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The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The disappearance and re-emergence ofspace and time in quantum theories of gravity
Christian Wüthrich(based on joint work with Nick Huggett)
University of California, San Diego
Black Forest Summer School:Physics and Philosophy of Time
26 July 2013
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Organization of lecture
1 The disappearance of space and time in QGSpace and time in canonical QGSpace and time in causal set theory
2 Empirical incoherence and the emergence of spacetimeThe threat of empirical incoherenceThe emergence of spacetime in causal sets
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The central problem... particularly if one insists on a fundamental role for spacetime
ThesisMany approaches to quantum gravity (QG) suggest or imply thatspace and time do not exist at the most fundamental ontological level.Thus deprived of their former status as part of the fundamentalfurniture of the world, together, perhaps, with quarks and leptons,they merely ‘emerge’ from the deeper physics that does not rely on,or even permit, their (fundamental) existence, rather like tables andchairs.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Time in physics
Time in physics does not have manyof the features we would usually thinkof as essential for time:
It is isomorphic to (an openinterval of) R, or imposes atotal order on events.
It contains aphenomenologically specialmoment—the ‘now’.
It has a dynamical quality—a‘flow’.
It is asymmetric, i.e., it has a‘direction’.
ThesisPhysics has denied all of these.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Physical spaceHenri Poincaré, Science and Hypothesis
In the first place, what are the properties ofspace properly so called?... The followingare some of the more essential:—1st it is continuous; 2nd, it is infinite;3rd, it is of three dimensions; 4th, it ishomogeneous—that is to say, all its pointsare identical one with another; 5th, it isisotropic. (1905, 52)
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Why we need a quantum theory of gravity
Source: http://blog.michaelkcooke.com/
General relativity (GR) can’t be thelast word: quantum effects must betaken into consideration in earlyuniverse and evolution of blackholes
⇒ must find theory that combinesquantum effects and (strong)gravitational fields
⇒ need a quantum theory of gravity
various approaches, string theoryand loop quantum gravity (LQG)most visible, causal set theory
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Mapping QGSource: http://universe-review.ca/F15-particle.htm
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
A road to unification for someSource: http://sciencereview.berkeley.edu/articleex.php?issue=18&article=features_04_quantum
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
The two main rivals: (1) String theory
Brian Giberson, “String Theory”
starts out from the standard modelof particle physics, a quantumtheory
sense in which gravity can be saidto be ‘naturally’ included; details farfrom clear
originally developed assuming aspacetime background (as inertcontainer), but questions havebeen raised about it (dualities:physically equivalent stringtheories use different spacetimes)
outstanding problems: ‘landscapeproblem’, repeated falsepredictions of ‘supersymmetricparticles’
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
The two main rivals: (2) Loop quantum gravity (LQG)
Source: http://www.cpt.univ-mrs.fr/∼rovelli/
starts out from GR,incorporates its core lessons(dynamical spacetime)
tries to incorporate quantumeffects by ‘quantizing’ GR
outstanding problems: (1)problem of time/ill-understooddynamics, (2) classicallimit/emergence of spacetimefrom basic quantum structure
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
The dissolution of spacetime in QG
In string theory as well as in loop quantum gravity (LQG) (andother approaches to QG), indications are coalescing that spaceand time are no longer fundamental entities as classicallyconceived (substantivally or relationally), but merely ‘emergent’phenomena that arise from the basic physics.
In particular (as in LQG): no continuous spacetime, discretestructure
In the language of physics: spacetime theories such as GR as‘effective’ and spacetime itself ‘emergent’, much likethermodynamics is an effective theory and temperature isemergent, as it is built up from the collective behaviour of gasmolecules.
Unlike the fact that temperature is emergent, the idea that theuniverse is not in space and time shocks our very idea ofphysical existence as profoundly as any scientific revolution.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
The project‘Emergent Spacetime in Quantum Theories of Gravity’
philosophers have only just started tostudy the implications of QG and thedisappearance of spacetime
Funded project with Nick Huggett(University of Illinois, Chicago):‘Emergent Spacetime in QuantumTheories of Gravity’
joint framework, Huggett studiesstring theory, Wüthrich does LQG,causal set theory
The remainder of this talk focuses onmy contribution to this project.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
On the conceptual foundations of LQG
LQG attempts to transpose the central lesson of GR into aquantum theory
central innovation of GR: spacetime isn’t fixed ‘background’,which determined the inertial forces, but a dynamical structurewhich interacts with matter
LQG is based on a reformulation of GR as a so-called‘Hamiltonian system’, which re-interprets spacetimes as(3 + 1)-dimensional (instead of 4-dim), with constraints
⇒ forces a ‘foliation’ of spacetime by an equivalence relation in3-dim ‘spatial’ hypersurfaces, parametrized by 1-dim ‘time’
3-dim ‘space’ considered as dynamical system which evolvesover ‘time’; 3-dim hypersurfaces represent (instantaneous)states of dynamical theory
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Source: http://sciencereview.berkeley.edu/articleex.php?issue=18&article=features_04_quantum
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
But hasn’t relativity taught us that no one shall put asunder whatMinkowski has joined together, i.e. that no dissection ofspacetime into space and time can be privileged?
⇒ True, but in order to make up for this, the Hamiltonian systems issubject to ‘constraints’, which mathematically ‘undo’ thedissection.
Limitations of approach: not all models of GR are considered(only globally hyperbolic vacuum models).
(Dynamical equations play somewhat different role, as we’ll see;matters of interpretation and observation will arise)
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
The problem of time in canonical quantum gravity(or at least one version of the problem)
Thesis (The problem of time)
If we only accept a few seemingly innocuous assumptions, then weare forced into accepting that, fundamentally, there cannot be time, orat least that there cannot be genuine change.
genuine physical magnitudes: Dirac observables
have to commute with Hamiltonian⇒ must be constants ofmotion
⇒ There cannot be change even in the very minimal sense that thephysical quantities can have different values at different times.
⇒ last remnant of a dynamical aspect of time is lost
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
The problem of time in canonical quantum gravity
Problem arises from treating Hamiltonian GR treating as westandardly treat gauge theories, i.e., from treating Diff(M) as agauge symmetry just as gauge groups in gauge theories
Technically: space of initial data is a presymplectic space withgauge symmetries, dynamical trajectories lie in gauge orbits
⇒ change is pure gauge, which means, if gauge symmetries areunderstood to relate mathematically distinct representations ofthe same physical possibility, that there is no change in any ofthe genuinely physical properties of the system
⇒ no change, perhaps no time
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Quantized space‘spin network states’: quantum states of the gravitational field;physical space is supposed to be quantum superposition of spinnetwork states
These spin network states can be represented by labelledgraphs embedded in some space:
2i
1i
1j
2j
3j
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Quantized space: spin network states
(To be precise: we should really be looking at abstract graphsand equivalence classes of spin network states, as how the spinnetworks are embedded in manifold is physically irrelevant.)
‘spin’-representations on vertices (represented by nodes ingraph) correspond to ‘size’ of the ‘space atoms’, those on edgescorrespond to ‘size’ of the surface connecting adjacent ‘chunks’of space
⇒ space is ‘granular’ at Planck scale
⇒ The smooth space of the classical theory is supplanted by adiscrete quantum structure.
⇒ Space, as it figures in our conceptions of the world, is emergentphenomenon.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Causal set theory and its motivations
In a nutshellCausal set theory is based on the assumption that the fundamental(spatio-temporal, gravitational?) structure is a discrete set of basalevents partially ordered by causality.
Theorems by Hawking et al. (J Math Phys 17 (1976): 174-181) andMalament (J Math Phys 18 (1977): 1399-1404): given causal structureand volume information, one finds metric gab of the manifoldM;same is true for dimension, topology, and differential structure
IOW: causal structure determines geometry (but not ‘size’)
⇒ motivates causal set approach: the fundamental structure is acausal set
discreteness of fundamental structure has technical andconceptual utility
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Causal set theory: the axioms
Axiom (Causal set theory)
The fundamental structure is a causal set, i.e. an ordered pair 〈C,≺〉consisting of a set C of elementary events and a relation, denoted bythe infix ≺, defined on C satisfying the following conditions:
1 ≺ induces a partial order on C:irreflexivity: ∀a ∈ C,¬(a ≺ a)antisymmetry: ∀a,b ∈ C,¬(a ≺ b and b ≺ a)transitivity: ∀a,b, c ∈ C, if a ≺ b and b ≺ c, then a ≺ c
2 local finitarity: ∀a, c ∈ C,card({b ∈ C|a ≺ b ≺ c}) <∞
Remarks:
antisymmetry⇒ no closed timelike curves
local finitarity⇒ structure must be discrete
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Representing causal sets: (almost) a Hasse diagram
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Two major challenges
1 Inverse problem (sometimes ‘entropy crisis’) present in discreterelational approaches to QG (causets, graphs, etc): vast majorityof basic structures are not approximated by a relativisticspacetime (in fact, the larger the causets the bigger the problem)
2 find dynamics in some principled and physically motivated waysuch that the dynamics will ‘drive’ the causal sets to those whichdo approximate a low-dimensional spacetime
so far, only classical dynamics (no quantum interferenceeffects taken into account)known dynamics do not tend to result in causal set whichgive rise to manifold-like spacetimes
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Time in causal set theory
fundamental relation is causal, not temporal; temporal relationsare supposed to emerge
⇒ Strictly speaking, there is no time in causal set theory.
Otherwise, ≺ looks somewhat like time (it doesn’t pick a Now ora Flow):
1 Its antisymmetry and irreflexivity conjointly implyasymmetry (∀a,b ∈ C,a ≺ b → ¬(b ≺ a)).
2 discrete, not continuous (not isomorphic to open interval ofR)
3 It orders events only partially, not totally.
⇒ looks like a discrete B-theoretic version of (special-)relativistictime without metric relations (durations)
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Space in causal set theory
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Space in causal set theory
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Space in causal set theory
set of all ‘spacelike-related’ events to ‘here-now’ h: events thatdo not causally precede or are causally preceded by ‘here-now’
Problem: in the green set, ∃ pairs of events which are causallyrelated
⇒ ‘space’ should be subset of these, such that no such pairsremain
Definition (Causal-set space)
Space as judged from the ‘here-now’ is the maximal set of events in Ccontaining h such that no two elements are causally related.
FactThe resulting subset of C is structureless—it is just a set of eventswith no relations defined on it. Technically, it is an ‘antichain’.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Recovering geometry: topology
S A Major, D Rideout, S Surya. Classical and Quantum Gravity 23 (2006): 4743.
D Rideout, P Wallden. Classical and Quantum Gravity 26 (2009): 155013.
⇒ A maximal antichain is completely characterized by itscardinality.
recovering topology: (future-)thicken antichain A to
An := {x ∈ Fut(A) ∪ A : |Past(x) \ Past(A)| ≤ n},
for some n ∈ N, similarly for (past-)thickened antichain. Defineopen sets of A as sets of events to the past (future) of events inAn \ A⇒ (coarse) topology
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Recovering geometry: metric
recovering metric (spacelike distances):
)y(+J\) x(+J
)y({J\) x({J
z
w
yx
problem: non-uniqueness
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
The problem of space in causal set theory
space by itself has no dimensions: usual dimension estimatorsfor embedding causal sets into d-dimensional Minkowskispacetime start out from length of causal chains, i.e., there is noteven a proxy for dimension
it has no affine or differentiable structure: causal sets arediscrete, so even the whole of a causal set has no such structure(except, perhaps, in trivial way in that structureless points can beconsidered a zero-dimensional affine space)
it also has no metric structure: there simply isn’t anything whichsupport ‘distances’ in this structureless set—metric emerges(together with the rest of the essential features) in the limit ofrelativistic spacetimes
non-locality: discreteness of the causal set and the Lorentzinvariance demanded of the emerging spacetime conspire torender the physics non-local in a way unfamiliar to generalrelativity.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Discreteness + Lorentz symmetry = Non-locality
p
r
q
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
Space and time in canonical QGSpace and time in causal set theory
Conclusion⇒We have a ‘problem of space’!
i.e., at the emergent, low-energy level, we seem to have a richlystructured space, for which, however, there is no analogue at thefundamental, high-energy level (at least not a structuredanalogue).
Recall that this is analogous to the problem of time/change(although, to repeat, I don’t claim that the problem of space isjust as puzzling as the problem of time).
Admittedly, structure (e.g. topological) is induced on space, i.e.,on the antichain, by causally ‘thick’ space.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
Are spacetime-less theories empirically incoherent?
Definition (Empirical incoherence)
A theory is empirically incoherent just in case the truth of the theoryundermines our empirical justification for accepting it as true. (cf.Barrett 1999, §4.5.2)
A theory denying the fundamental existence of spacetime is thusalleged to be empirically incoherent because
the empirical justification of a theory derives only from ourobservation of the effects of a localized something situatedin spacetime;yet if such a theory were true, there could be no suchlocalized something.
Barrett, Jeffrey A., The Quantum Mechanics of Minds and Worlds, Oxford University Press (1999).
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
Saving empirical coherence
Nick Huggett und CW, „Emergent spacetime and empirical (in)coherence”, Studies in the History andPhilosophy of Science (in press).
Avoiding the charge of empirical incoherence requires that atheory must be shown to admit emergent spacetime,
i.e., it must be shown how general-relativistic spacetimeemerges as an approximation in quantum theories of gravity
This task is also important in the ‘context of justification’, as itsdischarge provides an account of why the classical spacetimetheory (GR) was as successful as it was.
This is no mean feat, and quite generally the pièce de résistancefor formulating such a theory!
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
The emergence of spacetime in causal sets
?
iabg ;hMi¹, hC
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
The emergence of spacetime in causal sets
Lee Smolin. The case for background independence. In Dean Rickles et al. (eds.), The StructuralFoundations of Quantum Gravity. Oxford: Oxford University Press (2006), 196-239.
Definition
A causal set 〈C,≺〉 is said to approximate a classical spacetime〈M,gab〉 iff there is an injective function φ : C →M s.t.
1 the causal relations are preserved, i.e. ∀a,b ∈ C,a ≺ b iffφ(a) ∈ J−(φ(b));
2 on average, φ maps one element of C onto each Planck-sizedvolume of 〈M,gab〉.
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
The inverse problem for causal sets (Smolin 2006)
If the mapping φ is a random ‘Poisson sprinkling’ of events onto thespacetime manifold, then it is easy to find a causal set that approximates agiven globally hyperbolic general-relativistic spacetime with boundedcurvature (i.e. below the ‘discreteness scale’ of the causal set, the manifold isapproximately flat).
i¹, hCiabg ;hM
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
How sprinkling randomly saves Lorentz symmetryFrom David Rideout, http://www.phys.syr.edu/~rideout
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
The inverse problem for causal sets (Smolin 2006)
But almost no causal set approximates a low-dimensional spacetime.Theory does neither have the conceptual resources to select (inprincipled way), nor a dynamical principle to generate, those causalsets that approximate low-dimensional spacetime.
iabg ;hMi¹, hC
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
Classical sequential growth dynamics
David Rideout and Rafael Sorkin. Classical sequential growth dynamics for causal sets. Physical ReviewD61 (1999): 024002.
sequential growth dynamics, to produce a probability measure on set ofhistories
growth by stochastic Markov process obeying the following dynamicalconditions:
1 internal temporality: in auxiliary external order of their birth, thenthe events’ labels l ∈ N0 must obey the condition that if x ≺ y ,then l(x) < l(y)
2 discrete general covariance: labeling is unphysical, in thattransition probability between two causal sets related dynamicallyas along a directed path is path-independent
3 Bell causality: roughly, what transpires at spacelike separationfrom the node to which the newborn event is attached does not toinfluence the corresponding transition amplitude
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
Classical sequential growth dynamics
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
The hope is...
...that this framework permits a principled way of assigningtransition amplitudes such that ‘manifold-like’ causal sets willend up with a high probability, at the expense ofnon-manifold-like causets.
Some timid success so far, but much remains to be done.
Major problem: all of this classical; really, the dynamics wouldhave to be quantum, e.g., using quantum approach to sums overhistories
⇒ This will further complicate what space and time are in quantumcausal set theory...
Christian Wüthrich The disappearance and re-emergence of space and time
The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime
The threat of empirical incoherenceThe emergence of spacetime in causal sets
Conclusions
Space and time (or spacetime) seem to vanish in manyapproaches to quantum gravity.
I introduced examples of LQG and causal set theory, but theproblem is generic.
This threatens the empirical coherence of these theories.
Arguably, threat can only be averted by showing how relativisticspacetime emerges from fundamental structure.
This is necessary anyway, to explain why GR was as successfulas it was.
Christian Wüthrich The disappearance and re-emergence of space and time