν[0]AδK[0]

ν

δkµ= ξµ[0]A − `δ

µ0 ξ

1[0]A(q1 + p1) ,

xµA(s0) = −ξν[0]AδK[0]

ν

δpµ= ξµ[0]A − `δ

µ0 ξ

1[0]Ap1 + `δµ1 ξ

1[0]A(k0 − q0 − p0) ,

xµA(s1) = ξν[1]AδK[1]

ν

δpµ= ξµ[1]A − `δ

µ0 ξ

1[0]A(p′1 + p′′1) ,

yµA(s0) = −ξν[0]AδK[0]

ν

δqµ= ξµ[0]A + `δµ0 ξ

1[0]A(q1 + p1) + `δµ1 ξ

1[0]A(k0 − q0) ,

yµA(s2) = ξν[2]AδK[2]

ν

δqµ= ξµ[2]A − `δ

µ0 ξ

1[0]A(q′1 + q′′1) ,

x′µA (s1) = −ξν[1]AδK[1]

ν

δp′µ= ξµ[1]A − `δ

µ0 ξ

1[0]Ap

′1 + `δµ1 ξ

1[0]A(p0 − p′0 − p′′0) ,

x′′µA (s1) = −ξν[1]AδK[1]

ν

δp′′µ= ξµ[1]A − `δ

µ0 ξ

1[0]A(p′1 + p′′1) + `δµ1 ξ

1[0]A(p0 − p′′0) ,

y′µA (s2) = −ξν[2]AδK[2]

ν

δq′µ= ξµ[2]A − `δ

µ0 ξ

1[0]Aq

′1 + `δµ1 ξ

1[0]A(q0 − q′0 − q′′0) ,

y′′µA (s2) = −ξν[2]AδK[2]

ν

δq′′µ= ξµ[2]A − `δ

µ0 ξ

1[0]A(q′1 + q′′1) + `δµ1 ξ

1[0]A(q0 − q′′0) .

From this we immediately see that the action SA does not admit a relativisticdescription of distant observers (in relative rest), at least not in the sense intendedin Ref. [48]. And, as announced, the troubles originate from the finite worldlines,with two endpoints. For example, according to the observation reported in Ref. [48](and here summarized in Subsec. 6.2.1), one would like translation transformationssuch that the endpoints of the worldline of momentum p transform as follows:

xµB(s0) = xµA(s0) + bνδK[0]

ν

δpµ,

xµB(s1) = xµA(s1)− bν δK[1]ν

δpµ.

(6.58)

But we also must insist, if the transformation from Alice to Bob is to be relativistic,that the equations of motion written by Alice and Bob are the same, so thatin particular also for Bob xµB = 2Npδµ0 p0 − 2Npδµ1 p1. However, enforcing bothxµA = 2Npδµ0 p0 − 2Npδµ1 p1 for Alice and xµB = 2Npδµ0 p0 − 2Npδµ1 p1 for Bob imposeson our translation transformations that they be rigid translations of the endpoints,in the sense that for (6.58) one should have

δK[0]ν

δpµ= −δK

[1]ν

δpµ. (6.59)

And it is easy to see that this condition, while automatically verified at zero-th orderis in general not satisfied at O(`).

6.3 A Lagrangian description of relative locality with interactions 91

For example for the Majid-Ruegg connection one has that

K[0]1 =[k ⊕ (q)⊕ (p)]1 =k1−q1−p1−`(k0q1−q0q1+k0p1−q0p1−p0p1) ,

K[1]1 =[p⊕ (p′′)⊕ (p′)]1 =p1−p′′1−p′1−`(−p′′0p′′1−p′′0p′1−p′0p′1+p0p

′′1 +p0p

′1) ,(6.60)

from which it follows that

δK[0]1

δp0= `p1 ,

δK[0]1

δp1= −1− `(k0 − q0 − p0) ,

δK[1]1

δp0= −`(p′1 + p′′1) , δK[1]

1δp1

= 1 .(6.61)

which indeed confirms that the condition (6.59) is satisfied at zero-th order butviolated at O(`).

6.3 A Lagrangian description of relative locality withinteractions

6.3.1 R-DSR symplectic structure and translations generated bytotal momentum

In this section we show that it is possible to have a relativistic description of pairs ofdistant observers (in relative rest), in characterizations of interactions with particleexchanges (finite worldlines) formulated within the relative-locality framework ofRefs. [48, 49]. The main challenge we shall face in this section is the one characterizedin Subsection 6.2.4: relativistic descriptions of a single interaction with relativelocality are rather elementary, but when pairs of interactions a causally connectedthe availability of a relativistic description for distant observers is in no way assured,and actually before the study we are here reporting there was no known examplewhere it had been shown to work.

We shall also find reassuring that the Lagrangian description we obtain forinteracting particles, reproduces in an appropriate limit the results of chapter 4,concerning relative locality in a κ-Poincaré-inspired (R-DSR) Hamiltonian descriptionof free particles. In doing so we also provide an explicit analysis in which the non-trivial geometry of momentum space is analyzed while adopting a non-standardsymplectic structure.

Indeed, the first point of contact between our Lagrangian description and theHamiltonian description of chapter 4 is found in the choice of symplectic structureand on-shell condition characterizing the “free part” of the action”, which for thecase of 3 particles (of momenta kµ incoming and momenta pµ and qµ outgoing) takesthe form:

Sbulkκ =∫ s0

−∞ds(zµkµ−`z1k1k0+NkCκ [k]

)+∫ +∞

s0ds(xµpµ−`x1p1p0+NpCκ [p]

)+∫ +∞

s0ds(yµqµ − `y1q1q0 +NqCκ [q]

),

(6.62)


whereCκ[k] ≡ k2

0 − k21 + `k0k

21 ,

so that we implement the on-shell relation of the Hamiltonian R-DSR phase-spacesetup. We are adopting R-DSR Poisson brackets, so that for example for xµ

x1, x0

= `x1 ,

and from (6.62) one recognizes that our symplectic structure also matches the one ofthe Hamiltonian R-DSR phase-space9 setup Eq. (4.1); so that for example for xµ,pµ

x0, p0

= 1,x1, p0

= 0 ,

x0, p1

= `p1,x1, p1

= 1 .

For what concerns the conservation laws at interactions we shall adopt the Majid-Ruegg connection. But, as evident on the basis of the observation we reported inSection 6.2, once the conservation laws are specified the construction of this type ofrelative-locality theory still leaves open a choice among possible alternative ways ofimplementing such laws of momentum conservation through some boundary terms.We adopt a particular choice which we favor because it happens to be immune fromthe problem here highlighted in Subsection 6.2.4, which instead is found to affectseveral alternative possibilities. We qualify our choice of momentum-conservationconstraints as the ones that are suitable for a description of translations in which“translations are generated by the total momentum”, for reasons that will becomeclearer in the reminder of this section. The prescription we adopt will be generalizedas we go along, but let us here start with the case of a single interaction, whoseconservation law is

0 = k ⊕ (q)⊕ (p) .

As already stressed in Subsection 6.2.3, such a conservation law could be imple-mented by several inequivalent choices of K[0] for the constraints on the endpointsof worldlines, including

K[0] = k ⊕ (q)⊕ (p) ,

K[0] = (q)⊕ (p)⊕ k ,

K[0] = k − (p⊕ q) .

We find that this latter option K[0] = k − (p ⊕ q) admits a consistent relativisticdescription of distant observers. Evidence of this will be provided throughout thissection. But let us first notice that this sort of constraints is very intuitive: theyimplement the rather standard concept that the conservation law is such that thetotal momentum before an interaction should equal the total momentum after aninteraction. And we shall show that this form of the constraints allows one topreserve the usual notion that translation transformations are generated by the total

9The equations of Ch. 4 are recovered once the indexes of the spacetime coordinates are loweredwith the metric tensor ηµν = diag(1,−1).


momentum (though of course in our case the total momentum is obtained in termsof the nonlinear composition law), even when several interactions are analyzed andparticles are exchanged among some of the interactions.

k, z

p, x

q, y

K[0] = k − (p⊕ q)

Figure 6.3. The choice of K we adopt for the case of a single interaction with 1 incomingand 2 outgoing particles.

Essentially our proposal establishes that there is at least one way (at present weare unable to claim that it is unique) to address the challenge we earlier highlightedin Eq. (6.59). And the conceptual content of the solution we found for addressingthat challenge exemplified in Eq. (6.59) is, as shown below, rather simple: themost basic notion of relativistic translation transformation is as usual generatedby the total momentum acting on worldlines, but (as also shown in our discussionsurrounding Eq. (6.59)) the boundary terms used to enforce the conservation lawsrequire that endpoints transform under translations in ways governed by (or atleast conditioned by) the boundary terms. We handle the challenge illustrated byEq. (6.59) by essentially finding a way to render these two demands compatible:we enforce the conservation laws through boundary terms written in such a waythat when the worldlines are translated by the total momentum then the endpointsautomatically match the demands of the boundary terms.

Let us start seeing how this plays out for a case with a single interaction,considering, for the interaction in Figure 6.3, the action

Sκ=Sκbulk+Sκint=∫ s0

−∞ds(zµkµ−`z1k1k0+NkCκ [k]

)+∫ +∞

s0ds(xµpµ−`x1p1p0+NpCκ [p]

)+∫ +∞

s0ds(yµqµ−`y1q1q0+NqCκ [q]

)−ξµ[0]K

[0]µ (s0),

(6.63)

where indeed for K[0] we take

K[0]µ (s0) = kµ − (p⊕ q)µ = kµ − pµ + `δ1

µq0p1 . (6.64)

The equations of motion that follow from our action Sκ are of course the samefound in the Hamiltonian formulation of free R-DSR particles of Ch. 4:

pµ = 0 , qµ = 0 , kµ = 0 ,Cκ[p] = 0 , Cκ[q] = 0 , Cκ[k] = 0 , (6.65)

K[0]µ (s0) = 0


xµ=Np

(δCκ[p]δpµ

+`δµ0δCκ[p]δp1

p1

)=δµ0Np

(2p0−`p2

1

)−2δµ1Np (p1−`p0p1) ,

yµ=Nq

(δCκ[q]δqµ

+`δµ0δCκ[q]δq1

q1

)=δµ0Nq

(2q0−`q2

1

)−2δµ1Nq (q1−`q0q1) , (6.66)

zµ=Nk

(δCκ[k]δkµ

+`δµ0δCκ[k]δk1

k1

)=δµ0Nk

(2k0−`k2

1

)−2δµ1Nk (k1−`k0k1) .

And the interaction at s = s0 produces the boundary conditions:

xµ(s0) = −ξν[0]

(δK[0]

ν

δpµ+ `δµ0

δK[0]ν

δp1p1

)= ξµ[0] + `δµ0 ξ

1[0](p1 + q1) ,

yµ(s0) = −ξν[0]

(δK[0]

ν

δqµ+ `δµ0

δK[0]ν

δq1q1

)= ξµ[0] + `δµ0 ξ

1[0]q1 + `δµ0 ξ

1[0]p0 ,

zµ(s0) = ξν[0]

(δK[0]

ν

δkµ+ `δµ0

δK[0]ν

δk1k1

)= ξµ[0] + `δµ0 ξ

1[0]k1 .

(6.67)

The mechanism for relative locality which we already discussed above is evidentlyalso present here: the boundary conditions establish that if the observer is local tothe interaction, i.e. ξµ[0] = 0, then all endpoints of the semiinfinite worldlines arein the origin of the observer. If instead ξµ[0] 6= 0 the endpoints of worldlines do notcoincide.

It is also easy to check that our equations of motion (6.65), (6.66) and boundaryconditions (6.67) invariant under deformed translations generated by the totalmomentum acting on coordinates

x0B(s) = x0

A(s) + bµ(p⊕ q)µ, x0 = x0A(s)− b0 − `b1(p1 + q1) ,

x1B(s) = x1

A(s) + bµ(p⊕ q)µ, x1 = x1A(s)− b1 ,

y0B(s) = y0

A(s) + bµ(p⊕ q)µ, y0 = y0A(s)− b0 − `b1q1 ,

y1B(s) = y1

A(s) + bµ (p⊕ q)µ, y1 = y1A(s)− b1 − `b1p0 ,

z0B(s) = z0

A(s) + bµkµ, z0 = z0A(s)− b0 − `b1k1 ,

z1B(s) = z1

A(s) + bµkµ, z1 = z1A(s)− b1 .

(6.68)

It should be noticed that we essentially prescribe that a given point of a givenworldline is translated by acting with the total momentum written in the way thatis appropriate for that point of the worldline, so that, in the specific example hereunder consideration, all points with s < s0 are translated by kµ whereas all pointswith s > s0 are translated by (p⊕ q)µ.

The invariance of the equations of motion is easily seen by observing thatEq. (6.65) guarantees that pµ = 0, qµ = 0, kµ = 0 and that the translationtransformations depend only on momenta. Considering for example the worldlinexµ, and assuming of course that both observer Alice and observer Bob adopt the


equations of motion

xµ[A] = Np

(δCκ[p]δpµ

+ `δµ0δCκ[p]δp1

p1

),

xµ[B] = Np

(δCκ[p]δpµ


p1

),

(6.69)

one indeed finds that the translation transformations

x0B(s) = x0

A − b0 − `b1(p1 + q1) ,x1B(s) = x1

A − b1(6.70)

are such that xµ[B] = xµ[A] (since momenta are conserved).And the invariance of the boundary conditions is easily seen by directly checking

that the boundary conditions for Alice are mapped by the translation transformationsinto the (identical) boundary conditions for Bob. For example, we have for Alice

xµ[A](s0) = −ξν[0]A

(δK[0]

ν

δpµ+ `δµ0

δK[0]ν

δp1p1

)= ξµ[0]A + `δµ0 ξ

1[0]A(p1 + q1) , (6.71)

and the translation transformations (6.70) map this into

xµ[B](s0)=xµ[A](s0)−bµ−`δµ0 (p1+q1)=−ξν[0]A

(δK[0]

ν

δpµ+`δµ0

δK[0]ν

δp1p1

)−bµ−`δµ0 (p1+q1)

=ξµ[0]A+`δµ0 ξ1[0]A(p1+q1)−bµ−`δµ0 (p1+q1)=−(ξ[0]A−b)ν

(δK[0]

ν

δpµ+`δµ0

δK[0]ν

δp1p1

)

=−ξν[0]B

(δK[0]

ν

δpµ+`δµ0

δK[0]ν

δp1p1

).

(6.72)

Besides checking the invariance of the equations of motion and the boundaryconditions, which however already ensure that our translations are physical symme-tries, it is also valuable to apply the translation transformations (6.68) to the action(6.63), so that we can find the relation between the action of Alice and the action ofBob (distant from Alice). We find

SκB = SκA +∫ s0

−∞ds(−bµkµ

)+∫ ∞s0

ds(−bµpµ − `b1q1p0

)+∫ ∞s0

ds(−bµqµ − `b1p0q1

)+ ∆ξµ[0]K

[0]µ (s0) .

where ∆ξµ[0] = ξµ[0]B − ξµ[0]A . Substituting s′ = −s in the first integral and then

relabeling s′ → s , one then gets

∆Sκ = SκB − SκA =∫ ∞s0

ds(bµ(kµ − pµ − qµ

)− `b1 (q1p0 + p0q1)

)+ ∆ξµ[0]K

[0]µ (s0) .


Then using Eq. (6.64) we find

∆Sκ = SκB − SκA =∫ ∞s0

dsd

ds

[bµK[0]

µ

]+ ∆ξµ[0]K

[0]µ (s0) . (6.73)

The total derivatives contribute to the boundaries in such a way that, for thedifference (6.73) to be null, it must hold(

∆ξµ[0] − bµ)K[0]µ (s0) ,

from which we see that the ξµ[0] translate classically:

ξµ[0]B= ξµ[0]A

− bµ . (6.74)

And when the observer Alice is distant from the interaction, i.e. ξµ[0]A 6= 0, onecan always find through such translation transformations an observer Bob local tothe interaction and for whom the endpoints of worldlines match:

xµB(s0) = −ξνB

(δK[0]

ν

δpµ+ `δµ0

δK[0]ν

δp1p1

)= −(ξνA − bν)

(δK[0]

ν

δpµ+ `δµ0

δK[0]ν

δp1p1

)= 0 ,

yµB(s0) = −ξνB

(δK[0]

ν

δqµ+ `δµ0

δK[0]ν

δq1q1

)= −(ξνA − bν)

(δK[0]

ν

δqµ+ `δµ0

δK[0]ν

δq1q1

)= 0 ,

zµB(s0) = ξνB

(δK[0]

ν

δkµ+ `δµ0

δK[0]ν

δk1k1

)= (ξνA − bν)

(δK[0]

ν

δkµ+ `δµ0

δK[0]ν

δk1k1

)= 0 .

(6.75)

6.3.2 Causally connected interactions and translations generatedby total momentum

Our next challenge is to deal with causally-connected interactions. We show in figurethe case we here analyze as illustrative example.

k, z

p, x

p′, x′

p′′, x′′

q, y

K[1] = p⊕ q − (p′ ⊕ p′′ ⊕ q)

K[0] = k − (p⊕ q)

Figure 6.4. The case of causally-connected interactions analyzed in this subsection.

Of course, there is no difficulty generalizing to this case the bulk part of the


action:

Sκ(2)bulk =

∫ s0

−∞ds(zµkµ−`z1k1k0 +NkCκ [k]

)+∫ s1

s0ds(xµpµ−`x1p1p0 +NpCκ [p]

)+∫ +∞

s1ds(x′µp′µ−`x′

1p′1p′0 +Np′Cκ

[p′])

+∫ +∞

s1ds(x′′µp′′µ−`x′′1p′′1 p′′0+Np′′Cκ

[p′′])

+∫ +∞

s0ds(yµqµ−`y1q1q0+NqCκ [q]

).

(6.76)

For the description of the interactions we take a case characterized by thefollowing conservation laws:

k ⊕ (q)⊕ (p) = 0 ,p⊕ (p′′)⊕ (p′) = 0 . (6.77)

And we propose an implementation of these conservation laws that is compatiblewith a relativistic description of distant observers, based on adding to the actionconstraints with

K[0] = k − (p⊕ q) ,K[1] = (p⊕ q)− (p′ ⊕ p′′ ⊕ q) ,

i.e.

K[0]µ =kµ−(p⊕ q)µ=kµ−pµ−qµ−`δ1

µp0q1 ,

K[1]µ =(p⊕ q)µ−(p′ ⊕ p′′ ⊕ q)µ=pµ−p′µ−p′′µ−`δ1

µ

(−p0q1+p′0p′′1 +p′0q1+p′′0q1

).

(6.78)

For the constraints we are again implementing our prescription of writing themin terms of differences between the total momentum before the interaction and afterthe interaction. It may appear that in doing so we included in the constraints someirrelevant pieces (it is easy to verify that the conservation law K[1] = 0 is actuallyindependent of qµ, which is the momentum of the particle that is only a spectator ofthe interaction occurring at s = s1). However, as we shall see, those extra pieces,while irrelevant for the physical content of the conservation laws, do play a rolein the description of translation transformations and ensure the availability of arelativistic description of distant observers.

To show this let us then start by writing the full action for the two-interactioncase on which we are presently focusing:

Sκ(2) = Sκ(2)bulk + Sκ(2)

int =∫ s0

−∞ds(zµkµ − `z1k1k0 +NkCκ [k]

)+∫ s1

s0ds(xµpµ−`x1p1p0 +NpCκ [p]

)+∫ +∞



[p′])

+∫ +∞

s1ds(x′′µp′′µ−`x′′1p′′1 p′′0 +Np′′Cκ

[p′′])

+∫ +∞

s0ds(yµqµ − `y1q1q0 +NqCκ [q]

)− ξµ[0]K

[0]µ (s0)− ξµ[1]K

[1]µ (s1),

(6.79)


where indeed with K[0] and K[1] we take respectively k−(p⊕q) and (p⊕q)−(p′⊕p′′⊕q).It is again straightforward to derive the equations of motion (and constraints)

that follow from our action Sκ(2):

pµ = 0 , qµ = 0 , kµ = 0 , p′µ = 0 , p′′µ = 0 ,Cκ[p] = 0 , Cκ[q] = 0 , Cκ[k] = 0 , Cκ[p′] = 0 , Cκ[p′′] = 0 , (6.80)

K[0]µ (s0) = 0 , K[1]

µ (s1) = 0 ,

xµ = Np

(δCκ[p]δpµ


p1

)= δµ0Np

(2p0 − `p2

1

)− 2δµ1Np (p1 − `p0p1) ,

yµ = Nq

(δCκ[q]δqµ

+ `δµ0δCκ[q]δq1

q1

)= δµ0Nq

(2q0 − `q2

1

)− 2δµ1Nq (q1 − `q0q1) ,

zµ = Nk

(δCκ[k]δkµ

+ `δµ0δCκ[k]δk1

k1

)= δµ0Nk

(2k0 − `k2

1

)− 2δµ1Nk (k1 − `k0k1) ,

x′µ = Np′(δCκ[p′]δp′µ

+ `δµ0δCκ[p′]δp′1

p′1

)= δµ0Np′

(2p′0 − `p′21

)− 2δµ1Np′

(p′1 − `p′0p′1

),

x′′µ = N ′′p

(δCκ[p′′]δp′′µ

+ `δµ0δCκ[k]δp′′1

p′′1

)= δµ0Np′′

(2p′′0 − `p′′21

)− 2δµ1Np′′

(p′′1 − `p′′0p′′1

).

(6.81)

And also the conditions at the s = s0 and s = s1 boundaries produced by theinteraction terms are of rather standard relative-locality type:

zµ(s0) = ξν[0]

(δK[0]

ν

δkµ+ `δµ0

δK[0]ν

δk1k1

)= ξµ[0] + `δµ0 ξ

1[0]k1 ,

xµ(s0) = −ξν[0]

(δK[0]

ν

δpµ+ `δµ0

δK[0]ν

δp1p1

)= ξµ[0] + `δµ0 ξ

1[0](p1 + q1) ,

xµ(s1) = ξν[1]

(δK[1]

ν

δpµ+ `δµ0

δK[1]ν

δp1p1

)= ξµ[1] + `δµ0 ξ

1[1](p1 + q1) ,

yµ(s0) = −ξν[0]

(δK[0]

ν

δqµ+ `δµ0

δK[0]ν

δq1q1

)= ξµ[0] + `δµ0 ξ

1[0]q1 + `δµ1 ξ

1[0]p0 ,

x′µ(s1) = −ξν[1]

(δK[1]

ν

δp′µ+ `δµ0

δK[1]ν

δp′1p′1

)= ξµ[1] + `δµ0 ξ

1[1](p

′1 + p′′1 + q1) ,

x′′µ(s1) = −ξν[1]

(δK[1]

ν

δp′′µ+ `δµ0

δK[1]ν

δp′′1p′′1

)= ξµ[1] + `δµ0 ξ

1[1](p

′′1 + q1) + `δµ1 ξ

1[1]p′0 . (6.82)

However, thanks to our tailored choice of momentum-conservation constraints theboundary conditions at the two endpoints of the finite worldline exchanged by thetwo interactions (the finite worldline of particle coordinates xµ(s) and momentumpµ) match just in the right way to allow implementing as a relativistic symmetry


the following translation transformations, generated by the total momentum

z0B(s) = z0

A(s) + bµkµ, z0 = z0A(s)− b0 − `b1k1 ,

z1B(s) = z1

A(s) + bµkµ, z1 = z1A(s)− b1 ,

x0B(s) = x0

A(s) + bµ(p⊕ q)µ, x0 = x0A(s)− b0 − `b1(p1 + q1) ,

x1B(s) = x1

A(s) + bµ(p⊕ q)µ, x0 = x1A(s)− b1 ,

y0B(s) = y0

A(s) + bµ(p⊕ q)µ, y0 = y0A(s)− b0 − `b1q1 ,

y1B(s) = y1

A(s) + bµ(p⊕ q)µ, y1 = y1A(s)− b1 − `b1p0 ,

x′0B(s) = x′

0A(s) + bµ(p′ ⊕ p′′ ⊕ q)µ, x′0 = x′

0A(s)− b0 − `b1(p′1 + p′′1 + q1) ,

x′1B(s) = x′

1A(s) + bµ(p′ ⊕ p′′ ⊕ q)µ, x′1 = x′

1A(s)− b1 ,

x′′0B(s) = x′′

0A(s) + bµ(p′ ⊕ p′′ ⊕ q)µ, x′′0 = x′′

0A(s)− b0 − `b1(p′′1 + q1) ,

x′′1B(s) = x′′

1A(s) + bµ(p′ ⊕ p′′ ⊕ q)µ, x′′1 = x′′

1A(s)− b1 − `b1p′0 .

(6.83)

It is again straightforward to see that these transformations leave the equations ofmotion (6.81) unchanged by noticing, as done in the previous subsection, that theonly non trivial terms in the deformed translations (6.83) depend on momenta andthe momenta are conserved along the worldlines.

And it is also easy to verify that our translation transformations leave theboundary conditions unchanged. In order to give an explicit example let us checkthe case of x′′µ: substituting the translation calculated in Eq. (6.83)

x′′0B(s) = x′′

0A − b0 − `b1(p′′1 + q1) ,

x′′1B(s) = x′′

1A − b1 − `b1p′0 ,

ξµB = ξµA − bµ ,

in the boundary conditions (6.82)

x′′0B (s1) = −ξν[1]B

(δK[1]

ν

δp′′0+ `

δK[1]ν

δp′′1p′′1

)= ξ0

[1]B + `ξ1[1]B(q1 + p′′1) ,

x′′1B (s1) = −ξν[1]B

(δK[1]

ν

δp′′1

)= ξ1

[1]B(1 + `p′0) ,

we find

x′′0B (s1)− ξ0[1]B − `ξ

1[1]B(q1 + p′′1) = x′′

0A(s1)− ξ0

[1]A − `ξ1[1]A(q1 + p′′1) ,

x′′1B (s1)− ξ1[1]B(1 + `p′0) = x′′1A (s1)− ξ1

[1]A(1 + `p′0) .

which is evidently consistent with our boundary conditions.Thus we did succeed: even in the case of finite worldlines, causally connecting

pairs of interactions, our prescription for boundary terms does ensure translationalinvariance, addressing the challenge highlighted here in Section 6.2.4. And ourrelative-locality distant observers are connected by relativistic transformations gen-erated by the (⊕-deformed) total momentum.


The invariance of the equations of motion and boundary conditions under ourtranslations generated by the total momentum is also manifest in the propertiesof the action under these translation transformations. In fact, it turns out thatthese translation transformations do change the action Sκ(2), but only by terms thatdo not contribute to the equations of motion (once the constraints are taken intoaccount). In order to see this explicitly let us start by noticing that we can split theintegral for the worldline p, x in (6.79) in the following way:∫ s1

s0ds(xµpµ − `x1p0p1 +NpCκ [p]

)=

∫ ∞s0

ds(xµpµ − `x1p0p1 +NpCκ [p]

)−∫ ∞s1

ds(xµpµ − `x1p0p1 +NpCκ [p]

).

(6.84)

So we can separate in the action (6.79) the contributions relative to the interactionsat s0 and s1 (contributions with boundary at s0 and contributions with boundary ats1). The part relative to the vertex s0 is the same as the action (6.63) analyzed inthe previous section. We consider then only the contributions with boundary at s1:

∆Sκ(2)s1 = −

∫ ∞s1

ds(−bµpµ − `b1q1p0

)+∫ ∞s1

ds(−bµp′µ − `b1p′′1 p′0 − `b1q1p

′0

)+∫ ∞s1

ds(−bµp′′µ − `b1q1p

′′0 − `b1p′0p′′1

)+ ∆ξµ[1]K

[1]µ (s1) .

This evidently can be rewritten as

∆Sκ(2)s1 =

∫ ∞s1ds(bµ(pµ−p′µ−p′′µ

)−`b1

(−q1p0+p′′1 p′0+q1p

′0+q1p

′′0 +p′0p′′1

))+∆ξµ[1]K

[1]µ (s1),

which, taking into account Eq. (6.78), gives

∆Sκ(2)s1 =

∫ ∞s1

ds

(d

ds

[bµK[1]

µ

]− `b1K[1]

0 q1

)+ ∆ξµ[1]K

[1]µ (s1) .

The total derivative contributes as before to the translation of ξµ[1]: ξµ[1]B = ξµ[1]A− b

µ.In addition there is a left over bulk term,∫ ∞

s1ds`b1K[1]

0 q1 ,

but it is evidently inconsequential for what concerns the equations of motion. Infact, varying this left-over term one finds∫ ∞

s1ds`b1

(δK[1]

0 q1 +K[1]0 δq1

),

i.e. ∫ ∞s1

ds`b1(δK[1]

0 q1 − K[1]0 δq1 + d

dsK[1]

0 δq1

).

And the 3 terms in this expression contribute to equations of motion and boundaryconditions only terms which are already fixed to vanish because of constraints derivedfrom other parts of the action (specifically pµ = 0, qµ = 0, p′µ = 0, p′′µ = 0 andK[1]

0 = 0).

6.4 Time of arrivals 101

6.4 Implications for the times of arrival ofsimultaneously-emitted ultrarelativistic particles

In the previous section we established the basic notions and key characterizing resultsof our proposal of a first example of prescriptions for boundary terms ensuring arelativistic description of distant observers within the relative-locality framework ofRef. [48], with a Lagrangian formulation of interacting particles.

In this section we extend the scopes of our analysis slightly beyond basics, byfocusing on a first point of phenomenological relevance, concerning observations ofdistant bursts of massless particles.

In the process we shall also show that there is an appropriate limit where ourmore powerful formalism reproduces the previous results (discussed in Ch. 4) of theHamiltonian description of free R-DSR particles.

6.4.1 Matching Lagrangian and Hamiltonian description of R-DSRfree particles

Let us indeed start this section by showing that our proposal for translation trans-formations, besides fulfilling the demands of relativistic consistency verified in theprevious section, also has the welcome property of reproducing the previous results ofthe Hamiltonian description of free R-DSR particles, Sec. 4.1. Of course this occursin an appropriate limit of our framework, since in general our framework describesinteracting κ-momentum space particles. A key observation from this perspective isthat a particle is still “essentially free” when its interactions only involve exchangesof very small fractions of its momentum.

As an illustrative example of a situation where these concepts apply and thementioned “free Hamiltonian limit” is matched, we consider the situation shown inFigure 6.5.

p, x

k, z

p′, x′

q, y p′′, x′′

q′, y′

K[0]

K[1]

Figure 6.5. Schematics of a pion decaying into a soft and a hard photon, with the hardphoton ultimately detected through an interaction in which it exchanges a small part ofits momentum with a particle in a detector (hard worldlines in solid blue, soft worldlinesin dotted red)

Notice that the situation in Figure 6.5 is also relevant for the description ofobservations of gamma-ray bursts: the incoming blue worldline p, x could be, e.g.,a highly boosted pion, which decays at the source, producing a gamma ray (p′, x′)and a very soft photon (k, z); then the gamma ray propagates freely until its firstinteraction at the detector, where it exchanges a small amount of momentum with asoft particle (q, y). So we can ask if and how the time of detection of the gamma


ray depends on its momentum p′; thereby obtaining a prediction for the largeclass of studies which is considering possible energy/time-of-arrival correlations forobservations of gamma-ray bursts (see, e.g., Refs. [2, 10, 24, 127], and Secs. 5.1, 7).

An action which is suitable for the relative-locality description of the processshown in Figure 6.5 is

Sκ(2) =∫ +∞

s0ds(zµkµ − `z1k1k0 +NkCκ [k]

)+∫ s0

−∞ds(xµpµ − `x1p1p0 +NpCκ [p]

)+∫ s1



[p′])

+∫ +∞

s1ds(x′′µp′′µ−`x′′1p′′1 p′′0 +Np′′Cκ

[p′′])

+∫ s1

−∞ds(yµqµ−`y1q1q0 +NqCκ [q]

)+∫ +∞

s1ds(y′µq′µ−`y′1q′1q′0 +Nq′Cκ

[q′])

− ξµ[0]K[0]µ (s0)− ξµ[1]K

[1]µ (s1) ,

(6.85)

where

K[0]µ (s0)=(q ⊕ p)µ−(q ⊕ p′ ⊕ k)µ=pµ−p′µ−kµ−`δ1

µ(−q0p1+q0p′1+q0k

′1+p′0k1),

K[1]µ (s1)=(q ⊕ p′ ⊕ k)µ−(p′′ ⊕ q′ ⊕ k)µ=qµ+p′µ−p′′µ−q′µ−`δ1

µ(−q0p′1−q0k1−p′0k1+p′′0q′1+p′′0k1+q′0k1).

(6.86)

And, following the procedure we already used several times, from this action oneobtains easily the equations of motion and the constraints,

pµ = 0 , qµ = 0 , kµ = 0 , p′µ = 0 , p′′µ = 0 ,Cκ[p] = 0 , Cκ[q] = 0 , Cκ[k] = 0 , Cκ[p′] = 0 , Cκ[p′′] = 0 ,

zµ −Nk

(δCκ[k]δkµ

+ `δµ0δCκ[k]δk1

k1

)= 0 , yµ −Nq

(δCκ[q]δqµ

+ `δµ0δCκ[q]δq1

q1

)= 0 ,

y′µ −Nq′(δCκ[q′]δq′µ

+ `δµ0δCκ[q′]δq′1

q′1

)= 0 , xµ −Np

(δCκ[p]δpµ


p1

)= 0 ,

x′µ−Np′(δCκ[p′]δp′µ

+ `δµ0δCκ[p′]δp′1

p′1

)=0, x′′µ−Np′′

(δCκ[p′′]δp′′µ

+ `δµ0δCκ[k]δp′′1

p′′1

)=0,

and the boundary conditions:

zµ(s0) = −ξν[0]

(δK[0]

ν

δkµ+ `δµ0

δK[0]ν

δk1k1

), xµ(s0) = ξν[0]

(δK[0]

ν

δpµ+ `δµ0

δK[0]ν

δp1p1

),

x′µ(s0) = −ξν[0]

(δK[0]

ν

δp′µ+ `δµ0

δK[0]ν

δp′1p′1

), x′µ(s1) = ξν[1]

(δK[1]

ν

δp′µ+ `δµ0

δK[1]ν

δp′1p′1

),

x′′µ(s1) = −ξν[1]

(δK[1]

ν

δp′′µ+ `δµ0

δK[1]ν

δp′′1p′′1

), yµ(s1) = ξν[1]

(δK[1]

ν

δqµ+ `δµ0

δK[1]ν

δq1q1

),

y′µ(s1) = −ξν[1]

(δK[1]

ν

δq′µ+ `δµ0

δK[1]ν

δq′1q′1

).


For the first time in this chapter we are in this section interested not only inestablishing the relativistic properties acquired through our prescription for thechoice of boundary terms, but also on the predictions of the formalism for whathappens to particles. Evidently here the issue of interest is primarily contained in thedependence of the time of detection at a given detector of simultaneously-emittedparticles on the momenta of the particles and on the specific properties of theinteractions involved in the analysis. We shall analyze this issue arranging the setupin a way that renders transparent the comparison with the Hamiltonian treatmentof free particles of Section 4.1. We start by noticing that for the particle of worldlinexµ, we have

x1(s) = x1(s) + v1(x0(s)− x0(s)) , (6.87)

which in the massless case (and whenever m/p12 |`p1| takes the simple form

x1(s) = x1(s)− p1|p1|

(x0(s)− x0(s)) . (6.88)

In obtaining (6.88) we used the on-shell relation

p0 =√p2

1 +m2 − `

2p21 ,

and the fact that for m/p12 |`p1| (consistently again with our choice of conven-

tions10, which is such that v1 > 0 =⇒ p1 < 0)

v1 = x1

x0 = −p1p0

(1− `p0 + `p21

2p0) = − p1

|p1|.

Just as in Sec. 4.1, we have momentum-independent coordinate speeds formassless particles, so in particular according to Alice’s coordinates two masslessparticles of momenta ps1 and ph1 simultaneously emitted at Alice (in Alice’s spacetimeorigin) appear to reach detector Bob simultaneously, apparently establishing acoincidence of detection events. But, as stressed already in Sec. 4.1, the presenceof relative locality evidently requires that in order to establish the dependence ofthe time of detection on the momentum of the massless particles we must againtransform the relevant worldlines to the corresponding description by an observerBob local to the detection. Let us then return to the two-interaction process ofFig. 6.5 and take as our hard massless particle of momentum ph1 the particle in thatprocess which we had originally labeled as having momentum p′1. For the process of

10Remember that we are adopting conventions such that spacetime coordinates are connectedwith those of Ch. 4 by lowering the indexes with the metric ηµν = diag = (−1, 1). It thus followsthat in this chapter the spatial momentum p1 has opposite sign respect to the velocity dx1/dx0.


Fig. 6.5 our description of the transformation from Alice’s to Bob’s worldlines is

z0B(s) = z0

A(s) + bµ(q ⊕ p′ ⊕ k)µ, z0 = z0A(s)− b0 − `b1k1 ' z0

A(s)− b0 ,z1B(s)=z1

A(s)+bµ(q ⊕ p′ ⊕ k)µ, z1=z1A(s)−b1−`(p′0+q0) ' z1

A(s)−b1−`b1p′0,x0B(s) = x0

A(s) + bµ(q ⊕ p)µ, x0 = x0A(s)− b0 − `b1p1 ,

x1B(s) = x1

A(s) + bµ(q ⊕ p)µ, x1 = x1A(s)− b1 − `q0 ' x1

A(s)− b1 ,

x′0B(s)=x′

0A(s)+bµ(q ⊕ p′ ⊕ k)µ, x′0=x′

0A(s)−b0−`b1(k1+p′1)'x′0A(s)−b0−`b1p′1,

x′1B(s)=x1

A(s)+bµ(q ⊕ p′ ⊕ k)µ, x′1=x′1A(s)−b1−`q0 ' x′1A(s)−b1,

x′′0B(s)=x′′

0A(s)+bµ(p′′ ⊕ q′ ⊕ k)µ, x′′0=x′′

0A(s)−b0−`b1(q′1+k1+p′′1)'x′′

0A(s)−b0−`b1p′′1,

x′′1B(s) = x′′

1A(s) + bµ(p′′ ⊕ q′ ⊕ k)µ, x′′1 = x′′

1A(s)− b1 ,

y0B(s)=y0

A(s)+bµ(q ⊕ p′ ⊕ k)µ, y0=y0A(s)−b0−`b1(p′1+k1+q1)'y0

A(s)−b0−`b1p′1,y1B(s) = y1

A(s) + bµ(q ⊕ p′ ⊕ k)µ, y1 = y1A(s)− b1 ,

y′0B(s)=y′

0A(s)+bµ(p′′ ⊕ q′ ⊕ k)µ, y′0=y′A

0(s)−b0−`b1(k1+q′1) ' y′A0(s)−b0,

y′1B(s) = y′

1A(s) + bµ(p′′ ⊕ q′ ⊕ k)µ, y′1 = y′A

1(s)− b1 − `b1p′′0 .(6.89)

Using these transformation laws it is easy to recognize that, having dropped thenegligible “soft terms” from small momenta, indeed we are obtaining results thatare fully consistent with the ones obtained in the Hamiltonian description of freeparticles. To see this explicitly let us consider the situation where, simultaneouslyto the interaction emitting the hard particle x′, p′ in Alice origin, we also have theemission of a soft photon xs, ps.And as observer Bob let us take one who is reached in its spacetime origin by thesoft photon emitted by Alice. For the event of detection of the hard particle x′, p′we take one such that it occurs in Bob’s spatial origin.From a relative-locality perspective the setup we are arranging is such that “Aliceis local to an emitter” (the spatial origin of Alice’s coordinate system is local toan ideally compact, infinitely small, emitter) and “Bob is local to a detector” (thespatial origin of Bob’s coordinate system is local to an ideally compact, infinitelysmall, detector). The two worldlines we focus on, a soft and a hard worldline, bothoriginate from Alice’s spacetime origin (they are both emitted by Alice, in the spatialorigin of Alice’s frame of reference, and both at time tAlice = 0) and both end upbeing detected by Bob, but, while by construction the soft particle reaches Bob’sspacetime origin, the time at which the hard particle reaches Bob spatial origin is tobe determined through our analysis.

Reasoning as usual at first order in `, it is easy to verify that Bob describesthe “interaction coordinate” ξ[1]

B

µof the interaction at s = s1 as coincident with the

s = s1 endpoints of the worldlines x′, p′; x′′, p′′; q, y; q′, y′:

ξ[1]B

µ= x′B

µ(s1) = x′′Bµ(s1) = yµB(s1) = y′B

µ(s1) . (6.90)

We take into account that there are no relative-locality effects in the description givenby Bob whenever an interaction occurs “in the vicinity of Bob”: our leading-orderanalysis assumes the observatories have sensitivity sufficient to expose manifestation


of relativity of locality of order `phL (where L is the distance from the interaction-event to the origin of the observer and ph is a “suitably high” momentum), with Lset in this case by the distance Alice-Bob, so even a hard-particle interaction whichis at a distance d from the origin of Bob will be treated as absolutely local by Bob ifd L.According to this both “detection events” are absolutely local for observer Bob: ofcourse this is true for the event of detection of the soft photon xs, ps (which wedid not even specify since its softness ensures us of its absolute locality) and it isalso true for the interaction-event of “detection near Bob” of the hard particle x′, p′.Ultimately this allows us to handle the time component of the coordinate four-vector(6.90) as the actual delay that Bob measures between the two detection times:

∆t = ξ[1]B

0= x′B

0(s1) = x′′B0(s1) = y0

B(s1) = y′B0(s1) . (6.91)

From the equations (6.89) relative to the worldline x′, p′, it follows that

x′A1(s1) = x′B

1(s1) + b1 = b1 , (6.92)

from which, considering the worldlines (6.88), it follows that (assuming indeedm/(p′1)2 |`p′1|) Alice “sees” the s = s1 endpoint of the worldline x′, p′ at thecoordinates

x′µA(s1) = x′

µB(s1) + bµ = bµ = (b, b) . (6.93)

And then, from the equations (6.89) and (6.91), it follows that Bob measures thedelay

∆t = x′B0(s1) = x′

0A(s1)− b0 − `b1p′1 = `b|p′1| , (6.94)

in agreement with the result (4.34) found in the Hamiltonian description. Thesefindings are summarized in Figure 6.6.

Alice

Bob

p, x

p′, x′

p′′, x′′

q, y

q′, y′

k, z

ξµ[1]A = (b+ ℓb|p′1|, b)

ξµ[0]A = (0, 0)

x0A

x1A

ℓb|p′1|

Alice

Bob

p, x

p′, x′

p′′, x′′

q, y

q′, y′

k, z

ξµ[0]B = (−b,−b)

ξµ[1]B = (ℓb|p′1|, 0)

ℓb|p′1|

referencephoton

x1B

x0B

Figure 6.6. Schematic description of the time delay derived in this subsection. The variousagents in the analysis are shown both as described by Alice (left panel) and as describedby Bob (right panel). These are spacetime graphs (in a 2D spacetime) showing theactual worldlines of particles. In addition to the two hard interactions we are considering(qualitatively described already in Figure 6.5), we also show (as the orange-dottedworldline) a soft photon going from Alice’s origin to Bob’s origin. As shown in the figurewe have arranged the calculations in this section so that all emissions and detectionsoccur in the spatial origin of either Alice or Bob (but because of the relative localityaccording to Alice the hard detections at/near Bob would be nonlocal interactions andaccording to Bob the hard emissions at Alice would be nonlocal processes). We alsoshow, as the bulky green dots, the formal positions of the interaction points, as coded inthe formal “interaction coordinates” ξµ.


Since we are dealing with a momentum-space with torsion, i.e. the momentumcomposition law is noncommutative, it is interesting to check whether this resultestablishing agreement with the Hamiltonian description of free particles also holdsfor other choices of ordering of momenta in the conservation laws (and accordinglyin the boundary conditions).

An interesting alternative for the conservation laws and boundary conditions ofthe process in Figure 6.5 is the following

K[0]µ (s0) = (p⊕ q)µ−(k ⊕ p′ ⊕ q)µ = pµ−p′µ−kµ−`δ1

µ(−p0q1 + k0p′1 + k0q1 + p′0q1),

K[1]µ (s1) = (k ⊕ p′ ⊕ q)µ − (k ⊕ p′′ ⊕ q′)µ

=p′µ+qµ−p′′µ−q′µ−`δ1µ(−k0p

′1−k0q1−p′0q1+k0p

′′1 +k0q

′1+p′′0q′1).

(6.95)

Going from the previous version of the boundary conditions to this one does changeseveral things in the analysis, but it easy to see that it does not change anything aboutthe “free particle” x′, p′. With these conservation laws and boundary conditions therelationships between Alice’s worldline x′ and Bob’s worldline x′ are codified in

x′0B(s)=x′

0A(s)+bµ(k ⊕ p′ ⊕ q)µ, x′0=x′

0A(s)−b0−`b1(q1+p′1)'x′0A(s)−b0−`b1p′1,

x′1B(s) = x1

A(s) + bµ(k ⊕ p′ ⊕ q)µ, x′1 = x1A(s)− b1 − `b1k0 ' x′1A(s)− b1 .

(6.96)

And using the equation of motion (6.88) one easily checks that then the relevantparticle reaches Bob’s spatial origin at the time

∆t = x′B0(s1) = x′

0A(s1)− b0 − `b1p′1 = `b|p′1| , (6.97)

in perfect agreement with the result of Eq. (6.94), which had been obtained withthe other choice of ordering of momenta in the conservation laws.

So we find evidence of the fact that the properties of “free particles” (particlesexchanging only small fractions of their momentum) are insensitive to the orderingchosen for the law of composition of momenta.

And for what concerns bursts of simultaneously emitted massless particles, suchas in a gamma-ray-burst, this derivation predicts differences in times of arrivalgoverned by the formula

∆tarrival = `L|∆p1| ,

where L is the distance from source to detector (the corresponding translation fromobserver at the source to observer at the detector has bµ = (L,L)) and |∆p1| is thedifference of momentum among the two massless particles whose arrival times differby ∆tarrival.

The derivation in this subsection establishes this result for cases where theinteraction at the source emitting the particle of interest only involves one hardparticle in the in state and one hard particle in the out state (all other particlesinvolved in the interactions being soft).

6.4.2 More on observations of distant bursts of massless particles

In the previous subsection, in showing that our proposal (in an appropriate limit)matches the predictions of previous Hamiltonian descriptions of relative locality for


free particles, we also showed that, at least for certain types of emission and detectioninteractions, our proposal predicts time-of-detection delays ∆tarrival = `L|∆p1|between simultaneously-emitted massless particles with momentum difference |∆p1|.This is very interesting because, as established in several studies reported over thelast decade, such an effect is testable (see Sec. 5.1), even if ` is as small as thePlanck length11 (or even one or two orders of magnitude smaller than the Plancklength [2, 127, 24, 10]).

We derived this time-delay result assuming certain types of emission and detectioninteractions. But evidently the structure of our formalism is such that it would notbe surprising to find that the times of detection depended on the actual emissionand detection interactions involved. In this subsection we intend to establish thatthis is indeed the case, and that the torsion of momentum space plays a crucial rolein the relevant analysis.

It suffices to modify the analysis of the previous subsection in rather minorway for us to show that the times of detection of simultaneously emitted particlesdepend not only on the momenta of the particles but also on the actual nature ofthe emitting interaction. We find that in order for this to occur there must be atleast 3 hard particles in total, among in and out particles of the emission interaction.As an example of this we consider here explicitly the case of a ultraenergetic particleat rest decaying into two particles, both hard, one of which is the particle detectedat our observatory.

As shown in figure we arrange the analysis in exactly the same way as in theprevious section, with a tri-valent vertex for the emission interaction and a four-valent vertex for the detection. And the kinematics at the four-valent vertex is leftunchanged, involving a soft particle in the in state and a soft particle in the outstate. We only change the kinematics of the emission vertex, now assuming that allparticles involved are hard.

p, x

k, z

p′, x′

q, y p′′, x′′

q′, y′

K[0]

K[1]

Figure 6.7. Schematic description of a case where a hard ultrarelativistic particle originatingfrom a hard emission interaction (one hard particle in, two hard particles out) is detectedin a soft interaction (only one hard particle in and only one hard particle out). Solid-bluelines are for hard particles, dashed-red lines are for soft particles.

And we shall again consider two possible choices of conservation-enforcing bound-ary conditions, suitable for exploring the role of the noncommutativity of the law ofcomposition of momenta. The same two possible choices of conservation-enforcingboundary conditions already considered in the previous subsection.

11We introduced ` as a momentum-space property, with dimensions of inverse momentum. Whenwe mention the possibility of ` of order the Planck length we are essentially using jargon, a compactway to describe cases where `−1 is of order the Planck scale.


Let us start again by analyzing as first possibility

K[0]µ (s0)=(q ⊕ p)µ−(q ⊕ p′ ⊕ k)µ=pµ−p′µ−kµ−`δ1

µ(−q0p1+q0p′1+q0k

′1+p′0k1),

K[1]µ (s1) = (q ⊕ p′ ⊕ k)µ − (p′′ ⊕ q′ ⊕ k)µ

=qµ+p′µ− p′′µ−q′µ−`δ1µ(−q0p

′1−q0k1−p′0k1+p′′0q′1+p′′0k1+q′0k1).

(6.98)

The worldlines seen by observer/detector Bob, distant from the emission, that followfrom this choice of boundary terms have been already given in Eq. (6.88). Themain difference between the situation in the previous subsection and the situationwe are now analyzing is that the “primary”, the particle incoming to the emissioninteraction, is at rest, with p1 = 0, which also implies that the two outgoing particlesof the emission interaction must both be hard. For the worldlines involved in theemission interaction this leads to

x0B(s) = x0

A(s) + bµ(q ⊕ p)µ, x0 = x0A(s)− b0 − `b1p1 = x0

A(s)− b0 ,x1B(s) = x1

A(s) + bµ(q ⊕ p)µ, x1 = x1A(s)− b1 − `q0 ' x1

A(s)− b1 ,

x′0B(s) = x′

0A(s) + bµ(q ⊕ p′ ⊕ k)µ, x′0 = x′

0A(s)− b0 − `b1(k1 + p′1) = x′

0A(s)− b0,

x′1B(s) = x1

A(s) + bµ(q ⊕ p′ ⊕ k)µ, x′1 = x′1A(s)− b1 − `q0 ' x′1A(s)− b1 ,

z0B(s) = z0

A(s) + bµ(q ⊕ p′ ⊕ k)µ, z0 = z0A(s)− b0 − `b1k1 ,

z1B(s)=z1

A(s)+bµ(q ⊕ p′ ⊕ k)µ, z1=z1A(s)−b1−`(p′0+q0) ' z1

A(s)−b1−`b1p′0 .(6.99)

And from this one easily sees that the particle p′, x′, the particle then detectedat Bob, translates classically, without any deformation term. So this time we havethat no momentum dependence of the times of arrivalis predicted

tdetection = x′0B(s1) = x′0A(s1)− b0 = 0 .

Next we show that in this case with the emission interaction involving only hardparticles the noncommutativity of the composition law, which had turned out to beuninfluential in the previous subsection, does play a highly non-trivial role.To see this let us consider, as in the previous subsection, the following alternativechoice of K’s for the boundary terms

K[0]µ = (p⊕ q)µ − (k ⊕ p′ ⊕ q)µ = pµ − p′µ − kµ − `δ1

µ(−p0q1 + k0p′1 + k0q1 + p′0q1) ,

K[1]µ = (k ⊕ p′ ⊕ q)µ − (k ⊕ p′′ ⊕ q′)µ

=p′µ+qµ−p′′µ−q′µ−`δ1µ(−k0p

′1−k0q1−p′0q1+k0p

′′1 +k0q

′1+p′′0q′1).

(6.100)

Focusing again on the worldline x′, p′ detected at Bob we now find

x′0B(s)=x′

0A(s)+bµ(k ⊕ p′ ⊕ q)µ, x′0=x′

0A(s)−b0−`b1(q1+p′1)'x′0A(s)−b0−`b1p′1,

x′1B(s) = x1

A(s) + bµ(k ⊕ p′ ⊕ q)µ, x′1 = x′1A(s)− b1 − `b1k0 .

(6.101)

And from the equation of motion (6.88) one now deduces that

x′1B(s) = x′

0B(s)− `b1(k0 − p′1) ,


which in turn implies that the time of detection at Bob of the particle with worldlinex′, p′ is

tdetection = x′B0(s1) = −`b1(p′1 − k0) = 2`b1|p′1| . (6.102)

The dependence of the time of detection on the momentum of the particle beingdetected is back! And this dependence is twice as strong as the dependence onmomentum found in the previous subsection!

6.4.3 Nonmetricity, torsion and time delays

The results we obtained in this subsection are rather striking and deserve to besummarized and discussed in relation with previous related results.

For what concerns times of detection of simultaneously emitted massless particlesof momentum p′1, emitted from a source at a distance L from the detector weanalyzed 3 situations:(case A) the emission interaction involves only one hard incoming particle and onehard outgoing particle, all other particles in the emission interaction being soft:the times of arrival have a dependence on momentum governed by

tdetection = `L|p′1|

and this result is independent of the position occupied by the momentum p′µ in ournoncommutative composition law(case B) the emission interaction is the decay of a ultra-high-energy particle atrest, involves a total of 3 hard particles, and the momentum p′µ appears in thecomposition of momenta to the left of a hard particle:the times of arrival have no dependence on momentum

tdetection = 0

(case C) the emission interaction is the decay of a ultra-high-energy particle atrest, involves a total of 3 hard particles, and the momentum p′µ appears in thecomposition of momenta to the right of a hard particle:the times of arrival have the following dependence on momentum

tdetection = 2`L|p′1|

(twice as large as in the case A).We obtained this results working with our κ-Poincaré-inspired momentum space,

with nonmetricity and torsion.The fact that it would be possible with such a momentum space to have thatsimultaneously emitted massless particles are not detected simultaneously at the samedetector could be expected on the basis of the analysis reported in Ref. [124], whosemain message was that indeed nonmetricity should result in the possibility of havingsimultaneously emitted massless particles that are not detected simultaneously.The nature and scopes of our study were such that we could for the first timeinvestigate how the presence of both torsion and nonmetricity could affect thesetime-of-detection delays. And we evidently found that torsion can have strikingeffects, effects capable of changing the predicted new effect at order 1, and therefore


effects that are as much within reach of ongoing and forthcoming experiments asthe pure-nonmetricity (no torsion) effects.

It should be stressed that what we found might even underestimate the signif-icance of the effects of torsion on time delays (at least the effects on time delaysof torsion, when also nonmetricity is present). This is because we contemplated atotal of only 3 cases for what concerns the kinematics and the conservation lawsthat are relevant for such an analysis. But even within the confines of our prelimi-nary investigation we found a type of dependence of the time delays, not only onmomenta of observed particles but also on interactions that produced them, whichhad never been encountered before in the literature and would therefore provide avery distinguishing feature of the model of momentum space we here adopted asillustrative example.

6.5 A consistency check for translation invariance: thecase of 3 connected finite worldlines

We have established in Sec. 6.3, that at least in the simplest applications our pre-scriptions do provide the desired relativistic picture. In order to motivate our nextconsistency check it is useful to look at available results on relative locality from thefollowing perspective:? with the Hamiltonian description of relative locality for free particles given inCh. 4 (Refs. [11, 38]), one essentially obtains a characterization of relative localitylimited to infinite worldlines;? with the Lagrangian description of relative locality for interacting particles pro-posed in Ref. [48] the availability of a relativistic description of distant observershad been checked explicitly only for semi-infinite worldlines (a single interaction);? the results reported so far in Sec. 6.3 generalize the results for distant observersof Ref. [48] to the case where one of the worldlines is finite (a worldline exchangedbetween two interactions, establishing the causal relation between the two interac-tions).

In this section we provide evidence of the fact that our prescription is robust alsofor cases with several finite worldlines. We actually consider here a case which isvery meaningful from this perspective: the case shown in Figure 6.8, which includesa vertex where 3 finite worldlines meet.

k, z

k′, z′

k′′, z′′

p, x

q, y

p′, x′

p′′, x′′

q′, y′

q′′, y′′K[0]

K[1]

K[2]

K[3]

Figure 6.8. The case with 3 connected finite worldlines which we consider in this subsection.

Following the prescription we are advocating the situation in Figure 6.8 requires

6.5 3 connected finite worldlines 111

handling boundary terms with

K[0] = k − k′ ⊕ k′′ ,K[1] = k′ ⊕ k′′ − p⊕ q ⊕ k′′ ,K[2] = p⊕ q ⊕ k′′ − p′ ⊕ p′′ ⊕ q ⊕ k′′ ,K[3] = p′ ⊕ p′′ ⊕ q ⊕ k′′ − p′ ⊕ p′′ ⊕ q′ ⊕ q′′ ⊕ k′′ .

(6.103)

We therefore describe the chain of interactions in Figure 6.8 through the followingaction:

S(3conn) =∫ s0

−∞ds(zµkµ−`z1k1k0+NkC [k]

)+∫ +∞

s0ds(z′′µk′′µ−`z′′1k′′1 k′′0 +Nk′′C [k]

)+∫ s1

s0ds(z′µk′µ − `z′1k′1k′0 +Nk′C [k]

)+∫ s2

s1ds(xµpµ − `x1p1p0 +NpC [p]

)+∫ +∞


1p′1p′0+Np′C

[p′])

+∫ +∞

s2ds(x′′µp′′µ−`x′′1p′′1 p′′0 +Np′′C

[p′′])

+∫ s3

s1ds(yµqµ − `y1q1q0 +NqC [q]

)+∫ +∞

s3ds(y′µq′µ − `y′1q′1q′0 +Nq′C

[q′])

+∫ +∞

s3ds(y′′µq′′µ − `y′′1q′′1 q′′0 +Nq′′C

[q′′])

− ξµ[0]K[0]µ (s0)− ξµ[1]K

[1]µ (s1)− ξµ[2]K

[2]µ (s2)− ξµ[3]K

[3]µ (s3) .

(6.104)

We skip reporting the equations of motion and boundary conditions, which ,keeping in mind the ones in the previous section, are intuitive. And we just report


our notion of translation transformation to a distant observer, which is such that

x0B = x0

A + bµ(p⊕ q ⊕ k′′)µ, x0 = x0A(s)− b0 − `b1(p1 + q1 + k′′1) ,

x1B = x1

A + bµ(p⊕ q ⊕ k′′)µ, x1 = x′1A(s)− b1 ,x′0B = x′0A + bµ(p′ ⊕ p′′ ⊕ q ⊕ k′′)µ, x′0 = x′0A(s)− b0 − `b1(p′1 + p′′1 + q1 + k′′1) ,x′1B = x′1A + bµ(p′ ⊕ p′′ ⊕ q ⊕ k′′)µ, x′1 = x′1A(s)− b1 ,x′′0B = x′′0A + bµ(p′ ⊕ p′′ ⊕ q ⊕ k′′)µ, x′′0 = x′′0A (s)− b0 − `b1(p′′1 + q1 + k′′1) ,x′′1B = x′′1A + bµ(p′ ⊕ p′′ ⊕ q ⊕ k′′)µ, x′′1 = x′′1A (s)− b1 − `b1p′0 ,y0B = y0

A + bµ(p⊕ q ⊕ k′′)µ, y0 = y0A(s)− b0 − `b1(q1 + k′′1) ,

y1B = y1

A + bµ(p⊕ q ⊕ k′′)µ, y1 = y1A(s)− b1 − `b1p0 ,

y′0B = y′0A + bµ(p′ ⊕ p′′ ⊕ q′ ⊕ q′′ ⊕ k′′)µ, y′0 = y′0A(s)− b0 − `b1(q′1 + q′′1 + k′′1) ,y′1B = y′1A + bµ(p′ ⊕ p′′ ⊕ q′ ⊕ q′′ ⊕ k′′)µ, y′1 = y′1A(s)− b1 − `b1(p′0 + p′′0) ,y′′0B = y′′0A + bµ(p′ ⊕ p′′ ⊕ q′ ⊕ q′′ ⊕ k′′)µ, y′′0 = y′′0A (s)− b0 − `b1(q′′1 + k′′1) ,y′′1B = y′′1A + bµ(p′ ⊕ p′′ ⊕ q′ ⊕ q′′ ⊕ k′′)µ, y′′1 = y′′1A (s)− b1 − `b1(p′0 + p′′0 + q′0) ,z0B = z0

A + bµkµ, z0 = z0A(s)− b0 − `b1k1 ,

z1B = z1

A + bµkµ, z1 = z1A(s)− b1 ,

z′0B = z′0A + bµ(k′ ⊕ k′′)µ, z′0 = z′0A(s)− b0 − `b1(k′1 + k′′1) ,z′1B = z′1A + bµ(k′ ⊕ k′′)µ, z′1 = z′1A(s)− b1 ,z′′0B = z′′0A + bµ(k′ ⊕ k′′)µ, z′′0 = z′′0A (s)− b0 − `b1k′′1 ,z′′1B = z′′0A + bµ(k′ ⊕ k′′)µ, z′′0 = z′′1A (s)− b1 − `b1k′0 .

(6.105)

It is then easy to check that also in this case the equations of motion andboundary conditions are left unchanged by our notion of translation transformationto a distant observer. This is also verifiable by studying the implications of ourtranslation transformations for the action S(3conn) (see our work [39]).

113

Chapter 7

DSR in expanding spacetime

This chapter is devoted to the formulation of a relativistic theory of worldlines ofparticles with three observer-invariant scales: a large speed scale (“speed-of-lightscale”), a large distance scale (inverse of the “expansion-rate scale’), and a largemomentum scale (“Planck scale”).

As we have already mentioned, an improvement of the DSR formalism providingthe inclusion of the effects of curvature/expansion, represents a significant contribu-tion both from a conceptual point of view, and from a phenomenological perspective.Indeed, as we discussed in Ch. 5, some of the most compelling opportunities fortesting Planck scale effects of departure from standard special relativity, rely onan amplification of these effects provided by the distance traveled by the particlesinvolved in the observations. We mentioned in particular in Sec. 5.1 how the times-of-flight analysis for photons produced by gamma ray bursts can be used to testPlanck scale deformed dispersion relations. It is clear that in the case of particlespropagating from cosmological distances, the effect of the curvature/expansion ofthe universe cannot be neglected.

We first review, in Sec. 7.1, the description of the motion of a particle (incovariant Hamiltonian formalism) in “De Sitter relativity”. De Sitter relativitycan be thought of as a deformation of special relativity by the introduction of the(constant) expansion rate H as an observer-invariant scale. De Sitter spacetimeis indeed a particular case of the Friedman-Robertson-Walker (FRW) solution ofEinstein equations, in which the time dependence of the scale factor is given by theequation for the expansion rate (in comoving time)

a (t)a (t) = H , (7.1)

with H constant.It happens that the constancy of the expansion rate allows to define a class

of inertial observers characterized by the whole set of (H-deformed) spacetimesymmetries (translations, rotations and boosts). This is not the case for the generalFRW expanding spacetime, in which the time dependence of H breaks the invarianceunder time translations.

We discuss, in Sec. 7.1, a covariant Hamiltonian formulation of the motion of freeparticles in De Sitter, both in comoving-time t and in conformal-time coordinates η,

114 7. DSR in expanding spacetime

whose difference will be clarified in the following section. So that our notation willswitch from x0 to x0 → t or x0 → η depending on the choice of time coordinate.

The presence of the whole set of spacetime symmetries allow us to formulate aDSR deformation of De Sitter relativity starting, as done for the flat DSR case ofCh. 4, from the De Sitter algebra of symmetries. We thus show, in Sec. 7.3, how itis possible to combine the constant rate of expansion H and the inverse-momentumdeformation scale ` (and of course the implicit velocity scale c) as relativisticinvariants in a theory of particle worldlines. A crucial role for the construction ofour model and for its logical consistence is played by the results on relative localityexposed in the previous parts of this thesis.

We close Sec. 7.3 with a discussion of some phenomenological consequences relatedwith the observation of times-of-flight for photons produced by distant astronomicalsources. We point out also how one can get, already within our “DSR-DeSitter”formulation, a preliminary comparison with the estimates for the time delays derivedin the LSB scenarios.

We work at leading order in the DSR-deformation scale `, since (assumingit is of the order of the Planck length) that is the only realistic target for DSRphenomenology over the next few decades. And in order to keep things simple,without renouncing to any of the most significant conceptual hurdles, we opt to workin a 2D spacetime (one time and one spatial dimension), unless otherwise specified.

The results of Sec. 7.3 are reported in our work [44].

7.1 Covariant formulation of free particles motion inDe Sitter spacetime

We here review, in the language outlined in Ch. 2 and throughout this thesis, acovariant formulation of the motion of a free particle in De Sitter spacetime. I.e. wediscuss the motion of a particle in a relativistic theory with two observer-invariantscales, a velocity scale c, and a distance scale H−1.

The 3+1D De Sitter spacetime can be described in terms of the metric defined,in comoving coordinates, by the spacetime interval

ds2 = dt2 − a2 (t) d~x2 , (7.2)

where the scale factor a (t) is given by

a (t) = eHt , (7.3)

with H the constant expansion rate. So that the metric tensor is

gµν = diag(1,−e2Ht,−e2Ht,−e2Ht

). (7.4)

One can show that the algebra of De Sitter spacetime symmetry generators, i.e.the generators of the group of isometries of the metric (7.4), can be described bythe Poisson brackets

7.1 Covariant formulation of free particles motion in De Sitter spacetime 115

p0, pj = Hpj , Nj , p0 = pj +HNj , Rj , p0 = 0 ,pj , pk = 0 , Nj , pk = δjkp0 +HεjklRl , Rj , pk = εjklpl ,

Nj , Nk = −εjklRl , Rj , Rk = εjklRl , Rj , Nk = εjklNl . (7.5)

While the Casimir isC = p2

0 − ~p2 − 2H~p · ~N −H2 ~R2 . (7.6)

In comoving-time coordinates t, ~x, the phase space is defined by the relations

p0, t = 1 , p0, xj = −Hxj ,pj , t = 0 , pj , xk = −δjk , (7.7)

together withp0, pj = Hpj , t, xj = 0 , (7.8)

and the representation of rotation and boost generators are given by

Rj = εjklxkpl , (7.9)

Nj = xjp0 −1− e−2Ht

2H pj −12H~x

2pj . (7.10)

It’s easy to see that in the limit H → 0, we recover the special relativity set-updiscussed in Sec. 2.3. We can understand De Sitter relativity as an H-deformationof special relativity.

7.1.1 Worldlines in comoving-time coordinates

We restrict to the 1+1D dimensional case, so that the above relations simplify, forcomoving-time coordinates to

C = p20 − p2

1 − 2Hp1N . (7.11)

N, p0 = p1 +HN , N, p1 = p0 , (7.12)

p0, p1 = Hp1 , t, x = 0 ,p0, t = 1 , p0, x = −Hx ,p1, t = 0 , p1, x = −1 , (7.13)

N = xp0 −1− e−2Ht

2H p1 −12Hx

2p1 . (7.14)

We set the Hamiltonian constraint to be H = C −m2. We notice that in thecomoving-time representation, the on-shell relation H = 0, using (7.14) gives

H =p20 − p2

1 − 2Hp1

(xp0 −

1− e−2Ht

2H p1 −12Hx

2p1

)−m2

=p20 − e−2Htp2

1 − 2Hxp0p1 +H2x2p21 −m2 = 0 ,

(7.15)


from which, restricting to positive p0, it follows the dispersion relation

p0 =√e−2Htp2

1 +m2 +Hxp1 . (7.16)

The Hamiltonian constraint H = C −m2, determines the equations of motion(where X = dX

dτ )

t = H, t = 2 (p0 −Hxp1) ,

x = H, x = 2e−2Htp1 . (7.17)

We use the dispersion relation (7.14) to eliminate p0 in the first one of (7.17), whichbecomes

t = 2√e−2Htp2

1 +m2 . (7.18)

We define the velocity to be v = v (t) = dx(t)dt , so that, by the chain rule,

v (t) = dx (t)dt

= x

t= e−Htp1√

e−2Htp21 +m2

.

Integrating the last relations in t, considering also that dp1/dt = 0, we find theworldlines

xm,p (t) = x(t)

+

√m2 + e−2Htp2

1 −√m2 + e−2Htp2

1

Hp1. (7.19)

For massless particles the worldlines become

xm=0,p (t) = x+ p1|p1|

e−Ht − e−Ht

H. (7.20)

7.1.2 Finite translations in comoving-time coordinates

Consider an observer Alice, which describes a photon passing through her origin,and traveling in the positive direction of the x-axis (so that p1 > 0). In comovingcoordinates, according to (7.20) , Alice describes the worldlines

xA(tA)

= 1H

(1− e−HtA

). (7.21)

We want to consider now an observer Bob standing at a point along the photonworldline, represented by the coordinates

(tABob, x

ABob

)in Alice’s frame. Notice that,

for Bob’s origin to be along the photon worldline, the relation between tABob andxABob must be

xABob = 1H

(1− e−HtABob

). (7.22)

We can define the action of a finite time translation and of a finite spacetranslation on the phase space variables k = (p0, p1, t, x), respectively as (see Ch. 2)

Ta0 . kA = e−a0p0 . kA = kA − a0 p0, k|A + a2

0 p0, p0, k|A + · · · ,Ta1 . k

A = e−a1p1 . kA = kA − a1 p1, k|A + a21 p1, p1, k|A + · · · . (7.23)


From the phase space relations (7.13), we find

Ta0 . tA = tA − a0 , Ta0 . x

A = eHa0xA , Ta0 . pA1 = e−Ha0pA1 ,

Ta1 . tA = tA , Ta1 . x

A = xA − a1 , Ta1 . pA1 = pA1 . (7.24)

Combining space and time translations as shown in Sec. (2.2) (see in particular Eq.(2.50)), we find

Ta0,a1 . kA = (Ta0 Ta1) . kA = Ta1 .

(Ta0 . k

A), (7.25)

which means that Ta0,a1 is a map between Alice coordinates and the coordinates ofan observer reached by first translating spatially by a1, and then translating in timeby a0. We then find

Ta0,a1 . tA = tA − a0 ,

Ta0,a1 . xA = eHa0

(xA − a1

),

Ta0,a1 . pA1 = eHa0pA1 . (7.26)

One can verify that the worldlines (7.19) are covariant under the finite translations(7.26).

One can easily see, by substituting tA = a0 and xA = a1 in (7.26), that theframe’s origin of the observer connected by the translation Ta0,a1 , is in the point,in Alice’s frame, of coordinates (a0, a1). This means, recalling Eq. (7.22), that theclass of translated observers whose origin intersects the photon worldline (the onepassing through Alice’s origin), is defined by the equation, analogous to (7.22),

a1 = 1H

(1− e−a0

). (7.27)

It is interesting to notice that De Sitter relativity contains in its formalisma coordinate artifact of a similar nature of that of relative locality. Indeed, wenotice that Alice, in her coordinates (if she didn’t know the machinery of De Sittertranslations), would infer that Bob, who is along the photon worldline, would seethe photon moving slower close to him, as in Fig. 7.2.

This would be if Alice used ordinary special relativistic translations (in particularwith p0, x = 0), under which the worldlines (7.19) are not covariant. But since thetheory is relativistic, the correct translations are given by (7.26), under which theworldlines (7.19) are covariant. Then Bob, in his coordinates, describes the photonsmoving locally at the same velocity Alice, in her coordinates, attributes to photonsnear her origin, as in Fig. 7.4. This also connected with the fact that every De Sitterframe is locally Minkowskian, which would be in contradiction with Alice’s wronginference for which the photons travel slower than light for distant observers.

7.1.3 Worldlines in conformal-time coordinates

Conformal time η is defined by the requirement

dt = a (η) dη , (7.28)


tA

xA

Bob

Figure 7.1. Alice’s description

tB

xB

Alice

Figure 7.2. Alice’s inference of Bob’sdescription

the

tA

xA

Bob

Figure 7.3. Alice’s description

tB

xB

Alice

Figure 7.4. Bob’s description

where a (η) = a (t). Then

η (t)− η (0) =∫ t

0dta−1 (t) =

∫ t

0dte−Ht = 1− e−Ht

H(7.29)

setting η (0) = 0 we find

η (t) = 1− e−Ht

H(7.30)

anda (η) = 1

1−Hη , (7.31)

from which follows that a (0) = 1 and da (η) /dη = Ha2.The De Sitter metric in conformal-time coordinates is “conformal”:

ds2 = 1(1−Hη)2

(dη2 − dx2

).

The only element of the 1+1D Poisson structure that changes respect to comoving-time is

p0, η = 1−Hη . (7.32)


We rewrite for clarity the whole structure

C = p20 − p2

1 − 2Hp1N . (7.33)

N, p0 = p1 +HN , N, p1 = p0 , (7.34)

p0, p1 = Hp1 , t, x = 0 ,p0, η = 1−Hη , p0, x = −Hx ,p1, η = 0 , p1, x = −1 . (7.35)

The boost representation changes too as

N = xp0 −1− (1−Hη)2

2H p1 −12Hx

2p1 . (7.36)

We set the Hamiltonian constraint to be H = C −m2. We notice that in theconformal-time representation, the on-shell relation H = 0, using (7.36) gives

H =p20 − p2

1 − 2Hp1

(xp0 −

1− (1−Hη)2

2H p1 −12Hx

2p1

)−m2

=p20 − p2

1 + 2Hηp21 − 2Hxp1p0 −H2η2p2

1 +H2x2p21 −m2 = 0 ,

(7.37)

from which, restricting to positive p0, it follows the dispersion relation

p0 = Hp1x+√m2 + p2

1 − 2Hηp21 +H2η2p2

1 . (7.38)

The Hamiltonian constraint H = C −m2, determines the equations of motion(where k = dk

dτ )

η = H, η = 2 (1−Hη) (p0 −Hxp1) ,

x = H, x = 2 (1−Hη)2 p1 . (7.39)

We use the dispersion relation (7.36) to eliminate p0 in the first one of (7.39), whichbecomes

η = 2 (1−Hη)√m2 + (1−Hη)2 p2

1 . (7.40)

We define the velocity to be v = v (η) = dx(η)dη , so that, by the chain rule,

v (η) = dx (η)dt

= x

η= (1−Hη) p1√

m2 + (1−Hη)2 p21

.

Integrating the last relations in η, considering also that dp1/dη = 0, we find theworldlines

xm,p (η) = x+

√m2 + (1−Hη)2 p2

1 −√m2 + (1−Hη)2 p2

1Hp1

. (7.41)


xm=0,p (t) = x+ p1|p1|

(η − η) . (7.42)

We see that in conformal-time coordinates photon’s worldlines are the same asin special relativity.


7.1.4 Finite translations in conformal-time coordinates

We repeat the same analysis made for comoving-time coordinates. We considera first observer, Alice, who describes a photon passing through her origin. Inconformal-time coordinates, in according to (7.42), Alice describes the worldlines

xA(ηA)

= ηA . (7.43)

We consider a second observer Bob standing at a point along the photon worldline,represented, in Alice’s frame, by the coordinates

(ηABob, x

ABob

). Notice that, for Bob’s

origin to be along the photon worldline, the relation between ηABob and xABob mustbe, from (7.42),

ηABob = xABob . (7.44)

Referring again to Eqs. (7.23) for the finite translations, from the phase spacerelations (7.35), we find

Ta0 . ηA = eHa0ηA + 1− eHa0

H, Ta0 . x

A = eHa0xA , Ta0 . pA1 = e−Ha0pA1 ,

Ta1 . ηA = ηA , Ta1 . x

A = xA − a1 , Ta1 . pA1 = pA1 . (7.45)

Using again Eq. (7.25), we find, for the composed translation (space translationfollowed by time translation) Ta0,a1 ,

Ta0,a1 . ηA = eHa0ηA + 1− eHa0

H,

Ta0,a1 . xA = eHa0

(xA − a1

),

Ta0,a1 . pA1 = eHa0pA1 . (7.46)


From Eqs. (7.46), we can see that if we want to reach the point which in Alice’sframe has coordinates

(ηA, xA

), a0 and a1 must satisfy the relation

eHa0 ηA + 1− eHa0

H= 0 , a1 = xA . (7.47)

from which follows

a0 = − 1H

ln(1−HηA

), a1 = xA . (7.48)

Then, from Eq. (7.44), the class of translated observers whose origin is along thephoton worldline (the worldline of the photon passing through Alice’s origin), isdefined by the relation

a1 = xA = ηA = 1− e−Ha0

H, (7.49)

which coincides, as it must be, with Eq. (7.27).


the

ΗA

xA

Bob

Figure 7.5. Alice

ΗB

xB

Alice

Figure 7.6. Bob

7.1.5 Redshift

We close this section by reviewing briefly the definition of the redshift due tospacetime expansion in this formalism. It is defined as

z = anowathen

− 1 (7.50)

so that, in comoving coordinates, where a (t) = eHt, one has

z (t) = e−Ht − 1 . (7.51)

We notice that the redshift can be used to define the distance of the point ofemission reaching the observer’s origin. In that case

zem (tem) = e−Htem − 1 , (7.52)

where tem is the negative (coming from the past) time coordinate of the event ofemission.

From the relations (7.26), since

tBem = −a0 (7.53)

is the time, in Bob’s coordinates, at which the photon crosses Alice origin, we canexpress the redshift associated by Bob to the event of emission (assumed to be atAlice origin), in function of the translation parameter as

zem = eHa0 − 1 , (7.54)

or, reversing the relation, as

a0 = ln (1 + zem)H

. (7.55)

In conformal coordinates, one has

zem = −Hηem , (7.56)


which, expressed in function of a0 using Eq. (7.46), from which

ηBem = − 1H

(eHa0 − 1

), (7.57)

becomeszem = eHa0 − 1 , (7.58)

or, reversing the relation,

a0 = ln (1 + zem)H

, (7.59)

which coincide with (7.54) and (7.55).Notice that in Bob’s coordinates, we can relate the redshift to the scale factor

a (η) by the Eq. (7.50), so that we get the relation between the scale factor at theevent of emission and the translation parameter connecting an observer local to theemission with Bob as

a (ηem) = 11−Hηem

= 11 + zem

= e−Ha0 . (7.60)

7.2 A reformulation of the results of Sec. 4.5

Before discussing the case with curvature/expansion scale H, we review the resultsof Sec. 4.5 in a slightly different perspective. This will allow us to consider sometechnical generalizations that will be useful when studying to the case with spacetimeexpansion. We restrict for simplicity to the 1+1D dimensional case. As in Sec. 4.5,we work in coordinates x1, x0 such that1

x1, x0 = 0 , (7.61)

but here we omit the tilde (~) on the coordinates, since we will work only incoordinates with vanishing Poisson brackets between them throughout all thissection.

We start from the Casimir/Hamiltonian (H = C −m2) defined in Sec. 4.5,

Cα,β = p20 − p2

1 + `(αp3

0 + βp0p21

), (7.62)

where α and β are two numerical parameters. The point we want to stress isthat the difference between p0p

21 deformation and p3

0 deformation, if analyzed in aflat/non-expanding spacetime, carries very little significance (it should be inevitablyinsignificant at least for massless particles). But in our analysis of a first case withspacetime expansion we shall find that there are some significant difference betweenp0p

21 deformations and p3

0 deformations.We exhibit a corresponding DSR-deformed 1+1D Poincaré algebra of charges

compatible with the invariance of Cα,β , describable in terms of the following Poisson1We recall that, as explained in Ch. 4, and in particular in Sec. 4.3, this choice of coordinates

will not affect the results for the time of arrival of particles, which will be the main subject of ourinvestigation, since the canonical and R-DSR coordinates coincide in the origin.

7.2 A reformulation of the results of Sec. 4.5 123

brackets:

p0, p = 0 , N , p0 = p1 − ` (α+ β) p0p1 ,

N , p1 = p0 + 12`(αp2

0 + βp21

). (7.63)

The main difference with the analysis in Sec. 4.5, is that we don’t require thephase space to be canonical (even if (7.61) holds). By the discussion in Sec. 4.4, itfollows that we are thus allowing for relative locality effects in the translation sectorof transformations, even if the coordinates have vanishing Poisson brackets.

A conveniently intuitive picture of the deformation we are studying is obtainedby giving a representation of these symmetry generators in terms of time and spacecoordinates t, x and variables Π0,Π1 canonically conjugate2 to them

Π0, t = 1 , Π0, x = 0 ,Π1, t = 0 , Π1, x = −1 ,

t, x = Π0,Π1 = 0 , (7.64)

We find as representation the following:

p0 = Π0 + 12`((1− β)Π2

1 − αΠ20

), (7.65)

p = Π1 , (7.66)

N = −tp1 + xp0 + `

(12αxp

20 + tp1p0 + 1

2βxp21

). (7.67)

With this choice of representations, the phase space is not canonical:

p0, t = 1− `αp0 , p0, x = −`(1− β)p1 ,

p1, t = 0, p1, x = −1 . (7.68)

The main difference with the analysis in Sec. 4.5, is that we don’t require thephase space to be canonical (even if (7.61) holds). By the discussion in Sec. 4.4,it follows that we are thus allowing for relative locality effects in the translationsector of transformations, even if the coordinates have vanishing Poisson brackets.The coordinates x, t that we have chosen, even if the Poisson bracket between themvanishes, are not “absolute locality” coordinates for translations. But we remarkthat in the case of flat spacetime, it is always possible to define maps to coordinatesin which one has “absolute locality” for translations. In this case one has the maps

t→ t = t+ `αtp0 , x→ x = x+ ` (1− β) tp1 , (7.69)

so that in the coordinates t, x, one has the same framework of Sec. 4.5. We will seethat a similar map does not exist in the case with expanding spacetime.

We perform now the worldline analysis in coordinates t, x. We derive the world-lines by enforcing the Hamiltonian constraint H = C −m2. This straightforwardlyleads, as in Sec. 4.5, to the following worldlines

xm,p (t) = x0 +

p1√p2

1 +m2− `p1

(t− t0) , (7.70)

2With implicit Lorentzian metric.


which in the particularly interesting case of massless particles reduces, restrictingour focus on p1 > 0, to

xm=0,p (t) = x0 + (t− t0) (1− `|p1|) . (7.71)

As in the previous chapters, the charges p0, p1,N are conserved along the motion,in the sense that G = Cα,β, G = 0, and they generate respectively deformed timetranslations, spatial translations, and boosts, by Poisson brackets. By constructionwe have ensured that these transformations all are relativistic symmetries of thetheory, as one can explicitly verify [11] by acting with them on the worldlines (7.70),finding that the worldlines are covariant.

At this stage of the analysis the physical content of these worldlines is still hiddenbehind the relativity of locality. Another warning that this might be the case is seenin the fact that we are analyzing a case where the on-shell relation, in light of (7.62),is α and β dependent,

m2 = p20 − p2

1 + `(αp3

0 + βp0p21

),

whereas our worldlines are independent of α and β. Most importantly the coordinatevelocity one infers from those worldlines is independent of α and β. But we have seenin Ch. 4 that one of the manifestations of relative locality is a mismatch betweencoordinate velocity and physical velocity of particles.

So, following Ch. 4, let us probe the difference between coordinate velocityand physical velocity through the simple exercise of considering the simultaneousemission “at Alice” of two massless particles, one “soft” (with momentum ps smallenough that `-deformed terms in formulas fall below the experimental sensitivityavailable) and one “hard” (with momentum ph big enough that at least the leading`-deformed terms in formulas fall within the experimental sensitivity available).

We describe the relationship between the coordinates of two distant observersin relative rest in terms of the Poisson-bracket action of the translation generatorsE, p1, i.e. 1− atE, ·+ axp1, ·, with a0, a1 the translation parameters along tand x axes. In the specific case in which Bob detects the soft massless particle inhis origin, which restricts us to the possibilities a0 = a1 = L (L being the spatialdistance between Alice and Bob), one finds that Bob’s coordinates are related toAlice’s as follows:

tB0 = tA0 − L+ `Lαp0 , (7.72)xB0 = xA0 − L+ `L(1− β)p0 . (7.73)

The case we are considering, with a soft (p1 = ps) and a hard (p1 = ph) masslessparticle simultaneously emitted toward Bob in Alice’s origin, is such that the twoparticles are described by Alice in terms of the worldlines

xAm=0,ps(tA) = tA ,

xAm=0,ph(tA) = tA (1− `|ph|) .

And these worldlines, in light of (7.72)-(7.73), are described by Bob as follows:

xBm=0,ps(tB) = tB ,

xBm=0,ph(tB) = tB (1− `|ph|)− `L (α+ β) |ph| . (7.74)

7.2 A reformulation of the results of Sec. 4.5 125

This allows us to conclude that Bob, who is at the detector, measures the followingdifference of times of arrival between the soft photon (detected at Bob at tB = 0)and the hard photon

∆tB = `L|ph|(α+ β) . (7.75)

tA

xA

∆t = ℓL|ph|

detectorLβ

tB

xB

∆t = ℓL|pBh |(α+ )

-Lsource

Figure 7.7. We illustrate the results for travel times of massless particles derived in thissection by considering the case of two distant observers, Alice and Bob, in relativerest. Two massless particles, one soft (dashed red) and one hard (solid blue), areemitted simultaneously at Alice and they reach Bob at different times. Because of thenontriviality of translation transformations, in Bob’s coordinatization (bottom panel)the emission of the particles at Alice appears not to be simultaneous. And similarly, forthe difference in times of arrival at the distant detector (where Bob is located) Alice findsin her coordinatization (top panel) a value which is not the same as the difference intimes of arrival that Bob determines (bottom panel). The case in figure has α+ β = 4/3.For visibility we assumed here unrealistic values for the scales involved. For realisticvalues of the distance between observers and of the energy of the hard particle, taking às the inverse of the Planck scale, no effect would be visible in figure (the worldlineswould coincide).

So we see that the nontriviality of the translation transformations, as in thecase of Sec. 4.1, does affect the difference between coordinate velocity and physicalvelocity. And the relativity of locality produced by the nontriviality of the translationtransformations, while at first appearing to be counterintuitive, actually ensures theinternal logical consistency of the relativistic framework. Satisfactorily the physicalvelocity does depend on α and β just in the way one should expect on the basis of therole of α and β in the on-shell relation. But the dependence of the physical velocityon α and β is not very significant, since it comes only in the combination α + β(as already discussed in Sec. 4.5). It is because of this feature that the differencebetween p0p


0 deformation, if analyzed in a flat/non-expandingspacetime, carries very little significance. One can get the same physical velocity ofmassless particles by any mixture of p0p


0 deformation (includingthe cases where one of the two is absent) as long as α + β keeps the same value.There is nothing extraordinary or surprising about this: we are working at leading


order in `, so in terms which already have an ` factor we can use p0 = |p1| (formassless particles, and using the fact that at zero-th order in ` the massless shell isp0 = |p|). Evidently within these approximations here of interest a correction termof form `p0p

21 is indistinguishable from a correction term of form `p3

0. While this isnot surprising it was still worth devoting to it this section since one of the mainfindings of the generalization proposed in the next sections is that essentially inpresence of spacetime expansion correction terms of form `p0p

21 are very significantly

different from corrections term of form `p30.

7.3 DSR De Sitter

7.3.1 DSR-deformed de-Sitter relativistic symmetries and equa-tions of motion

We discuss here the formulation of a theory of classical-particle worldlines with `-deformed de-Sitter-relativistic symmetries. It is interesting that de Sitter relativisticsymmetries can themselves be viewed as a deformation of the special-relativisticsymmetries of Minkowski spacetime such that the expansion-rate parameter H isan invariant (see Sec. 7.1). And we have shown in Sec. 7.1 that the invariance ofthe expansion rate comes at the cost of some velocity artifact. With the pointssummarized in the previous section we can easily contrast the two deformationsof the special-relativistic symmetries of Minkowski spacetime which provide thestarting points for our work. On one side we have `-deformed Lorentz(/Poincaré)symmetries, a deformation by a large momentum scale `−1 which produces velocityartifacts connected with the novel feature of relative locality. And on the other sidewe have de-Sitter-relativistic symmetries, a deformation by a large distance scaleH−1 which produces velocity artifacts connected with spacetime expansion. Thetheory we are seeking must be such that both of these features can be accommodated,while preserving the relativistic nature of the theory. So both ` and H shall berelativistic invariants (for a total of 3, including the speed-of-light scale, here mutebecause of our choice of units). And we shall have both expansion-induced andrelative-locality-induced velocity artifacts.

We start by specifying that our `,H-deformed relativistic symmetries shall leaveinvariant the following combination of the energy E, momentum p and boost Ncharges of particles:

CH,α,β = p20 − p2

1 − 2HNp1 + `(αp3

0 + βp0p21

). (7.76)

Evidently for `→ 0 this reproduces the standard invariant of de Sitter symmetries(7.6). We assume the form (7.76) to be a fairly general deformation of the DeSitter Casimir (7.6) (at first order in `) by analogy with the DSR cases studiedin the previous chapters. Something even more general than our correction term`(αE3 + βEp2) could here be envisaged, but we are not seeking results of maximum

generality. On the contrary we want to make the case as convincingly and simplyas possible that there are examples of the novel class of relativistic theories we arehere proposing. Moreover the correction term `

(αE3 + βEp2) does have enough

structure for us to investigate the interplay of Ep2 deformations and E3 deformationswith spacetime expansion.

7.3 DSR De Sitter 127

The path drawn in the previous section guides us to observe that the following`-deformed (2D) de Sitter algebra of charges is compatible with the invariance ofCH,α,β

p0, p1 = Hp1 − `αHp0p1 ,

N , p0 = p1 +HN − `αp0(p1 +HN )− `βp0p1 ,

N , p1 = p0 + 12`αp

20 + 1

2`βp21 . (7.77)

One easily sees that for `→ 0 this reproduces the standard properties (7.5) of theclassical de Sitter algebra of charges while for H → 0 it reduces to (7.63).

We give a convenient representation of these symmetry generators in terms of“conformal (canonical) coordinates”, with conformal time η and spatial coordinate x,and variables Π0,Π1 canonically conjugate to these conformal coordinates:

Π0, η = 1 , Π0, x = 0 ,Π1, η = 0 , Π1, x = 1 ,

η, x = 0 . (7.78)

We find the following representation

p0 =Π0 −HηΠ0 +HxΠ1 + `

2(1− β)Π21

+ `

2HηΠ21 −

`

2α(Π0 −HηΠ0 +HxΠ1)2 , (7.79)

p1 =Π1 , (7.80)

N =xp0 −(η − H

2 η2 + H

2 x2)p1

+ `

(12xαp

20 + ηp0p1 + 1

2xβp21 −Hxηp2

1

).

(7.81)

We recall that the fact that CH,α,β is an invariant of the (deformed-)relativisticsymmetries implies that the charges that generate the symmetry transformationsare conserved over this evolution:

p0 =CH,α,β, p0=0 , p=CH,α,β, p=0 ,N =CH,α,β,N=0 .

Importantly by consistency with the chosen (deformed) form of the invariant CH,α,βwe have obtained phase space (and thus translation transformations) which aresignificantly deformed, specifically for the time direction. In fact, from (7.79) and(7.80) one finds

p0, η = 1−Hη − `α (1−Hη) p0 ,

p0, x = −Hx− ` (1− β +Hη) p1 + `αHxp0 ,

p, η = 0 , p, x = −1 . (7.82)

One can show that if H 6= 0, there is not a map like the one (7.69) defined in theprevious section, for which the phase space (7.82) reduces to the canonical phase


space. So that in this formulation of DSR-De-Sitter relativity of locality occursnecessarily also for coordinates with vanishing Poisson brackets (x, η = 0) Thisprepares us to deal with relative-locality effects, of the sort described in the previoussection but intertwined with the additional complexity of spacetime expansion.

Let’s first derive the worldlines in this frameworks. We set as the Hamiltonianconstraint to be H = CH,α,β −m2. We derive the worldlines by using the auxiliaryvariables Π0,Π1, in which the calculations are much easier, and express finally theresult in the “physical” momenta p0, p1. Expressed in the coordinates Π0,Π1, theCasimir (7.76) takes the form

C = (1−Hη)2(Π2

0 −Π2 + `Π0Π2). (7.83)

We notice that the on-shell relation H = 0, then gives the equation

Π0 =√

Π21 + m2

(1−Hη)2 −`Π2

12 . (7.84)

From (7.78), the Hamiltonian flow generates the equations of motion

η = H, η = 2 (1−Hη)2 Π0 + ` (1−Hη)2 Π21 ,

x = H, x = 2 (1−Hη)2 Π1 − 2` (1−Hη) Π0Π1 . (7.85)

We use the dispersion relation (7.36) to eliminate Π0 in the last equations, and thendefine coordinate the velocity to be v = v (η) = dx(η)

dη , so that, by the chain rule,

v (η) = dx (η)dt

= x

η= (1−Hη) Π1√

m2 + (1−Hη)2 Π21

− `Π1 .

We notice that since by (7.80), we have the expression for the coordinate velocity interms of the physical momentum p1 by the simple substitution Π1 = p1 in the lastequation.

Integrating the last relations in η, considering also that dp1/dη = 0, we find theworldlines

xm,p (η) = x+

√m2 + (1−Hη)2 p2

1 −√m2 + (1−Hη)2 p2

1Hp1

− ` (η − η) p1 . (7.86)


xm=0,p (t) = x+ p1|p1|

(1− ` |p1|) (η − η) . (7.87)

In conformal coordinates the worldlines of our DSR-DeSitter framework coincidewith the worldlines of the flat DSR case (see the precious section).

By construction these worldlines (7.86),(7.87), are covariant under the (deformed-)relativistic transformations generated by the charges p0, p1,N , as one can also verifyexplicitly.


7.3.2 Travel time of massless particles

As summarized in the previous section (and shown in the previous chapters), theDSR-deformed relativistic symmetries introduce (as most significant among manyother novel features) a dependence on energy of the travel time of a massless particlefrom a given source to a given detector. In the analysis we provided so far of our novelproposal of DSR-deformed relativistic symmetries of an expanding spacetime (withconstant expansion rate) an indication of dependence on energy of the travel time ofa massless particle from a given source to a given detector is found in Eq. (7.87): theequation governing the properties of the worldline of a massless particle in conformalcoordinates is `-corrected and the correction term introduces a dependence on theenergy(/momentum) of the particle. However, as also expected on the basis ofprevious results on the case without spacetime expansion, we are evidently workingin a framework where locality is relative (evidence of which has been provided sofar within our novel framework implicitly in Eqs. (7.82). So the analysis of theequations of motion written by one observer is inconclusive for what concerns thenotion of “travel time”, i.e. the correlation between emission time and detectiontime. As illustrated for the non-expanding case in the previous section (and in theprevious chapters), we must guard against the coordinate artifacts associated torelative locality by analyzing the emission of the particle in terms of the descriptionof an observer near that emission point and we must then analyze the detection ofthe particle in terms of the description of an observer near the detection point.

This is indeed our next task. We consider, as in the previous chapters, the caseof two distant observers: Alice at the emitter and Bob at the detector. We keepthe analysis in its simplest form, without true loss of generality, by consideringthe case of simultaneous emission at Alice of only two massless particles, one with“soft” momentum ps and one with “hard” momentum ph. Evidently, on the basis ofthe analysis reported in the previous subsection, Alice describes the two particlesaccording to

xAps(ηA) = ηA ,

xAph(ηA) = ηA(1− `|pAh |

), (7.88)

where we specified xA = ηA = 0, so that the emission is at (0, 0)A. Since translationsare a relativistic symmetry of our novel framework, we already know that the sametwo worldlines will be described by the distant observer Bob in the following way

xBm=0,ps(ηB) = xBs + ηB − ηBs ,

xBm=0,ph(ηB) = xBh +(ηB − ηBh

) (1− `|pBh |

), (7.89)

i.e. the same type of worldlines but with a difference of parameters here codified inxBs , ηBs , xBh , ηBh . Indeed, Alice’s worldlines (7.88) and Bob’s worldlines (7.89) haveexactly the same form, but for the ones of Alice we had by construction (by havingspecified simultaneous emission at Alice) that xAs = ηAs = xAh = ηAh = 0 whereasBob’s values of the parameters, xBs , ηBs , xBh , ηBh , should be determined by establishingwhich (`-deformed) translation transformation connects Alice to Bob.


As in Sec. 7.1, we describe the action of a finite translation on the coordinatesby its exponential representation. So that, referring again to Eqs. (7.23), from thephase space relations (7.82), we find

Ta0 . ηA = eHa0ηA + 1− eHa0

H+ `αa0e

Ha0(1−HηA

)pA0 ,

Ta0 . xA =eHa0xA + `

(1− e−Ha0

) (1 + eHa0HηA

)H

pA1

− `αa0eHa0HxApA0 − `β

sinh (Ha0)H

pA1 ,

Ta0 . pA1 = e−Ha0pA1

(1 + `αHa0p

A0

). (7.90)

Ta1 . ηA = ηA , Ta1 . x

A = xA − a1 ,

Ta1 . pA1 = pA1 , Ta1 . p

A0 = pA0 −Ha1p

A1 . (7.91)

Using again Eq. (7.25), we find, for the composed translation (space translationfollowed by time translation) Ta0,a1 , substituting Eqs. (7.91) in Eqs. (7.90),

Ta0,a1 . ηA = eHa0ηA + 1− eHa0

H+ `αa0e

Ha0(1−HηA

) (pA0 −Ha1p

A1

),

Ta0,a1 . xA =eHa0

(xA − a1

)+ `

(1− e−Ha0

) (1 + eHa0HηA

)H

pA1

− `αa0eHa0H

(xA − a1

) (pA0 −Ha1p

A1

)− `β sinh (Ha0)

HpA1 ,

Ta0,a1 . pA1 = e−Ha0pA1

(1 + `αHa0

(pA0 −Ha1p

A1

)). (7.92)


We can now use translations (7.92) to evaluate the parameters xBs , ηBs , xBh , ηBhin Bob’s description of the photons (7.89). The worldline parameters of observerAlice are fully specified (xAs = ηAs = xAh = ηAh = 0) by its being at the point ofsimultaneous emission of the two particles. We assume, as in the previous chaptersanalysis, without true loss of generality, that the soft particle reaches Bob in hisspacetime origin: so we are free to enforce xBs = ηBs = 0.

We recall that soft particle is defined so that the momentum ps has beenchosen to be small enough to render the `-deformed effects inappreciable withinthe experimental sensitivities available to Alice and Bob. Then, the fact that bothxAs = ηAs = 0 and xBs = ηBs = 0, i.e. the fact that the soft photon passes throughboth Alice’s origin and Bob’s origin, means that they are connected by a translationwith parameters satisfying Eq. (7.49):

a1 = 1− e−Ha0

H. (7.93)

Through this we are essentially exploiting the fact that the deformation is ineffectiveon the soft particle as a way for us to focus on a distant observer Bob whose


relationship to Alice (translation parameters connecting Alice to Bob) can be specifiedusing only known results on the undeformed/standard relativistic properties3.

We can use now the equations (7.92), together with the condition (7.93), to eval-uate the remaining parameters in Bob’s hard photon worldline (7.89). Substitutingthe conditions (7.93) and (xAs = ηAs = xAh = ηAh = 0) in Eqs. (7.92) for the hardphoton (p1 = ph), and using also that up to first order in `, for m = 0 and p1 > 0,the relation `p0 = `p1 holds, we find

ηBh = 1− eHa0

H+ `αa0p

Ah ,

xBh =1− eHa0

H+ `

1− e−Ha0

HpAh

+ `αa0(1− e−Ha0

)pAh − `β

sinh (Ha0)H

pAh ,

pBh = e−Ha0pAh + `αHa0(pAh

)2. (7.94)

Substituting the last equations in Bob’s hard photon worldline (7.89) (assuming alsop1 > 0), we find, after some algebra,

xBm=0,ph(ηB) = ηB − `pBh

(ηB + αa0 + β

e2Ha0 − 1H

). (7.95)

In turn this allows us to obtain the sought result for the dependence on en-ergy(/momentum) of the travel times of massless particles: by construction of theworldlines and of the Alice→Bob transformation the soft massless particle emittedin Alice’s spacetime origin reaches Bob’s spacetime origin, whereas from (7.95) wesee that the hard massless particle also emitted in Alice’s spacetime origin reachesBob at a nonzero conformal time. Specifically one can derive the arrival time of thehard photon in Bob’s space origin, i.e. the arrival time of the hard photon at thedetector respect the arrival time of the soft photon, by setting Eq. (7.95) to 0 andsolving for ηB. So that the difference in conformal travel times between the soft andhard photon is

∆ηB = ηBh

∣∣∣xBh

=0= `|pB|

(αa0 + β

e2Ha0 − 12H

). (7.96)

3This also implicitly requires [38] that the clocks at Alice and Bob are synchronized by exchangingsoft massless particles.


ηA

xA

∆η = ℓL|pAh |

detectorL

βηB

xB

∆η = ℓL|pBh |(α 1

H ln( 11−HL ) +

L−HL2

2

(1−HL)2

)

− L1−HLsource

Figure 7.8. We illustrate the results for travel times of massless particles derived in thissection, adopting conformal coordinates. We consider the case of two distant observers,Alice and Bob, connected by a pure translation, and two massless particles, one soft(dashed red) and one hard (solid blue and solid violet), emitted simultaneously at Alice.And we consider two combinations of values of α and β producing the same α+ β: thecase α = 2/3, β = 2/3 (Bob’s hard worldline in blue) and the case α = 1/3, β = 1 (Bob’shard worldline in violet). As in the case without expansion (Fig. 7.7) we have here thatin Bob’s coordinatization (bottom panel) the emission of the particles at Alice appearsnot to be simultaneous. Similarly for the difference in times of arrival at the distantdetector Bob Alice finds in her coordinatization (top panel) not the same value as thedifference in times of arrival that Bob determines. Comparison of the blue and violetworldlines shows that in the case with spacetime expansion the travel time does dependindividually on α and β (not just on α+ β as in the case without spacetime expansion).Again for visibility we assumed here unrealistic values for the scales involved.

We summarize the relativistic properties of this travel-time analysis, in conformalcoordinates, in Fig 7.8. By comparison with Fig. 7.7 one sees that in conformalcoordinates the qualitative picture of the energy dependence of travel times isvery similar to the one of the case without spacetime expansion. But here, withspacetime expansion, there are tangible differences between p0p

21 deformations and

p30 deformations.

7.4 Implications for phenomenology 133

In Fig. 7.9 we characterize the results of our travel-time analysis in comovingcoordinates, which for most studies of spacetime expansion are the most intuitivechoice of coordinates. [The differences between Fig. 7.8 and Fig. 7.9 all are astraightforward manifestation of the relationship η = H−1(1− e−Ht) between theconformal time η and the comoving time t.]

tA

xA

∆t = ℓL|pAh |

detectorL

βtB

xB

∆t = ℓL|pBh |(α 1

H ln( 11−HL ) +

L−HL2

2

(1−HL)2

)

− L1−HLsource

Figure 7.9. We here illustrate the results for travel times of massless particles derivedin this section, adopting comoving coordinates. The situation here shown is the samesituation already shown (there in conformal coordinates) in Fig. 7.8.

7.4 Implications for phenomenologyIn this section we discuss some consequences, emerging from our analysis of DSR ina De Sitter expanding spacetime, related to the observation of gamma ray bursts,introduced in Sec. 5.1.

Our first step toward the phenomenological issues of interest in this sectionis to reformulate the results for travel times of massless particles derived in theprevious section in a way that is in closer correspondence with the type of factsthat are established experimentally, when our telescopes observe distant sources ofgamma rays. Many of the analyses performed at our telescopes amount to timingthe detection of photons emitted from a source at a known redshift z. Postponingfor a moment the fact that the relevant contexts are not such that we could assumea constant expansion rate, let us observe that the results we derived in the previoussection are easily reformulated as the following prediction for the differences indetection times of photons of different energies(/momenta) emitted simultaneouslyby a source at redshift z. Using Eq. (7.59), the delay (7.96), becomes

∆t = `|p|(α

ln (1 + z)H

+ βz + z2

2H

). (7.97)

Here again ∆t is the difference in detection times between a hard gamma-ray photonof momentum p and a reference ultrasoft photon emitted simultaneously to the


hard photon at the distant source. Notice that with ∆t we denote the comovingtime but in the relevant timing sequences of detections at telescopes the differencebetween comoving and conformal time is intangible: the telescopes are operated fora range of times within the t = 0 of their resident clock which is relatively small,much smaller than the time scale H−1, so that conformal time and comoving timeessentially coincide (one has η ≡ H−1(1− e−Ht) ' t for t H−1). Moreover, sincewe are working at first order in `, and since, from (7.96), ∆η = `f (z), it follows that

∆t = th − ts = th = − ln (1−Hηh)H

= − ln (1−H`f (z))H

= `f (z) = ∆η . (7.98)

A first aspect of phenomenological relevance which must be noticed in our result(7.97) is the dependence on the parameters α and β, i.e. the difference between p0p

21

deformations and p30 deformations. The fact that for the non-expanding spacetime

case this dependence only goes with α+ β is of course recovered in our result (7.97)in the limit of small redshift z. This is expected since for small redshift the expansiondoes not manage to have appreciable consequences. What is noteworthy is thatalready at next-to-leading order in redshift the dependence on α and β is no longerfully of the form α+ β, as seen by expanding our result (7.97) to second order in z:

∆t∣∣∣z1' `|p|

H

[(α+ β)z + (β − α)z

2

2

]. (7.99)

In a non-expanding spacetime the difference between p0p21 deformations and p3

0deformations would not have significant implications on travel-time determinations(since it depends on α + β one gets the same result reducing the amount of p0p

21

deformation in favor of an equally sizable increase of the amount of p30 deformation).

The situation in expanding spacetimes is evidently qualitatively different in thisrespect since p0p


0 deformations produce corrections to the traveltimes that have different functional dependence on redshift. So we are learning thatfor determinations of travel times from distant astrophysical sources the differencebetween p0p


0 deformations is a phenomenologically viable(phenomenologically determinable) issue.

Related to this observation is also the other point we want to make on the phe-nomenology side, which concerns the comparison with analogous studies of scenarioswhere relativistic symmetries are actually “broken” (allowing for a preferred/“aether”frame), rather than DSR-deformed. Broken-Lorentz-symmetry theories with a pre-ferred frame are far simpler conceptually than DSR-deformed relativistic theories,and indeed the issue of the interplay between scale of Lorentz-symmetry break-down and spacetime expansion has been usefully investigated already for severalyears [98, 128, 99], even producing a rather universal consensus on the proper for-malization that should be adopted [99, 100, 10]. Again thanks to the simplicity ofbroken-Lorentz-symmetry theories these results apply to the general case of varyingexpansion rate, and for the massless particles they take the form

∆t = λLIV |p|∫ z

0

dz

a(z)H(z) , (7.100)

where a is the scale factor, H is the expansion rate, and λLIV (“LIV” standing forLorentz Invariance Violation) is the counterpart of our scale `: just like ` for us is


the inverse-momentum scale characteristic of the onset of the DSR-deformation ofrelativistic symmetries, λLIV is the inverse-momentum scale characteristic of theonset of the LIV-breakdown of relativistic symmetries. Of course, there is a crucialdifference between the properties of ` and the properties of λLIV : `, as characteristicscale of a deformed-symmetry picture, takes the same value for all observers, whileλLIV , as characteristic scale of a broken-symmetry picture, takes a certain valuein the preferred frame and different values in frames boosted with respect to thepreferred frame.

Setting momentarily these differences aside we can compare our results for DSR-deformed symmetries with constant rate of expansion with the special case of thebroken-symmetry formula (7.100) obtained for constant rate of expansion:

∆t = λLIV |p|z + z2

2H

. (7.101)

Comparing this to our result (7.97) we find that fixing α = 0 in the deformed-symmetry case one gets a formula (valid in all reference frames) which is the sameas the formula of the broken-symmetry case in the preferred frame. So if α = 0the differences between deformed-symmetry and broken-symmetry cases would betangible only by comparing studies of travel times of massless particles between twotelescopes with a relative boost: the difference there would be indeed that ` takesthe same value for studies conducted by the two telescopes whereas for λLIV thetwo telescopes should give different values.

We must stress however that within our deformed-symmetry analysis we foundno reason to focus specifically on the choice α = 0. And if α 6= 0 in the deformedsymmetry case even studies conducted by a single telescope could distinguish betweenthe case of symmetry deformation and the case of symmetry breakdown.


MKN501PKS2155-304

GRB090510Α=1.32,Β=0

Α=

0,Β=0.65

0.05 0.10 0.50 1.00 5.00 10.00

0.001

0.01

0.1

1

z

Dt

Dp

@sG

eV-

1 D

Figure 7.10. Here we show the dependence on redshift of the expected time-of-arrivaldifference divided by the difference of energy of the two massless particles. Two suchfunctions are shown, one for the case α = 0, β = 0.65 (violet) and one for the caseα = 1.3, β = 0 (blue). We also show the upper limits that can be derived from datareported in Refs. [129, 96, 10], setting momentarily aside the fact that our analysisadopted the simplifying assumption of a constant rate of expansion (whereas a rigorousanalysis of the data reported in Refs. [129, 96, 10] should take into account the non-constancy of the expansion rate). The values α = 0, β = 0.65 and α = 1.3, β = 0have been chosen so that we have consistency with the tightest upper bound, the oneestablished in Ref. [10]. The main message is coded in the fact that at small values ofredshift the blue and the violet lines are rather close, but at large values of redshift theyare significantly different (this is a log-log plot). In turn this implies that at high redshiftthe difference between adding correction terms of form p0p

21 and adding correction terms

of form p30 can be very tangible.

In Fig. 7.10 we compare the dependence on redshift of our deformed-symmetryeffect among two limiting cases of balance between α and β, and we also comparethese results to bounds on travel-time anomalies [129, 96, 10] obtained in studiesof sources at redshift smaller than 1 (where the assumption of a constant rate ofexpansion is not completely misleading).


In Fig. 7.11 we illustrate the “constraining power” of our results and of foreseeablegeneralizations of our analysis gaining access to cases with non-constant rate ofexpansion. For better visibility and as a way to offer more intelligible visual messageswe restrict our focus to the case in which both α and β are positive.

PKS2155-304

z=0.116

0 5 10 15 200

5

10

15

20

Α

Β

GRB090510

z=0.9

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

1.2

Α

ΒFigure 7.11. Here in the left panel we show the constraint on the α, β parameter space

that can be obtained from Ref. [96], concerning a source at the relatively small redshiftof z = 0.116, where we can confidently apply our results for constant rate of expansion asa reliable first approximation. Through this we show that even within the confines of ouranalysis Planck-scale sensitivity (values of |α| and |β| smaller or comparable to 1) is notfar. In the right panel we show the much tighter (indeed “Planckian”) constraint on theα, β parameter space which would be within our reach if we could assume our analysis toapply also to redshifts close to 1, as for GRB090510 observed by the Fermi telescope [10].By comparing the left and the right panel one also finds additional evidence of how thedifference between adding correction terms of form p0p

21 and adding correction terms of

form p30 becomes more significant at higher redshifts: in the left panel (data on source

at small redshift of z = 0.116) the bound on the α parameter is nearly as strong as onthe β parameter, whereas in the right panel (data on source at redshift of z ' 0.9) theconstraint on the alpha parameter is significantly weaker than the constraint on the βparameter.

In the left panel of Fig. 7.11 we show how the bound obtained from Ref. [96], atthe relatively small redshift of z = 0.116 (where we should really be able to applyour results for constant rate of expansion as a reliable first approximation), providesa constraint on the α, β parameter space. For these purposes we can of course fixthe value of ` to be exactly the Planck length (the inverse of the Planck scale) sinceany rescaling of ` can be reabsorbed into an overall rescaling of α and β. Withinthis choice of conventions the target “Planck-scale sensitivity” would manifest itselfas the ability to constrain values of α and β of order 1. As shown in the left panel ofFig. 7.11, even restricting our focus on cases with redshift much smaller than 1 (asneeded because of the present limitation of applicability of our approach to constant,or approximately constant, rate of expansion) this Planck-scale sensitivity is notfar. In the right panel of Fig. 7.11 we show the much tighter constraint on the α, βparameter space which would be within reach if we could assume our analysis to applyalso to redshifts close to 1, as for GRB090510 observed by the Fermi telescope [10].Analyzing data from sources at redshift of ' 1 assuming a picture with constantrate of expansion cannot produce conservative experimental bounds, but the contentof the right panel of Fig. 7.11 serves the purpose of providing evidence of the factthat full Planck-scale sensitivity will be within reach of improved versions of ouranalysis, extending our results to the case of expansion at non-constant rate.

139

Chapter 8

Spacetime fuzziness

So far in this thesis, we investigated only the classical (non-quantum) aspects ofparticle kinematics in DSR theories. We introduced in Sec. 3.1 the κ-Poincaré/κ-Minkowski description of spacetime and spacetime symmetries, and we often referredto a classical (non-quantum) version of that formulation for our illustrative examplesof DSR effects. Still, the idea of spacetime non-commutative, on which κ-Poincaré/κ-Minkowski is based, and for which the commutator between spacetime coordinatesdoes not vanish as in

[xj , x0] = i`xj , [xj , xk] = 0 , (8.1)

does not make sense in a setting based on classical mechanics, where the formalismprovides no room for non-commutativity of coordinates.

This in itself is not so alarming, since classical mechanics should anyway onlyemerge as an approximate regime of a quantum mechanics, and the limiting proce-dure from quantum mechanics to classical mechanics may well be such that also thenoncommutativity of spacetime coordinates is removed in the classical limit. Thereal problem is that even giving a formulation of κ-Minkowski spacetime noncom-mutativity in a quantum-mechanics setup is not straightforward. This is due tothe fact that in κ-Minkowski the time coordinate is a noncommutative observable,whereas in the standard formulation of quantum mechanics the time coordinate ismerely an evolution parameter (a necessarily classical evolution parameter). Time,according to κ-Minkowski, should be an operator that does not commute with thespatial-coordinate operators, but in the standard setup of quantum mechanics weare not in the situation of time being described by an operator that commutes withthe spatial-coordinate operators: in the standard setup of quantum mechanics timeis not an observable at all, it just plays the role of evolution parameter.

We believe that it was indeed because of this mismatch between the nature oftime in quantum mechanics and the properties of the κ-Minkowski time coordinatethat progress in formulating observable spacetime consequences of κ-Minkowskinoncommutativity remained stalled. Several results were obtained on the indirectimplications of (8.1) for the structure of momentum space and the role played by theκ-Poincaré Hopf algebra [55, 56, 57] in describing the symmetries of κ-Minkowski.In this perspective, there have been considerable results in studying some of thequantum features arising in a formulation of field theories based on κ-Minkowski

140 8. Spacetime fuzziness

non-commutative spacetime (see e.g. [130, 131, 132]). These studies focused mainlyon introducing a deformed “star-product” between fields, reflecting the properties ofthe commutation rules (8.1), and then deriving the Feynman rules. These previousinvestigations did provide some insight on the implications of k-Minkowski non-commutativity for the deformed momentum space. But for the original objective ofspacetime noncommutativity, the one of providing a characterization of spacetimefuzziness at the Planck length, the implications of (8.1) remain unclear.

In this chapter, based on our work [43], we propose a way to address this issue thatrelies on results which were not mature when κ-Minkowski was first introduced butbecame increasingly solid over the last decade. These are results [133, 134, 53] on acovariant formulation of ordinary quantum mechanics. In this powerful reformulationof quantum mechanics both the spatial coordinates and the time coordinate playthe same type of role. And there is no “evolution”, since dynamics is codifiedin a constraint, just in the same sense familiar for the covariant formulation ofclassical mechanics. Spatial and time coordinates are well-defined operators on a“kinematical Hilbert space”, which is just an ordinary Hilbert space of normalizablewave functions [53]. And spatial and time coordinates are still well-defined operatorson the “physical Hilbert space”, obtained from the kinematical Hilbert space byenforcing the constraint of vanishing covariant-Hamiltonian. Dynamics is codifiedin the fact that on states of the physical Hilbert space, because of the implicationsof the constraint they satisfy, one finds relationships between the properties ofthe (partial [53]) observables for spatial coordinates and the properties of the time(partial) observable. In this way, for appropriate specification of the state onthe physical Hilbert space, the covariant pure-constraint version of the quantummechanics of free particles describes “fuzzy worldlines” (worldlines of particlesgoverned by Heisenberg uncertainty principle) just in the same sense that thecovariant pure-constraint formulation of the classical mechanics of free particlesdescribes sharp-classical worldlines.

So, over this last decade, the community has developed a formulation of quantummechanics in which both time and the spatial coordinates are operators on a Hilbertspace, which of course commute (they do not commute with their conjugate momenta,but commute among themselves [53]). Our proposal is that this is the correct startingpoint for formulating κ-Minkowski noncommutativity: the commuting time andspatial-coordinate operators of the covariant formulation of quantum mechanicsshould be replaced by time and spatial-coordinate operators governed by the κ-Minkowski noncommutativity.

The key ingredient of the strategy of analysis we propose is a novel type of“pregeometric representation” of κ-Minkowski. This idea of pregeometric represen-tation had already been discussed for κ-Minkowski (see, e.g., Ref. [19]) and wasoriginally conceived also as conceptual tool: one could conjecture the emergence ofκ-Minkowski from quantum gravity at some level of effective description, and fromthis perspective it might be natural to describe κ-Minkowski noncommutativity interms of a standard Heisenberg quantum mechanics introduced at some deeper levelof the description. One would then seek a relationship between κ-Minkowski coordi-nates xj , x0 and the phase space coordinates qµ, πµ of the pregeometric formulation,with q, π forming standard Heisenberg pairs of conjugate observables on a Hilbertspace. It was already established in previous works that this could be done, but

141

focusing exclusively on giving such a description of the κ-Minkowski coordinates.We here find a new pregeometric description capable of accommodating not onlythe κ-Minkowski coordinates but also the associated differential calculus and theκ-Poincaré symmetry generators.

As evidence of the empowerment produced by our novel pregeometric description,we are capable of analyzing for the first time in quantitative manner the implicationsof the feature which is most fascinating (and puzzling) of κ-Minkowski: the form of(8.1) suggests heuristically that κ-Minkowski noncommutativity becomes stronger atlarger distances (since xj is on the right-hand-side of the nontrivial commutationrelation),but distances from what?And if it happens to be the distance from the origin of the observer’s frame is thisthen a preferred-frame picture or somehow still a fully relativistic picture?

Indeed naively (8.1) appears to imply a preferred frame picture, a picturespecialized to a preferred observer in a scenario where relativistic symmetries (andparticularly translational invariance) are broken. But this naive interpretation iseven more puzzling considering the many (though partial) successes of the descriptionof relativistic transformations in κ-Minkowski as given in terms of the κ-PoincaréHopf algebra.

Within our description these crucial issues can be analyzed without relyingon heuristic/naive reasoning: we can formalize fuzzy points in κ-Minkowski asstates on our pregeometric Hilbert space. And our pregeometric description of thedifferential calculus and the κ-Poincaré generators allows us to describe relativistictransformations very explicitly, in terms of actions on the pregeometric states. Wethereby obtain conclusive evidence of the fact that κ-Minkowski is a fully relativisticspacetime. The feature that was customarily described naively as “uncertaintiesgrowing with distance” is here properly described as a novel feature of relative locality.In this first example of relative locality in a quantum spacetime, which we hereprovide through our analysis of κ-Minkowski, one has a fully relativistic descriptionof how the fuzziness of events may appear to take different shape depending on thedistance from the events: with a given network of events and a given network ofobservers one would find that all observers describe as less fuzzy those events thatare near to them whereas they infer increased fuzzyness for events that are far fromthem. All this occurs in a fully relativistic manner, and can be understood as mainlya manifestation of the peculiarities of translational symmetries in κ-Minkowski,which we shall here analyze in detail.

We close this chapter with a discussion of the connection of these results withspacetime fuzziness and the associated blurring of images of distant astrophysicalsources, discussed preliminarily in Sec. 5.2. Our results are not yet suitable for a fullanalysis of blurring of quasar images, but allows us to investigate some of the keyassumptions made in the heuristic arguments which have so far driven the relevantphenomenology. They confirm earlier heuristic arguments suggesting that spacetimefuzziness, while irrelevantly small on terrestrial scales, could be observably largefor propagation of particles over cosmological distances. But for what concerns thedependence of the fuzziness of a particle worldline on the energy of the particle ourresults differ significantly from what had been suggested heuristically and couldprovide guidance for reshaping accordingly the phenomenology.


In the following we work for simplicity in a 2D (1+1-dimensional) κ-Minkowskispacetime, adopting throughout conventions for the Minkowski metric tensor ηµν =(1,−1). And we adopt units such that the speed-of-light scale (speed of masslessparticles in the infrared limit) and the Planck constant are 1 (c = 1,~ = 1).

8.1 Preliminaries on κ-Minkowski differential calculusand translation generators

We here review briefly some of the properties of κ-Poincaré/κ-Minkowski translationsin time-to-the-right basis, introduced in Sec. 3.1, for the 2D case here of interest.This formulation of κ-Poincaré/κ-Minkowski is based on the work of Majid andRuegg [55]. We moreover introduce some of the properties characterizing the κ-Poincaré/κ-Minkowski differential calculus.

The generators of κ-Poincaré translations are conveniently characterized throughthe following rule of action on exponentials

Pµ . eik1x1e−ik0x0 = kµe

ik1x1e−ik0x0 . (8.2)

These translation generators have “non-primitive coproduct” ∆Pµ = Pµ ⊗ 1 +e−`δ

1µP0 ⊗ Pµ, in the sense that their action on products of functions is governed by

a deformed Leibniz rule

Pµ . f(x)g(x)

= (Pµ . f(x)) g(x)+(e−`δ

1µP0 . f(x)

)(Pµ . g(x)) ,

(8.3)

With this coproduct one can show that the commutator (8.1) is conserved in thefollowing sense

Pµ . [x1, x0] = i`Pµ . x1 . (8.4)

However, the generators Pµ are not the only nontrivial structure needed forimplementing translation transformations in κ-Minkowski. One of course wantsthe coordinates x′µ of a translated observer to be linked to the coordinates of theobserver from which the translation is made by a rule of the type x′µ = xµ − aµ,while enforcing κ-Minkowski noncommutativity also for the translated coordinates[x′j , x

′0

]= i`x′j ,

[x′j , x

′k

]= 0. This can be done by describing a translation acting

in the familiar formT = 1 + d , d = −ia`µPµ , (8.5)

but only if the “translation parameters” have themselves some noncommutativityproperties [135, 79, 80, 81, 136], which in particular can take the form[

a`1, x0]

= ià`1 ,[a`µ, x1

]= 0 , [a`0, x0] = 0 . (8.6)

One can show that these (8.6) satisfy the conditions for having a quantum differentialcalculus, in the sense first introduced by Woronowicz [82]. And the description oftranslations based on (8.5)-(8.6) proved robust also in work establishing [137] thepresence of Noether charges in theories formulated on κ-Minkowski with κ-Poincarésymmetries.

8.2 κ-Minkowski pregeometry 143

8.2 A novel pregeometric representation of κ-Minkowskicoordinates, differential calculus and translations

The notion of pregeometric description which we are here adopting was proposedin Ref. [19]. Conceptually it can be inspired by the idea (or it can suggest that)spacetime noncommutativity arises from a more fundamental theory: the morefundamental theory would be needed to analyze more general quantum-gravityissues but in certain limiting cases (regimes) a description based solely on spacetimenoncommutativity would arise. Technically a pregeometric description allows toreformulate the complexity of the κ-Minkowski commutation relations in terms of(a few copies of) the familiar Heisenberg algebra, so it can often provide a usefulexpedient for relying on the large number of results available on the Heisenbergalgebra.

Actually one can have interesting examples of pregeometric description evenbased on deformed Heisenberg algebras. In particular in Ref. [19] (developing onresults previously reported in Ref. [138]) it was noticed that one could take asstarting point a two-parameter (ρ, ~0) family of commutation relations

[q, π] = i~0(1− e−qρ ) (8.7)

with co-algebraic structure:

∆q = q ⊗ 1 + 1⊗ q ∆π = π ⊗ 1 + e− qρ ⊗ π (8.8)

In this setup the link between the “pregeometric observables” q, π and the k-Minkowski coordinates emerges in the limit1 ~0, ρ → 0, ~0

ρ → `, where one cantake x0 = π and x1 = q.

Other pregeometric representations of κ-Minkowski were developed (either explic-itly of implicitly advocating a pregeometric viewpoint) in Refs. [139, 140, 141, 142].We shall not dwell on the details of these other pregeometric representations. Itsuffices to notice that they all described the k-Minkowski coordinates in terms of(a few copies of) the undeformed Heisenberg algebra. And it is also importantfor us to stress that these previous pregeometric descriptions did not make roomfor accommodating the elements of the κ-Minkowski differential calculus, whereasachieving a pregeometric representation of the κ-Minkowski differential calculus iscrucial for our purposes. Moreover, these previous studies, while providing impor-tant breakthroughs on the technical side, left largely unaddressed the key issue forphysical applications of spacetime noncommutativity: taking as starting point ourcurrent theories where and how should we make room for the noncommutativity ofcoordinates? And in which way would this noncommutativity lead to observableeffects?

We argue here that the procedure we outlined for the study of κ-Poincaré/κ-Minkowski quantum mechanics description, provide us with a notion of “geometry ofκ-Minkowski”. We advocate the viewpoint that the kinematical Hilbert space plays arole within the covariant formulation of quantum mechanics that is closely analogous

1Note that in this pregeometric setup of Ref. [138] the “pregeometric Planck constant” ~0 is ingeneral unrelated to the physical Planck constant ~.


to the role of Minkowski spacetime in the classical mechanics of special-relativisticparticles. Within the covariant formulation of quantum mechanics the kinematicalHilbert space codifies the geometry of spacetime. Indeed, just like Minkowskispacetime is the arena where the dynamics of relativistic classical particles unfolds,produced by enforcing the Hamiltonian constraint, the kinematical Hilbert space isthe arena where the dynamics of relativistic quantum particles unfolds, producedby enforcing the Hamiltonian (quantum-operator) constraint. Minkowski spacetimeon its own is not really equipped with any physically observable property: theobservables we occasionally label as “spacetime observables of Minkowski spacetime”truly are operatively defined through the experimental study of the propertiesof classical particles in Minkowski spacetime. But understanding the properties,and particularly the relativistic symmetries of Minkowski spacetime is an exerciseof much more that mere academic interest, since the formal properties of emptyMinkowski spacetime strongly affect then the physical properties of theories ofparticles in Minkowski spacetime. Similarly the properties of observable-operatorson the kinematical Hilbert space of the covariant formulation of quantum mechanicsare not themselves subjectable to measurement, but they usefully characterize thespacetime arena where then the quantum dynamics of particles on the physicalHilbert space takes place.

So In this section we shall study the properties of the noncommuting coordinatesof κ-Minkowski spacetime at the level of the kinematical Hilbert space of a covariantformulation of quantum mechanics. These properties of the κ-Minkowski coordinateswill characterize κ-Minkowski spacetime as an arena for the dynamics of particles,which will be discussed in Sec. 8.5. We describe the properties of the coordinates of2D κ-Minkowski spacetime on the kinematical Hilbert space by providing a suitable“pregeometric representation”, given in terms of standard (undeformed) phase-spaceobservables,

[π0, q0] = i , [π0, q1] = 0[π1, q0] = 0 , [π1, q1] = −i , (8.9)

for the covariant formulation of 2D quantum mechanics.For our representation of the κ-Minkowski coordinates we view q0 and q1 of

(8.9) as operators for the pregeometric position in time and space, indeed operatorsordinarily studied [53] on the kinematical Hilbert space of the covariant formulationof quantum mechanics. We then describe the κ-Minkowski coordinates x0, x1, from(8.1), as follows

x0 = q0 , x1 = q1e`π0 , (8.10)

which indeed satisfies2 (8.1).And we do find in this pregeometric description also opportunities for describing

the κ-Minkowski differential calculus and the κ-Poincaré translation generators. For

2It is worth to notice that the x0, x1 defined by (8.10) act on the functions of momenta f(πµ)with primitive coproduct, as expected [55]. For example x1 . f(πµ)g(πµ) = (x1 . f(πµ)) g(πµ) +f(πµ) (x1 . g(πµ)).

8.3 Boosts and a fully pregeometric picture 145

the translation generators by posing

P0 . f(x0, x1)←→ [π0, f(q0, q1e`π0)] ,

P1 . f(x0, x1)←→ e−`π0 [π1, f(q0, q1e`π0)] , (8.11)

one does reproduce all the properties of κ-Poincaré translation generators, summa-rized in Sec. 8.1.

And most crucially we also notice that the properties of the elements a`µ of thedifferential calculus given in Eq. (8.6) can be reproduced by combining ordinary(numerical) parameters aµ and the observable π0:

a`0 = a0 , a`1 = a1e`π0 . (8.12)

8.3 Boosts and a fully pregeometric pictureThe main objectives of the analysis we are here reporting concern translationtransformations, which, as shown in the next section, when formulated according toour proposals and pregeometric formulation, shed light on several grey areas of ourprevious understanding of κ-Minkowski. But before we get to that let us pause inthis section for introducing our pregeometric description of boosts in κ-Minkowski,not only for showing the completeness of our pregeometric representation, but alsofor completing the characterization of the kinematical Hilbert space on which thismanuscript focuses.

We have already implicitly specified that the states of our kinematical Hilbertspace for κ-Minkowski will admit representation (in the “pregeometric momentum-space representation”) as square-integrable functions of variables π0 and π1. Butthe prescription of square-integrability is meaningful only once a measure on thiskinematical Hilbert space is introduced. Understanding the properties of boostsin κ-Minkowski, as formulated in our pregeometric picture, will allow us to specifythis measure and we shall see that in that respect our kinematical Hilbert space isnot exactly the same as the one (see, e.g., Ref. [53]) of the covariant formulation ofquantum mechanics.

Essentially the task we must accomplish is providing a pregeometric descriptionof the boost sector of the κ-Poincaré Hopf algebra. Working again consistently withthe choice of conventions introduced in Sec. 3.1, which we adopt throughout, in our2D κ-Minkowski spacetime boost generators should satisfy the following propertiesof commutation with translation generators and of coproduct:

−i[N , P0] . f(x) ≡ P1 . f(x) , (8.13)

−i[N , P1] . f(x) ≡(

1− e−2`P0

2` − `

2P21

). f(x) , (8.14)

∆N = N ⊗ 1 + e−`P0 ⊗N , (8.15)

Notice that in the 2D κ-Minkowski the coproduct of boost generators has the sameform as the coproduct of translation generators (here shown in Sec. 8.1). This is apeculiarity of the 2D case which simplifies the description of boost transformations.


Generalizing our results for the pregeometry from our 2D case to a 4D κ-Minkowskifor what concerns translation transformations is completely elementary. For booststhe 4D generalization is also conceptually straightforward but technically requiresthe added structure of the specific properties of κ-Poincaré boosts in the 4D case,where the coproduct of boost generators no longer has the same structure of thecoproduct of translation generators (see Sec. 3.1).

This fact that 2D κ-Poincaré boost generators have the same coproduct as 2Dκ-Poincaré translation generators immediately leads us to also specify the propertiesof boost transformation parameters. In fact, as observed for example in Ref. [143],the noncommutativity properties of transformation parameters are directly linkedto the coproduct properties of the generators of the transformations. Also for boosttransformations we can give a formulation analogous to the one of Eq. (8.5), withthe action of an infinitesimal boost taking the form

B = 1 + dN , dN = iξ`N , (8.16)

and noncommutative boost-transformation parameter such that [143][ξ`, x0

]= i`ξ` ,

[ξ`, x1

]= 0 . (8.17)

Our task then is to provide a pregeometric representation of the boost generatorN and of the noncommutative boost-transformation parameter ξ` reflecting theproperties (8.14), (8.15), (8.17). We find that this is indeed possible. The prege-ometric description of the noncommutative boost-transformation parameter ξ` isgiven in terms of an ordinary (numeric, commutative) boost parameter ξ and the π0observable

ξ` = ξe`π0 . (8.18)

For the boost generator we find the pregeometric prescription

N . f(x) ≡ e−`π0 [η, f(x)] (8.19)

withη ≡

(e2`π0 − 1

2` + `

2 π21

)q1 − π1q0 (8.20)

These pregeometric representations provide the basis for studying boost transforma-tions in κ-Minkowski. Notice that from (8.18) and (8.19), the action (8.16) can beexpressed in terms of the operator η as an ordinary (adjoint) action by commutator,and it can be exponentiated as

B . O → B†OB = eiξη†Oe−iξη . (8.21)

At this point we have exhibited the full strength of our pregeometric description:whereas previous pregeometric descriptions only accommodated the κ-Minkowskicoordinates (plus, in some cases, some κ-Poincaré generators) we gave a pregeometricdescription of all the most used tools of the literature on κ-Minkowski, includingthe differential calculus (which also play the role of noncommutative translationparameters), the translations generators, the noncommutative boost parameter andthe boost generator.

8.3 Boosts and a fully pregeometric picture 147

And we are now well equipped for returning to the issue highlighted at thebeginning of this subsection, concerning the specification of the measure on ourkinematical Hilbert space. We shall characterize our scalar products in momentumspace, as

〈O〉 = 〈ψ|O|ψ〉 =∫D(πµ)ψ?(πµ)O(πµ)ψ(πµ) , (8.22)

and in order to get a boost invariant scalar product, we want the measure D(πµ) tobe invariant under the action of boosts (8.16). From the definitions (8.16),(8.18-8.20),together with (8.9), one easily finds that under the action of boosts (8.16)

π′0 = π0 − ξπ1 ,

π′1 = π1 − ξ(e2`π0 − 1

2` + `

2π21

). (8.23)

Guided by the criterion that the measure D(πµ) should be invariant under thesetransformations we are led to adopt

D(πµ) = dπ0dπ1e−`π0 . (8.24)

One easily sees that this deformed measure (8.24) also ensures that η is hermitian,so that the boost operator B defined in (8.21) is unitary and preserves the scalarproduct:

〈ψ′|ψ′〉 = 〈ψ|eiξηe−iξη|ψ〉 = 〈ψ|ψ〉 . (8.25)

While, as mentioned, we postpone to Sec. 8.5 the introduction of physical/on-shell particles in our κ-Minkowski spacetime, let us pose briefly for observing thatthe properties of boosts also strongly characterize the form of the on-shell condition,which in turn (through an appropriate “Hamiltonian constraint” [53]) governs therelationship between the kinematical Hilbert space and the physical Hilbert space.On the basis of the properties derived above one easily finds that the demand ofinvariance under boosts leads to adopting the following “deformed d’Alembertianoperator”

` =(2`

)2sinh2

(`π02

)− e−`π0 π2

1 . (8.26)

Let’s examine now the implications of the integration measure D(πµ) of (8.24)for the properties of the κ-Minkowski coordinates. One easily sees that the spatialcoordinate x1 is a hermitian operator on our kinematical Hilbert space equippedwith the integration measure D(πµ), since, by the momentum space representationq1 ≡ i∂π1 , it follows that

q1e−`π0 = e−`π0 q1 .

For the operator q0 one easily sees that it is not hermitian but it misses beinghermitian by a constant term of order `

q†0 = q0 + i` .

Indeed, since q0 ≡ −i∂π0 , one finds that

q0e−`π0 = e−`π0(q0 + i`) .


We shall not be too concerned about this peculiar lack of hermitianity of q0.One could easily obtain from q0 a hermitian operator that can serve the purposeof κ-Minkowski time coordinate3, such as x∗0 ≡ q0 − i`/2. But we feel we can stilltake x0 = q0 since the properties of x0 on our kinematical Hilbert space are nottruly observable: they merely provide a way for characterizing the abstract notionof the geometry of empty κ-Minkowski spacetime. As we shall show in Sec. 8.5, thephysical properties of κ-Minkowski spacetime, the ones affecting the analysis of thephysical Hilbert space, will have to be formulated in terms of operators that commutewith the Hamiltonian constraint, written in terms of (8.26), and the κ-Minkowskitime coordinate is not one such observable. Moreover, when we are interested inthe κ-Minkowski time coordinate as a partial observable [53] on the physical Hilbertspace we shall inevitably find that the most meaningful features are to be phrasedin terms of differences among values of this operator (reflecting the evident fact thatthe physical content of the notion of “time” all resides in time differences/intervals),so that the choice between x0 = q0 and x∗0 ≡ q0 − i`/2 is intangible.

8.4 Fuzzy points, translation transformations and rela-tive locality

In the previous sections we introduced a novel pregeometric description of all theingredients needed for describing “points” in κ-Minkowski and for examining thesefuzzy points from the perspectives of pairs of distant observers in relative rest,observers connected by a pure translation. In this section we shall show that κ-Minkowski, contrary to what might appear when looking naively at its commutationrelations, affords us a fully relativistic description of distant observers, and providesthe first ever example of relative locality in a quantum spacetime.

8.4.1 Fuzzy points

First we need to give a description of points in κ-Minkowski. We of course expectthat they should not be the sort of sharp points available in a classical geometry.Evidently within our pregeometric description a point will be identified with a statein the pregeometric Hilbert space that gives rather well determined values to x0 andx1. It is indeed easily seen that no state in the pregeometric Hilbert space givesabsolutely sharp values to x0 and x1: in light of x0 = q0, x1 = q1e

`π0 a sharplyspecified x0 requires an eigenstate of q0 but on such eigenstates of q0 one has thatπ0 is infinitely fuzzy (δπ0 ∼ ∞) which in turn implies that x1 = q1e

`π0 cannot besharp. So all points in κ-Minkowski must be fuzzy4.

3Note that x∗0 is a good choice of κ-Minkowski time coordinate, since [x1, x∗0] = i`x1. And

one also easily verifies that κ-Minkowski described by x1, x∗0 has good properties under boosts,

[B.x1, B.x∗0] = i`B.x1 (or N.[x1, x

∗0] = i`N .x1), and under translations, [T .x1, T .x

∗0] = i`T .x1

(or Pµ . [x1, x∗0] = i`Pµ . x1).

4We shall pay little attention to the fact that actually there is an exception to this “fuzzinesstheorem”: the interested reader can easily verify that the origin of the observer, x0 = x1 = 0, canbe sharp. This can be straightforwardly added as a limiting case for the discussion we offer in thefollowing, and in particular one finds that even a point that is absolutely sharp in the origin of oneobserver is described by a distant observer as a fuzzy point.

8.4 Fuzzy points, translation transformations and relative locality 149

A class of pregeometric states which is well suited for exploring the properties ofκ-Minkowski fuzziness is the one of gaussian states on our pregeometric Hilbert space.We adopt a “pregeometric-momentum-space description” of these gaussian states,denoted by Ψπµ,σµ,qµ(πµ) so they are given in terms of functions of the variables πµparametrized by πµ , σµ, and qµ:

Ψπµ,σµ,qµ(πµ)=Ne− (π0−π0)2

4σ20− (π1−π1)2

4σ21 eiπ0q0−iπ1q1 , (8.27)

where N is a normalization constant. Essentially π0, π1 have the role of expectedvalues for the pregeometric momenta π0, π1, whereas σ0, σ1 characterize the uncer-tainties for π0, π1. Moreover, as we shall see, q0, q1 determine the expected valuesfor the pregeometric position coordinates q0, q1.

Of course, our main focus of attention will be on establishing how the κ-Minkowskiscale ` affects the results. This is going to be our indicator of the difference betweenclassical Minwkoswki spacetime and κ-Minkowski. We start by noting down howthe scale ` intervenes in the normalization factor N . By imposing 〈Ψ|Ψ〉 = 1, andtaking into account the integration measure (8.24), one easily finds that

N2 = e`π0e−`2σ2

02

2πσ0σ1. (8.28)

We characterize the properties of points of κ-Minkowski spacetime by evaluatingin our gaussian pregeometric states (8.27) the mean values and uncertainties of theoperators x0, x1 as x0 = 〈q0〉 and x1 = 〈q1e

`π0〉, while for the uncertainties δx0,δx1

we can resort to δx0 =√〈q2

0〉 − x20 and δx1 =

√〈(q1e`π0)2〉 − x2

1.Unsurprisingly the κ-Minkowski scale ` turns out to play a particularly significant

role in the properties of the coordinate x1, for which we find

〈x1〉 = 〈q1〉⟨e`π0

⟩= q1e

`π0e−`2σ2

02 ,

δx1 = e`π0

[ 14σ2

1+ q2

1

(1− e−`2σ2

0)]1/2

. (8.29)

Instead for the κ-Minkowski time coordinate x0 there is no `-deformation, with theexception of the constant imaginary contribution of order ` which should be expectedon the basis of the remarks at the end of the previous section (and to which weattach little significance, for reasons also stressed at the end of the previous section):

〈x0〉 = q0 − i`

2 , δx0 = 12σ0

, (8.30)

We notice already at this stage that for fixed values of q0, π0, σ0, σ1 one finds largerfuzziness of x1 at large values of q1, because of the contribution to δx1 by the termwith q2

1 in (8.29). It is however of very limited interest to compare different fuzzypoints in κ-Minkowski: at any distance from the origin we can anyway get pointsas fuzzy as we might desire. The key feature we need to uncover concerns how thesame point is seen by observers close to it and by distant observers.


8.4.2 Translations and Relative Locality

The pregeometric description given in Sec. 8.2 already provides us all that is neededfor implementing a translation transformation on one of our fuzzy κ-Minkowskipoints. Evidently the action for our translation will be of the form T = 1 + dP ,with dP = −ia`µPµ, and we established in Subsec. 8.2 that P0 . f(x) ≡ [π0, f(x)]and P1 . f(x) ≡ e−`π0 [π1, f(x)] whereas a`0 = a0, a`1 = a1e

`π0 . In particular, ourconstruction exposed the simplicity of the differential operator dP , which in previousworks on κ-Minkowski remained hidden behind the rather virulent properties of thegenerators Pµ and of the elements a`µ of the noncommutative differential calculus:within our pregeometric description one has that

dP . f(x0, x1)←→ −iaµ[πµ, f(q0, q1e

`π0)], (8.31)

so this action involves only familiar commutative transformation parameters aµ andstandard translations (acting by commutator) at the pregeometric level.

This allows us to implement translation transformations straightforwardly. Wefind

T . x0 = x0 − a`0 = q0 − a0 ,

T . x1 = x1 − a`1 = e`π0 (q1 − a1) . (8.32)

We can now evaluate the mean values and uncertainties of T . xµ on the gaussianstate (8.27), to find5

〈T . x0〉 = q0 − a0 − i`

2 (8.33)

δ (T . x0) = 12σ0

, (8.34)

and

〈T . x1〉 = (q1 − a1) e`π0e−`2σ2

02 (8.35)

δ(T . x1)=e`π0

[ 14σ2

1+(q1−a1)2

(1−e−`2σ2

0)]1/2

. (8.36)

The interpretation here of course is such that the xµ are operators characterizingthe distance of a given (fuzzy) point from the frame origin of some observer Alice,and then T . xµ are the operators that characterize the distance of that point fromthe frame origin of an observer Bob, purely translated with respect to Alice. Andaccordingly one can deduce the relation between the mean values and uncertaintiesin positions among two distant observers in relative rest by comparing (8.30) to(8.33)-(8.34) and comparing (8.29) to (8.35)-(8.36).

5Of course the same results for mean values and uncertainties of κ-Minkowski coordinates canbe obtained by acting with T on the pregeometric state and evaluating xµ and δxµ in the statethereby obtained. The equivalent alternative we follow, by acting with T on xµ and evaluating themean value and the uncertainty of T . xµ in the original state just allows the derivation to proceeda bit more speedly.

8.4 Fuzzy points, translation transformations and relative locality 151

The main message is contained in our Fig. 8.1. There we show two fuzzy pointsin κ-Minkowski as described by two distant observers. One of the points is nearobserver Alice, while the other one is near observer Bob, purely translated withrespect to Alice. What is shown in figure is indeed obtained by comparing (8.30) to(8.33)-(8.34) and comparing (8.29) to (8.35)-(8.36). There are two main features:(i) the same point appears to be more fuzzy to a distant observer than to a nearbyobserver, and(ii) the point at Alice is not described as being at Alice in the coordinatization ofspacetime of observer Bob, and vice versa the point at Bob is not described as beingat Bob in the coordinatization of spacetime of observer Alice.The second feature, (ii), is essentially the feature of relative locality in the classicallimit we found in chapter 4: one can have consistently relativistic theories wherepairs of points found to be coincident by a nearby observer (or, as in the case hereconsidered, a point found to coincide with the origin of that observer) are insteaddescribed as noncoincident if one uses the inferences about those points by a distantobserver. Feature (i) established for the first time in the literature in our work [43],is a feature of relative locality for the fuzziness of points in a quantum spacetime.And, as shown in figure, it is also a fully relativistic effect: on the basis of thecontent of Fig. 8.1 there is no way to distinguish between Alice and Bob. Aliceattributes to the point at Bob more fuzziness than observed by Bob, and also Bobattributes to the point at Alice more fuzziness than observed by Alice. This unveilsthe nature of κ-Minkowski as a fully relativistic spacetime. Without our morepowerful characterization of κ-Minkowski one could (see e.g. Ref. [139]) think that“κ-Minkowski has a special point, a sort of center, and the origin of the κ-Minkowskipreferred frame should be made coincide with that special point”. Instead we cannow clearly see that no point is special and no observer is special/preferred inκ-Minkowski: all observers in κ-Minkowski have the property that they perceivetheir origin as the point of lowest fuzziness and attribute to distant points fuzzinessproportional to the distance (but then an observer located at one of those distantpoints will again describe that point as the one of minimal fuzziness).


xA1

xA0

L

xB1

xB0

L

Figure 8.1. We illustrate the features of relative locality we uncovered for the κ-Minkowskiquantum spacetime by considering the case of two distant observers, Alice and Bob, inrelative rest (with synchronized clocks). In figure we have only two points in κ-Minkowski,each described by a gaussian state in our Hilbert space. One of the points is at Alice(centered in the spacetime origin of Alice’s coordinatization) while the other point is atBob. The left panel reflects Alice’s description of the two points, which in particularattributes to the distant point at Bob larger fuzziness than Bob observes (right panel).And in Alice’s coordinatization the distant point is not exactly at Bob. Bob’s description(right panel) of the two points is specular, in the appropriately relativistic fashion, to theone of Alice. The magnitude of effects shown would require the distance L to be muchbigger than drawable. And for definiteness in figure we assumed π0 ' 2σ0 and σ1 ' σ0.

8.5 From kinematical to physical κ-Minkowski Hilbertspace

In section 8.3 we established the form of the “on-shellness operator” (the quantumHamiltonian constraint) H in terms of a deformed d’Alembertian operator (8.26), as

H =(2`

)2sinh2

(`π02

)− e−`π0 π2

1 . (8.37)

But we did not explore the implications of enforcing the Hamiltonian constraint inobtaining the physical Hilbert space. Since we are here interested in the fuzziness ofthe worldline of a physical particle we must progress to that next level. More preciselywe characterize physical observables in our theory of free relativistic quantum particlesin κ-Minkowski spacetime following the covariant prescription adopted in Ref. [53]:we obtain the needed feature of invariance of physical observables under the action ofH by introducing a new scalar product [53], that projects all the orbit of the gaugetransformation generated by H on the same state. This allows to formally refer tostates in the kinematical Hilbert space (but only as representatives of an orbit) andalso allows to describe the physical scalar product as a straightforward modification

8.5 From kinematical to physical κ-Minkowski Hilbert space 153

of the scalar product on the kinematical Hilbert space. For massless particles thisamounts to inserting the covariant-Hamiltonian constraint [53] as follows

〈ψ|φ〉H = 〈ψ|δ (H) Θ(π0)|φ〉 (8.38)

where Θ(π0) specifies a restriction [53] to positive-energy solutions of the on-shellnessconstraint. Accordingly in the “momentum-space representation” one has (also takinginto account of the deformed measure (8.24))

〈ψ|φ〉H =∫e−`π0dπ1dπ0δ (H) Θ(π0)ψ∗(π)φ(π)

We focus here on the case of a localized massless particle, describable in termsof the classic example of a gaussian state6

Ψq0,q1(πµ;πµ, σµ)=Ne− (π0−π0)2

4σ20− (π1−π1)2

4σ21 eiπ0q0−iπ1q1 ,

where N is a normalization constant

N−2 =∫e−`π0dπ1dπ0δ (H) Θ(π0)|Ψq0,q1(πµ;πµ, σµ)|2

and Ψq0,q1 is evidently written in the momentum-space representation, with pa-rameters π0, π1, σ0, σ1, q0, q1 (these parameters all have the same standing, butour notation is such that q0, q1 are highlighted, since the issue of localization ofthe particle, here particularly crucial, is predominantly connected with those twoparameters).

Our Ψq0,q1 labels a state on our physical Hilbert space of the covariant formulationof relativistic free-particle quantum mechanics, so it physically identifies a fuzzyworldline [53], as it shall be evident also from what follows. The expectation for ameasurable quantity in our physical state identified by Ψq0,q1 will be given by

〈Ψq0,q1 |O|Ψq0,q1〉H

where O is the self-adjoint operator describing the relevant observable.The next hurdle we must face concerns the identification of a well-defined

observable suitable for the characterization of the fuzziness of the worldline. Theapparently obvious choices, x1 and x0, are actually not suitable for this task, sincethey are not self-adjoint operators on our physical Hilbert space (in particular theydo not commute, with H). We propose to remedy this by focusing on the following“intercept operator” A:

A = e`π0

(q1 − V q0 −

12[q0, V]

)(8.39)

where V is short-hand for

V ≡(∂H∂π0

)−1∂H∂π1 .

6Of course in the massless-particle limit, here of interest, one must proceed cautiously:Ψq0,q1(πµ;πµ, σµ) must replaced by Ψα

q0,q1(πµ;πµ, σµ) = exp(−α/π2

0)Ψq0,q1(πµ;πµ, σµ) with al-pha a small infrared regulator which never actually matters in the results we here exhibit.


One may notice that A is describable as an `-deformed Newton-Wigner opera-tor [89]. And it is well known that within special-relativistic quantum mechanicsthere is no better estimator of localization than the Newton-Wigner operator (it canonly be questioned for localization comparable to the Compton wavelength of theparticle [89], but this merely conceptual limit of ideal localization is evidently irrele-vant for the level of localization achieved by particle production at, e.g., quasars).For our purposes it is important to notice that A is a good observable on our physicalHilbert space (self-adjoint, commuting with H) and in the classical limit A reducesto the observable x1,cl − vclx0,cl (label “cl” on quantities pertaining to the classicallimit). So for the case of free particles we are here considering it gives the interceptof the particle worldline with the x1 axis. We adopt the interpretation that A givesthe intercept also in our quantum theory.

Let us focus, for conceptual clarity, on the analysis of the properties of A for thecase of Ψ0,0, i.e. for q0 = 0, q1 = 0. One then easily finds that

〈Ψ0,0|A|Ψ0,0〉H = 0

so this is a case where the particle intercepts the observer Alice in her origin.The fact that this intercept is fuzzy reflects the fuzziness of the worldline described

by Ψ0,0, and in particular the leading `-dependent contribution to this fuzziness ischaracterized by

δA2[`] =

(〈Ψ0,0|A2|Ψ0,0〉H

)[`]≈ `〈π0〉σ−2/2 (8.40)

where for simplicity we assumed (as we shall so throughout) that σ1 is small enough,in comparison to σ0, π1, to allow a saddle point approximation in the π1 integration;then σ (without indices) is the effective gaussian width after the saddle pointapproximation in π1: σ−2 ≡ σ−2

1 + < V >2 σ−20 .

In our proposed interpretation of the formalism Eq. (8.40) gives the fuzzinessof the worldline “at Alice” (at the point of crossing the origin of Alice’s referenceframe). The main objective of this study is to establish whether observers reachedby the particle at cosmological distances from Alice will observe bigger fuzziness. Wecharacterize such observers as those who are connected to Alice by a pure translation,so that for them the state of the particle is Ψa0,a1 , and are, like Alice, such that< A >= 0, i.e. 〈Ψa0,a1 |A|Ψa0,a1〉H = 0. Finding these observers amounts to findingthe translation parameters a0, a1 such that 〈Ψ0,0|T−1AT |Ψ0,0〉H = 0, where T is theone defined in Sec. 8.4.2 (see in particular Eq. (8.31)). Of course, this leads to aone-parameter family of solutions (the family of observers “on the worldline”), whichunsurprisingly takes the form a1 = 〈V〉a0.

Crucial for us is that these observers with vanishing expectation value for theintercept find different values for the uncertainty in the intercept δA:

δA2[`] =

(〈Ψa0,〈V〉a0 |A2|Ψa0,〈V〉a0〉H

)[`]≈

≈(`〈π0〉2σ2 + `2a2

0σ2)

(8.41)

So we do have here a quantum-spacetime picture, and an interpretation of relativisticquantum mechanics in such a spacetime, providing the main ingredient of the heuristic

8.5 From kinematical to physical κ-Minkowski Hilbert space 155

arguments that have so far inspired spacetime-fuzziness phenomenology: one canin fact interpret our observer Alice, the observer on the worldline for whom thefuzziness of the intercept takes the minimum value, as the observer at the source(where the particle is produced), and then the intercept of the particle worldlinewith the origin of the reference frames of observers distant from Alice (where theparticle could be detected) has bigger uncertainty.

While this most crucial qualitative agreement is present, our quantum-spacetimepicture provides a quantification of the relevant effects that differs from what hadbeen suggested heuristically. In Sec. 5.2 we mentioned two heuristic formulas asbeing considered most actively for the description of the gravity-induced contributionto the uncertainty in the localization of a particle after propagating over a distancex:

δx[`] ∼ `α x1−α (8.42)

andδx[`] ∼ `αEα x , (8.43)

with α, assumed [28, 29, 30, 31, 32, 26, 27] to take values between 1/2 and 1, beingthe single parameter which should discriminate in this respect among differentproposals for a quantum spacetime. While, for what concerns energies, the heuristicarguments assume [28, 29, 30, 31, 32, 26, 27] that there should be an irreduciblequantum-gravity-induced contribution to the uncertainty, governed by a law of theform

δE[`] ∼ `α E1+α . (8.44)

However, our quantum-spacetime picture provides a quantification of the rel-evant effects that differs from what had been suggested heuristically. A pivotalrole in the characterization of these difference is played by our result for the en-ergy uncertainty δE. In Refs. [43] we established that π0 does have a standardrole of energy for particles in our κ-Minkowski spacetime. And by evaluating〈Ψa0,〈V〉a0 |π2

0|Ψa0,〈V〉a0〉H − 〈Ψa0,〈V〉a0 |π0|Ψa0,〈V〉a0〉2H we find

δE2 ' σ2 − 2Èσ2 (8.45)

So, contrary to what had been unanimously assumed on the basis of heuristicarguments, in our quantum spacetime there is no irreducible Planck-scale contributionto the energy uncertainty (since in our gaussian states σ can be made arbitrarilysmall).

Phenomenologically, the fact that we found no irreducible Planck-scale contribu-tion to the energy uncertainty renders completely inapplicable to κ-Minkowski theexperimental bounds on spacetime fuzziness derived in Refs. [28, 29, 30, 31, 32, 26, 27].As mentioned before the link from Planck-scale spacetime fuzziness to Planck-scale-induced energy uncertainties was conjectured heuristically on the basis of howoperatively one would derive the notion of energy from (in this case fuzzy) spacetimemeasurements, so essentially the arguments were of the sort of error propagationin a classical theory. But we know that quantum uncertainties often do not followthat logic: classically one would assume uncertainties in the components of angu-lar momentum should produce an uncertainty for total momentum, but quantummechanically one has total-momentum eigenstates which are not eigenstates of the


components of angular momentum. It seems κ-Minkowski is an example wherespacetime is irreducibly fuzzy but energy can be sharp.

Our result (8.45) also plays an important indirect role in characterizing, withinthe κ-Minkowski framework, the dependence of worldline fuzziness on energy anddistance of propagation. Indeed combining (8.41) and (8.45) we find that thePlanck-scale contribution to the localization uncertainty can be approximated asfollows:

δA[`] ' `D δE (8.46)

where we only included the contribution growing with the propagation distance (whichof course eventually dominates) and we used D = a0 since the translation parameteris just the propagation distance (any difference from this would only manifest itselfat subleading orders in `). The key aspect to be noticed in this result (8.46) is itsdependence on the energy uncertainty. the literature on heuristic characterizations ofspacetime fuzziness had been so far inspired by the intuition [28, 29, 30, 31, 32, 26, 27]that the fuzziness of worldlines should either be independent of any aspect relatedto the energy of the particle, following a law of type ∼ `α D1−α, or depend on theenergy itself with laws of the type `αDEα. Since our result (8.46) prescribes a lineardependence on the Planck length it should be compared with the cases α = 1 ofthese previous expectations. But quantitatively it provides a picture of worldlinefuzziness which is not like anything that had been envisaged by previous heuristicarguments: whenever D > 1/δE it predict effects stronger than fuzziness at level` (corresponding ∼ `α D1−α for α = 1) but, since typically δE < E, weaker thanthose expected for the estimate `DE (i.e. `αDEα for α = 1).

157

Chapter 9

Conclusions

This thesis work was devoted to address three significant challenges that were facedby research on Planck-scale-deformed relativistic symmetries before our work started:

• (I) The introduction of a scale of spacetime curvature in theories with deformedrelativistic symmetries.

• (II) The formalization of DSR theories in a quantum mechanical framework,i.e. the inclusion of ~ (the Planck constant) in the theory.

• (III) The clarification of the fate of locality in DSR theories.

Our main results relevant for issue (I) we reported in Ch. 7, based on the study weannounced in Ref. [44]: the compatibility of DSR with the introduction of a constantexpansion-rate (as in De Sitter spacetime) was establishing the first ever formulationof a relativistic theory of worldlines of particles with three nontrivial relativisticinvariants, a large speed scale c (“speed-of-light scale”), a large distance scale H−1

(inverse of the “expansion-rate scale”), and a large momentum scale `−1 (“Planckscale”). This led us to consider some of the phenomenological consequences relatedto the observation of times of arrival of arrival of ultra-energetic photons emittedfrom cosmological distances (as in gamma ray bursts). We find that at distances forwhich the assumption of a constant rate of expansion is a good approximation (andwhere significant data are available) the predictions for the time-of-arrival featuresallow us to discriminate between different models.

Our main results relevant for issue (II) were reported in Ch. 8, based onthe study we announced in Refs. [43, 46]. We proposed a quantum mechanicalformalization of DSR theories for the specific case of κ-Minkowski non commutativespacetime and the related κ-Poincaré description of spacetime symmetries. Weachieved such a characterization relying on a covariant formulation of quantummechanics [53], in which space and time coordinates can all be described as operatorsacting on a kinematical Hilbert space, while the physical dynamic is obtainedenforcing the on-shell constraint. We have found that to obtain a satisfactoryquantum mechanical description of κ-Minkowski/κ-Poincaré one needs to definea pregeometric representation both of κ-Minkowski/κ-Poincaré Hopf algebra andof the their non-commutative differential calculus. We obtained in this way aconstructive characterization of spacetime fuzziness, which has interesting links with

158 9. Conclusions

the possibility of testing Planck scale effects through the observation of blurring ofimages from distant quasars [28, 26].

For obtaining these results, a crucial role was played by our understanding ofhow DSR theories affect spacetime locality. It was not until our works [11, 38](see also [47]), that it was realized that some known puzzling features of locality inDSR theories (item (III)) needed to be understood as part of a logically-consistentrelativistic picture, the “relative locality” scenario. Relative locality is the spacetimecounterpart of the DSR-deformation scale ` just in the same sense that relativesimultaneity is the spacetime counterpart of the special-relativistic scale c (scale ofdeformation of Galilean Relativity into Special Relativity). And one of the strikingconsequences of relative locality is that events established to be coincident by nearbyobservers may appear to be non-coincident in the description of those events givenby distant observers on the basis of their inferences about the events (such as theirobservation of particles originating from the events). Relative locality is introducedin Ch. 4, where it is illustrated for the simplest case of free particles motion in DSRtheories.

Among the results obtained in this thesis, we contributed also to clarify thedescription of interactions in DSR theories, which is the subject of Ch. 6. It has beenshown in [48], that it is possible to describe interactions in DSR theories througha Lagrangian approach, in which the DSR relative locality features are encoded inthe curvature of momentum space, and the deformed momenta conservation lawsare introduced through boundary terms in the action. This approach, in which themain focus is on the momentum space rather then on the spacetime, has been called“principle of relative locality” by the authors of [48, 49]. Our main contribution hasbeen that of clarifying how translational invariance can be enforced, even in thedeformed case, in presence of sequences of causally connected interaction events.

While addressing the challenges (I),(II),(III) is surely as step in the direction ofdeveloping more robustly the DSR framework, we believe that a satisfactory DSRapproach to Quantum Gravity should ultimately accommodate a geometrodynamicdescription of spacetime and gravitation. This goal is still far, but our contributionperhaps brings it somewhat closer. And there is at least one more intermediatestep that can be taken on the basis of the results we here reported. The twomain extensions of DSR formalism we carried out in this thesis, the inclusion of acurvature/expansion scale, and the inclusion of ~, have been obtained independentlyof one another. The next step in the direction of a quantum gravity would beto combine these two features, thereby obtaining a DSR description of quantumparticles in a curved/expanding spacetime.

It would also be interesting (even temporarily not taking into account quantumeffects) to generalize the results for constant rate of expansion here reported in Ch. 7to the case of a FRW-type time-dependent expansion rate. This would be a valuablestep toward possible applications of the DSR proposal in cosmology.

And the results we here reported in Ch. 6 concerning causally-connected interac-tions in a flat spacetime (with curved momentum space) should provide the basis forproducing a similar description of causally-connected interactions in an expandingspacetime.

159

Acknowledgements

161

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