my.ilstu.edumy.ilstu.edu/~lmiones/LMI_DWT1p.pdf · The Digital World Theory c 1 (The Quantum...

The Digital World Theory c© 1

(The Quantum Matrix)

Version 1.0 : An Invitation!

Lucian Miti Ionescu

Department of Mathematics, Illinois State University,

IL 61790-4520, USA

E-mail address: [email protected]

Virtual Institute for Quantum Entropy and Space-Time,

www.VIReQuEST.com

E-mail address: [email protected]

V I ReQuEST

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Mathematics

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Physics 2.0

The MPCS-Alliance

2006

1www.TheDigitalWorldTheory.com

1991 Mathematics Subject Classification. Primary:81; Secondary:18

Key words and phrases. Quantum physics, quantum mechanics,quantum computing, quantum field theory, Feynman process, entropy,

mind-matter, quantum gravity etc.;abbreviated: QP, QM, QC, QG, QFT etc.

(To my family & ... www!)

3

Abstract. The Digital World Theory (DWT) is intended as atrademark of a new approach in modeling interacting and inter-active systems, from fundamental interactions including quantumgravity, to complex systems like communication networks, i.e. mod-els natural and artificial reality alike!

It gathers various scattered new trends consistent with the au-thor’s own (re)search and philosophy, into a “project description”playing the role of a grant proposal for “A New Kind of Science”([Wolfram] indeed!), prone to “open science development” (seeAnnex B).

The major ”upgrades” of DWT are:

1) There is no apriori space nor time, not even a “Space-Time”:

1800 1900 2000Space × Time Space-Time Causal structure

direct product ”warped” “multi-extension”

The causal structure is rather a “multi-dimensional time ex-tension of space”, and a “linear time” might not exist even locally.

There is no “time flow”, but: Quantum Information Flows !

2) The causal structure with “variable geometry” is a dis-crete model (q-digital!), not a continuum model (“the reals are

not real”): the Quantum Dot Resolution (graded of finite type:

“no functions, but generating functions!”)

3) A new Energy − Information Unifying Principle binding

Einstein’s energy-matter and Information is introduced, based onthe well known balance between entropy and energy. It provides a

common framework for a unified description of Mind and Matter .

4) It is a concerted modeling effort unifying contributions

from Physics and Computer Science at the application level,under the auspices of Mathematics at the implementation level:Computer-Physics 2.0 for [Human 2.0]! It is mandated by theduality:

Quantum Interaction ∼= Quantum Computing ,

at the hardware (matter - quantum computer) and software (physicsmodel - computer science interpretation) levels. It provides an“umbrella description” of interacting systems (System-System) andinteractive systems (System-Observer, i.e. measurements and Observer-Observer, i.e. communications).

This last upgrade is probably the most valuable ideologicalunification, representing the GUT character of the DWT.

Contents

Introduction 111. The DWT unifying ideas 122. What was “Space-Time”? 132.1. It’s all about “Time” ... 152.2. ... measurements. 152.3. What is a Causal Structure? 162.3.1. Zooming in and out on a System 172.3.2. A “new” algebraic-geometry principle 173. About the book 173.1. About the merger of QFT and GR 183.2. String theory and entangled thumbs! 193.3. “Paradoxes” of Quantum Mechanics 203.4. What is new, really 204. Acknowledgments 22

Part 1. User’s Manual 23

Chapter 1. The Search for a New Unifying Principle 251. Models, models, models! 251.1. What do we mean when we ask “What is “time”?”? 251.2. “Is it a particle or a wave?” 261.3. Interpretations of quantum mechanics 261.4. The main “lesson” 262. Three ... revolutions: returning to principles! 272.1. First revolution: 272.2. The second revolution: General Relativity or Quantum

Mechanics? 272.3. Space-Time: Is “motion” possible? 282.4. What is an “Event”? 292.5. Quantum Field Theory 302.6. External/Internal Degrees of Freedom: The Automaton

Picture 302.7. Is there a “time”, after all? 313. A “New” Principle 33

5

6 CONTENTS

3.1. “Mind versus matter” 343.2. Are black holes prototypical? 354. Conclusions 36

Chapter 2. Miscellaneous: a “warm-up” 371. On the measurement “paradox” 372. Modeling a complex process 372.1. The duality: internal - external 372.2. Structural modeling (SM) 382.3. Why Path Models? 383. What is the Information Flow? 394. The “understanding” of QM 405. Is it all in the brain? 416. Classical or quantum mechanics?

... what’s the difference anyway! 417. To think or not to think classically? 438. Add a pinch of ... Einstein relativity! 44

Part 2. Reference Manual 45

Chapter 3. The Principles of DWT 471. Why Path Models? 472. The Causal Resolution 473. The Fundamental Principles of DWT 49

Chapter 4. Causality: Information versus Time flow 51

Chapter 5. Modeling classical information flow 531. Measurement: a “paradox”, or what? 532. Shannon or Von Newmann? 543. Shannon’s measure of information 553.1. What is a (good model of a) quantum system”? 553.2. What is a (good model of a ) classical system? 563.3. Decision trees 563.4. Shannon’s entropy: a framework 583.4.1. It’s all about ... colorful decisions! 593.4.2. Shannon’s axioms for classical entropy 603.4.3. An invariant of trees 604. Probabilities and Partition Functions 624.1. Micro and macro states 624.2. The Law of Large Numbers 644.3. Shannon entropy and typical configurations/sequences 644.4. Partition function 65

CONTENTS 7

4.4.1. Probabilities and real projective space 664.5. Taking absolute values and limits: old habits! 675. The thermodynamics legacy: a hidden message? 675.1. Internal Energy U 685.1.1. What internal energy is not 685.1.2. Is it heat “virtual” work? 685.1.3. Vacuum causal bubbles 695.2. Enthalpy and free energy 695.2.1. Pressure and volume 695.2.2. Enthalpy and free energy 705.2.3. They’re just state functions after all 715.2.4. The energy conservation: a homotopy? 725.2.5. To minimize or to maximize, this is the question 725.3. Entropy - statistical considerations 735.3.1. Ensembles 735.3.2. Statistical entropy 745.4. What do yousay, Mr. Feynman? 745.4.1. Energy: dark or white? 755.4.2. The partition function - revisited 755.4.3. What is energy after all? 765.4.4. The other observables: piece of cake 766. Feynman path integrals and entropy: graphs invariants! 776.0.5. FPI and probabilities 776.1. Entropy and Information Charge/Potential 796.1.1. The “right invariant” goes by the right rule! 806.1.2. Invariants: discrete versus continuum 826.2. Concluding speculations 837. Relative entropy and information channels 847.1. Heisenberg commutation relations 857.2. What is relative entropy 857.2.1. Transition cobordisms 857.2.2. Entropy as a derivation 867.2.3. Entropy as a functor 877.2.4. Relative entropy 877.2.5. Entropy in the Energy Picture 877.3. Overview: Z2/R/C/H-physics 88

Chapter 6. Modeling quantum information flow 911. Constructivism: the critique of the critique ...? 912. Quantum entropy 932.1. Quantum transitions 932.2. Mixed and pure states 93

8 CONTENTS

2.3. Von Newmann entropy 932.4. Non-commutativity of measurements 933. Two good examples 943.1. Example 1 943.2. Example 2 944. Quantum information 94

Chapter 7. Classical and Quantum logic 951. Propositional calculus 952. The lattice: creation and annihilation! 953. The modular identity and ... entropy! 964. Relation to projective geometry 975. Conclusions 975.1. What is the experimental meaning? 985.1.1. Creation or annihilation? 985.1.2. 2nd Law finally tamed! 985.2. What about “tertium”: “datur”, or not? 98

Chapter 8. Quantum dots and bits 1011. Quantum dots: theory and practice 1011.1. What are position and momentum? 1011.2. Quantum jumps 1021.3. Position and momentum incompatibility 1032. Reversibility: information flow and time reversal 1042.1. Category theory and information flow 1043. Categorifying classical mechanics 1053.1. Phase spaces: classical and quantum 1053.2. Classical limits 1053.2.1. ... and generating functions 1063.2.2. ... and actions 1063.2.3. ... and Heisenberg relations 1073.2.4. What about measurements? 1073.3. Exercises 108

Chapter 9. Quantum gravity and information 1091. Artificial Intelligent Geometry 1091.1. Monte-Carlo simulations 1102. Comments on Loop Quantum Gravity 1113. Is “reality” 1-D? 1123.1. Are quaternions the new coefficient-material? 1123.2. All good things come as triples; why? 1133.3. Matter or antimatter? 1144. The Laws of Black Hole Radiation - revisited 115

CONTENTS 9

4.1. Entropy “resides on the surface” 1154.2. QFT and black hole thermodynamics 1154.3. What is missing? 1164.4. Unruh’s Law 1164.5. The first law of BH mechanics 1174.6. The Hwking effect 118

Chapter 10. Conclusions and further developments 1211. Dimensions: 1,2,3 ... ∞! 1222. The Holographic Universe 1223. Time flow is Subjective 1234. Cosmological constant: nowadays Cinderella 1245. What next? 1256. Epilog 126

Bibliography 129

Appendix A. A research diary ... 1331. Why is mass energy: E = mc2? 1332. Information flow revisited 1373. 4D Hyperdynamics 1374. Space-Time duality and Hypersymmetry 1395. QG as a Deformation Theory 1395.1. String Theory as a Deformation Theory 1416. Quantum Digital Gravity 1426.1. Energy-momentum tensors: external and internal 1436.2. More clues ... 1466.2.1. An other possibility ... 1486.3. Quantizing I/E DOFs 1486.3.1. The string field and entropy 1496.4. Currents in String Theory 1496.5. Gravity and electromagnetism: the revival of an old story!1506.5.1. The Quantum Temperature-Entropy Field Theory 1516.5.2. Is the merger of EM and GM possible? 1526.5.3. Variation of the action: manifold too! 1536.6. Bohmian mechanics: a “hidden” message 1536.7. Heat transfer and entropy flux 1556.7.1. Is it you, String Theory? 1566.8. The Bohmian mechanics interpretation 1576.8.1. The quantum dice 1586.8.2. Entropy and String action 1596.8.3. Information current revisited II 1616.8.4. Again, why is energy mass? 162

10 CONTENTS

6.8.5. Bohmian mechanics’ Schrodinger equation 1626.8.6. Interpreting the new “Klein-Gordon equation” 1646.9. Entropy entanglement and Riemann surfaces 166

Appendix B. VIReQuEST: a Virtual Institute 1711. Mathematical-physics and top-down design 1712. DWT ver.2 implementation goals 173

Introduction

The book is an invitation to a collective effort for a new kind oftheoretical approach to understand reality, whether “natural” or arti-ficial.

It is not The Theory of Everything nor a Final Theory! It is a pro-posal for a “Grand Unification”, not only as a unified description of thefundamental forces and of complex systems alike, but also of theoreticalcontributions from Mathematics, Physics and Computer Science.

The DWT expresses a personal point of view; it is a “Big Picture”getting its shape from lots of pieces of the reality puzzle, put togetherwith a lot of ... wishful thinking (Ahem ... “design specifications” Imeant to say :-). Therefore “I conjecture/believe/etc.” are suppressedas ... the default!

One cannot understand any of the foundational concepts, e.g. space,time, matter, information, mind, conscience etc., without finding a uni-fied approach to understand all of them.

To understand Quantum Gravity for instance, we adopt the view ofGauss, Euler, Grothendieck etc. to broaden the goal, rather then tryingto simplify our task and focus on a unification of quantum mechanicsand general relativity only, without modeling the other interactions(e.g. Loop Quantum Gravity [Rovelli-1]; see Section 2, p.111).

It is time for a concerted effort bringing together not the tradi-tional mathematical-physics “team” (with its glory but also deficiencies[Kirrilov], [Gosson]), but also Computer Science the “New World”of science, as it become more and more obvious with the advent ofquantum mechanics and quantum computing, starting perhaps withFeynman (see [Milburn]).

The DWT is also a (grant) proposal towards building the teamwhich will design a unified scientific model of mind and matter, hu-man and computer, life under its various guises. The present “projectdescription” sketches a few new ideas which, as usual, can be later rec-ognized being implicit in various scattered places ... It is not the timefor “priorities”; the priority is for a “new kind of science” [Wolfram].

11

12 INTRODUCTION

The DWT message is that “there is no Time”, universal or not,global or local, meaning that in order to progress beyond the currentstage, science has to develop models which are not based on a “lineartime”, with its companion assumption of an “orthogonal space” (“ ...if it’s not parallel it must be sequential”!?), i.e. we have to develop theTheory of Causal Structures, beyond Markov and Feynman’s pictures,causal structures which are not a (warped) product of space and time,not even locally.

The historic regret after the “lost paradise” of deterministic science,should be compensated by the excitant of recovering the “freedom” ofinformation ... flow, with the hope of taming the “apparent chaos”which is the trademark of efficient communication protocols ...

The double parenting of the “new science” (Physics and ComputerScience) 1 brings the dual interpretation of “change” as physical in-teraction and communication, e.g. particles and waves or informationsources and messages, timing of events or process scheduling and in-formation flow. Obedient as always (or rather with hind site!) math-ematics moved away from static Space-Time models to models with“variable geometry”.

And speaking of the “Science Drift Theory”, the Computer Science- Mathematics alliance might be able to stop the current drift of thescience’s focus/interest from physics to biology [ST].

1. The DWT unifying ideas

The main unifying and upgrading ideas fall into two categories:I) Methodology:

1) a broad goal and theoretical technology used (unified math/phys/ CS approach): in order to design the theory to last a few decades(not just another resonance / “theoretical physics cycle”).

2) Building theories for machines, for higher processing power(e.g. lattice QCD for computability [Mackenzie]), with care for theHuman Interface, (with a conceptual high-level language descriptionfor physicists). It is thought of as “Physics 2.0” for [Human 2.0], i.e.new software for the new human-machine merger in the near future;

3) A top-down design approach (necessary for the stratified com-plexity of the future’s theoretical constructs), in order to make it“upgradable” and to allow each specialist to design his/her own layer,within his/her own level of expertize, and with care for the interfacebetween layers;

II) Content:

1Therefore it needs a new name: Compu-Mathical-Physics!? :-(

2. WHAT WAS “SPACE-TIME”? 13

1) The overall philosophy can be summaries as:

(Modeling)Reality IS Quantum Computing.

This reflects the dual interpretation of mathematical models: QuantumPhysics and Computer Science. More specifically:

Quantum interaction = Quantum communication.

2) A New Unifying Principle, beyond Einstein’s E = mc2 unificationof energy and matter, stating the balance of entropy/information andenergy.

If almost a century of efforts to incorporate gravity as an exchangeinteraction (force-field) failed, perhaps it is time to try an alternativeapproach. Gravity is a theory of space-time, i.e. of causality, andquantum mechanics emphasized the role of measurement and that ofthe observer.

The idea that gravity is related to the measurement paradox [Penrose]and it is an organizational principle acting against the entropic princi-ple [Davis1] is promoted to a goal:

Gravity and Mass are Entropic Effects.

Interpreting an interaction as a quantum communication requires a uni-fying point of view such that any interaction, system-system, system-observer and observer-observer, be treated on an equal footing, i.e. asa quantum communication.

Therefore a quantum theory of space-time must incorporate entropyand information at its foundations. Then space-time is not a “fixedmodel”, independent of the observer’s modeling goals; it is a device tokeep track of degrees of freedom (DOFs / memory to store information).This is only half of the picture: the external DOFs; the other half,keeping track of internal DOFs, comes with a duality ... then energyand information are the two faces of the same coin!

2. What was “Space-Time”?

The author’s search for a substitute of space-time as a way to resolvethe incompatibility between general relativity and quantum mechanicsevolved through several “stages of enlightenment”.

The categorical language as an “object-oriented language” was clearlythe starting point, and a categorical analog for the “spectrum of thealgebra of observables as a substitute for space seemed highly desirableand sketched in [Ion00] to express Feynman’s ideology that one shouldthink of quantum processes as “histories”, or paths. Of course, this (thePath Model) is a disguise of a states and transitions approach, i.e. the

14 INTRODUCTION

Automaton Picture (again quantum computing!). It became clear thatFeynman’s paths are in fact a substitute for “space-time”, and later on[VIRequest-UP], that all one needs (and can hope for) is a “causalstructure” (there is no “Time”, really - in the modern models, that is:there should be no global “internal time”).

While studying Kontsevich’s graphs as a tool in deformation quanti-zation [K97] and Kreimer’s use of graphs in renormalization [Kreimer](“IHES illumination period”), the author realized the additional im-portance of graphs, not just as a perturbational device in QFT, butone which enables a Scaling Principle (“zoom in/out” on a system).Graphs were providing something like a resolution of a ... “point”(!?). The “missing words” (of encouragement) were found in Gelfandand Manin’s book [GM], p.6) (where else?); the message (for me) was“Forget the space(-time), all you need is a resolution”!

Now, classical “time”, representing our bold conviction that we canput a linear order on phenomena, was longtime forgotten, in the searchof a less strict requirement to represent causality (causal structures, e.g.Feynman graphs etc.), in a homological algebra vain by “generators andrelations”. So a new principle of modeling “space-time”, and in generalcausal structures, thought of as the structure modeling the “externaldegrees of freedom” (EDOF for short), was adopted.

Definition 2.1. A Causal Structure is a resolution of a “point”(“The System” S), with duality between external and internal degreesof freedom, called the Quantum Dot Resolution (QDR).

It provides a substitute for “Space-Time” (we avoid further techni-cal terms at this point). The duality refers to the capability of “trad-ing” internal (I) and external (E) DOFs, since collapsing subgraphs(“zoom out” on the system’s model) leads to “loosing” EDOFs (andtherefore information!), which must be compensated by IDOFs.

This Scaling Process implemented by collapsing of graphs enables ageneralized micro-macro observable correspondence (see Boltzmann’sapproach to entropy), requiring/enabling the introduction of a “bal-ance between entropy and energy, and showing that a “flexible model”of “space-time” (no more fixed configuration spaces) is both an “effec-tive theory” and an “exact theory” at each scale level. This approachshould allow, it is hoped, to connect with the black-hole radiation laws,which hint to a “discrete structure” of space-time (which loop quantumgravity actually gets to [Rovelli-1] - with a lot of work, though!).

The differences between “deformations” (perturbation) and resolu-tions, both aspects present in the FPI perturbative approach to QFT

2. WHAT WAS “SPACE-TIME”? 15

are discussed, hopefully providing a link between the perturbative andnon-perturbative approaches.

The more important lesson learned from the above, is that entropyand information gain/loss enter into the picture early on, probably atthe axiomatic level, and the CS-interpretation is crucial to model andunderstand the implications.

2.1. It’s all about “Time” ... Returning to the existence of“Time”, the internal structure of a model of a quantum system re-veals the presence of an information flow as a substitute. The “globaltime” corresponds to a labeling of the external “symbols”/ states re-ceived/observed by another interacting system/observer, under the dualPhysics-Computer Science interpretation (PCS-interpretation for short).

At this point it is productive to think about the EDOFs, which in-corporate the symmetries of “space-time” also as “internal DOFs” byduality, as qbits, due to the “rich coincidences”: quaternions H viewedas C⊕C and SU(2) ∼= SL(1, 1) etc. A “(anti)particle” traveling up anddown in time (Feynman), or using the PCS-correspondence, the infor-mation flowing “up-and-down” relative to the observer’s global time, isa heavy parallel and sequancial (quantum) computing process, whichcorresponds in the physics language to s-correlation and t-correlation,the cousins of String Theory’s s and t channels. They are “coordi-nate dependent” concepts, allowing to split the quantum computa-tion into a parallel and sequential computational blocks (“space” and“time”), subject to some gauge symmetry related to conformal invari-ance (again some “coincidences” to keep in mind: Conf ∼= Diff(S1) ∼=Diff(S1) ∼ V irasoro algebra and qbit symmetries).

2.2. ... measurements. As an additional benefit of the PCS-alliance, we mention the possibility of resolving the quantum mea-surement paradox by modeling the measurement process as an “eaves-dropping on a quantum communication”. Recall that in our DWT, inprinciple there is no difference between “observers” O and “systems” S(remember the Human 2.0 merger?), since “quantum interactions” are“quantum communications”. Now modeling/interpreting a prepara-tion/observation measurement process as a “classical communication”,one might expect to be able to “explain” (model) the measurementprocess as the eavesdropping by Eve, the “Observer”, on the quan-tum communication between “Alice”, the preparation apparatus, and

16 INTRODUCTION

“Bob”, the measurement apparatus:

AliceQ−Interaction

QC// Bob

EveCC

ddHHHHHHHHH CC

;;wwwwwwww

The abbreviations QC and CC stand for quantum and classical com-munications.

So God/Nature does not play dice, but rather (quantum) talks tous out-laud, and the miracle is that we occasionally get an idea aboutwhat He/She is talking about :-)!

This brings us to the question regarding how the observer “thinks”.A “brainstorming start” consists in believing that our conscience is aclassical computer, since we need to interprete things in our familiarclassical terms (models), while probably the inner subconscience mech-anisms are quantum computations. We get a glimpse upon them whenwe remember our dreams (decoherence, as in any Output operation)... It is a speculation only, but it’s hard to believe that God/Naturemissed such a remarkable implementation opportunity.

Of course, such “dangerous speculations” should be “confined” tothe introduction, and mentioned only to disclose the possible implica-tions of a fully developed DWT, unifying the powers of Math-Physicsand Computer Science in order to “create” the unifying theory of Mindand Body. The potential benefits to the many other sciences studyinglife and mind etc. are obvious.

2.3. What is a Causal Structure? Causal structures appearedlong time ago in QFT in the disguise of approximating schemes: Feyn-man graphs (perturbative approach to QFT). The present author sug-gested in [Ion00], still in search of the appropriate mathematics modelfor Space-Time, that there is more to it than it meets the eyes: Feyn-man graphs are a substitute for the possible paths in a space-time. Theidea that there should be a more general structure including cobordismscategories besides Feynman graphs emerged. The term generalizedcobordism categories, denoting the would be more general structure,was coined during graduate school before the author became aware ofthe advent of operads and PROPs. Moreover, later on, it became clearthat there should be no “fixed causal structure”, allowing to account forthe “scaling problem”, and also a duality keeping a balance, allowingto trade “geometry” for “physics, as suggested in [VIRequest-UP](“packing and unpacking DOFs”).

3. ABOUT THE BOOK 17

2.3.1. Zooming in and out on a System. The appropriate technicaltool seems to be the insertion and collapse of subsystems (zoom-in andout on a system: how natural!). The precise mathematical implemen-tation is a generalization of Kreimer’s insertion and deletion operationson Feynman graphs, operations which allowed Kramer in his work withConnes [Kreimer, CK1, CK2] to rephrase and finally tame renor-malization.

The present author provides the physical interpretation beyond themathematical tool itself: the Quantum Dot Resolution (QDR) 2.1 isbelieved to be the promising framework for developing (classifying)QFTs, thought of as “Space-Time with variable geometry” (causalstructures). In a way the interpretation came post-factum, as some-times happened in the past. Indeed, as relativity as a conceptual breakthrough, appeared based on the already existing physics and mathe-matics of Lorentz, Minkowski etc. causal structures have already ap-peared as operads and PROPs, related to the operator product ex-pansions (OPE). OPEs establish a relation between causality and theexternal global time. In some sense, introducing a macroscopical timeconsistent with the classical limit corresponds to OPEs and quantiza-tion.

2.3.2. A “new” algebraic-geometry principle. The concept of “vari-able geometry” imposed by the modern homological algebra-geometryprinciple “forget the space, use a resolution ...” (see 2.1) had alreadyfound a “back-door entry” into discrete models (lattice models andMonte-Carlo simulations [DJ], p.254; see also [Ion01]).

The Feynman causal structure is not just a perturbative approach,it is rather a substitute, and in fact a generalization of the conceptof Space-Time; moving from (loop) degree to degree, as in Hilbert’s“syzygy theorem”, is an approximating procedure.

Kreimer’s insertion-elimination operations should be interpreted as“Space-Time bubble fluctuations” of the causal structure: “Dirac’s vac-uum”. At a more technical level, the general framework is that of a2-category of “generalized cobordisms”, where extensions (e.g. graphextensions) play the role of “homotopies”.

3. About the book

The book is organized into parts. The User’s guide targets a largeraudience, being an exposition of the main ideas and of the correspond-ing development projects. The corresponding implementation descrip-tions are provided in the Reference Guide. Annex B presents the Vir-tual Institute ReQuEST as an interface between sponsors and researcher

18 INTRODUCTION

teams developing the proposed projects. It is also a permanent (non-standard) grant proposal.

The focus is on putting the ideas on the table (usually representingabout 1% of a project), with care not to get swamped by technicaldigressions. It is not a question whether the envisioned theory is trueor false, but rather, as the author believes, how hard it is to implementit at a level accounting for the predictions of the present theories (99%).The benefits of the ambitious unifications proposed should overwhelmthe reasons not to depart from the traditional technology and currenttheoretical products. On the other hand, since the past and presenttheories provide key ideological jewels which could be imported (generalrelativity, quantum loop gravity, lattice gauge theory etc.), it is rathera matter of time to put together this bigger “puzzle” using alreadyforged pieces ...

3.1. About the merger of QFT and GR. There are various“indications” that something is “wrong” with General Relativity, be-yond the well-known (implicit) incompatibility with Quantum Theory(it is a “local theory”, not a “correlation theory” [Ion00]), in spiteof its impressive age and “stature” among “Venerable Theories”. Onesuch indication is termed “dark energy”, as a wide carpet where manyinconsistency may be swiped under. Let’s not forget that there is al-ways an alternative when changing our models, for example Newton’sF = ma theory, where RHS sets the theoretical framework and LHSallows for the experimental input. In the context of General Rela-tivity, one can “fine tune” the model of the experimental input, i.e.the Energy-momentum tensor (postulate some “new matter” source,etc.; that’s easy) or “adjust” the T = κG theory, i.e. the theoreticalframework. Now, that’s another story; twitching with the cosmologicalconstant (e.g. [Was Einstein right all along?], p.18) is not enough!

An attempt to merge QFT and general relativity is made by incor-porating the concept of information to the very foundations of space-time-matter viewed as an interaction process (see also Ch.8 §3).

The principle involves in an essential way the dichotomy “observer-system observed”, which is the new key feature of quantum physics (Iguess, despite what some people say; see [Gosson], p.35).

An avenue for involving new cosmological phenomena (dark matteretc.) opens up when taking into account entropy, since probably thefirst thing to do is to try to interpret mass and inertia etc. as anentropic effect, while keeping in mind that gravity has a clusteringeffect, so it is in fact an organizational principle at the level of space([Penrose], Ch.30; [Davis1]).


3.2. String theory and entangled thumbs! The pros and consregarding String Theory are a great divide: one thumb up and the otherone down!

To start with the bad news, the Nobel laureate David Gross’ exas-peration regarding String Theory “We don’t know what we are talk-ing about.” [ST], it is due to so many decades of unfulfilled hope;string theory looked like a promising theory “... and promising ... andpromising”, yet without any soon delivery, in terms of experimentalpredictions or verification.

It is well agreed: new ideas are needed [ST, Davis1], starting withString Theory’s lack of explanations regarding “ ... where space andtime come from”, while providing a sense of an academic exercise sincethe equations “describe nothing we could recognize”.

The “good news” is: the idea (within string theory) counts! TheFeynman philosophy pointed towards the mathematical structure I callFeynman Process, as a representation of a Feynman causal structure,and the benefits of using Riemann surfaces, a historical load we haveto reexamine (e.g. “fewer” equivalent transitions which are naturallyrelativistic since interactions are not point-wise localized etc.) are over-come by the complexity leading to “rigidity”; on top of this they donot poses a “computer friendly interface” (In fact they do, but underthe guise of ribbon graphs etc.).

But the killer feature is that they need to float in some backgroundspace (cheaper by the dozen!).

What can we do to save the day? Comparing with the principles ofDWT, the missing conceptual principle is the duality between internaland external DOFs. With Riemann surfaces, internal DOFs come asvertex operators, and after representing the Riemann surfaces “prop”(PROP), one gets a “clean” algebraic structure: Vertex Operator Al-gebra (VOA). What lacks is a “graph differential” allowing to insert /collapse EDOFs, in duality with a corresponding differential (L-infinitystructure?) of the VOA.

The present stated principles cope nicely with the simpler, toymodel structure of graphs. The difficulty of putting a space-time struc-ture on them, in order to have Poincare invariance and therefore rela-tivity, is avoided by categorifying classical physics first 3. The idea isto “forget” manifolds, and have a categorical substitute for the phasespace with external symmetries (Poincare group) dualised as internalDOFs. The current proposal is crude, yet promising (and ... promising...etc.?).

20 INTRODUCTION

3.3. “Paradoxes” of Quantum Mechanics. The historical para-doxes of quantum mechanics are revisited: a model of the measurement“paradox” is (tentatively) suggested and the dichotomy quantum de-scription of system versus classical description of the apparatus is de-emphasized, as rather being a modeling issue.

The information-matter duality is introduced, as an unification be-yond Einstein’s energy-matter unifying principle. In order to do this,thermodynamics have to be accommodated within QFT. The solution:interacting systems, whether complex networks or fundamental parti-cles present two fundamental dualities. The dichotomy between ob-served and observer in correspondence with the duality between inter-nal (“hidden” DOFs) and external (“observed” EDOFs).

The black-hole laws, estimated to play an important role in theimplementation of the theory, are discussed in the perspective of aderivation from the “thermodynamics of the causal structure” with(future) “technical help” from Loop Quantum Gravity.

In the context of DWT, we speculate regarding some “coincidences”,e.g. the number of fermion generations and the dimensionality of“space” etc., to envision deeper connections in the context of the DWT.

3.4. What is new, really. This, of course, is “reader dependent”and it is too soon to “classify” ...

The author pointed out that the concept of event and thereforeof field as a function defined point-wise on a predefined “space-time”,constitute the source of the incompatibility between general relativityand QFT [Ion00].

The author proposed a global categorical picture where the inter-action is the primary concept, and the “system” (whatever it mayrefer to: universe etc.) is viewed as an automaton, or in mathematicalterms, a representation of a geometrical category capturing causalityand thought of as replacing the “old” concept of space-time. The spacepart consists of the objects (source - target), while the morphisms rep-resent the causality (correlations/transitions etc.), replacing the “old”time arrows (e.g. generalized TQFTs, QFT on Feynman graphs, StringTheory on Riemann surfaces etc.)

This picture is necessarily incomplete if entropy-information is notincluded, as part of the quantum “interaction” observer-observed sys-tem (quantum measurement). The important duplex interpretationinteraction = communication is introduced (or emphasized).

Including the qbit as the quanta of information and treating it as avirtual particle “completes” the quantum description of reality (a timidcompletion, for the time being). The “deterministic evolution of the


system governed by (say) Schrodinger equation should now be compat-ible with the collapse of the wave function due to a measurement, sinceinformation is transfered from the observed system (S) to the observingsystem (O). In this way unitarity of the evolution of S is sacrificed, andnon-local features are introduced in the new theory, yet the enlargedpicture describing the evolution of both S and O is expected to beconsistent at a closer inspection (the quantum computation should beconsidered together with the quantum program controlling the quan-tum computation).

The generic space-time foam, beyond the particular implementationusing (say) triangulations etc., represents also information flow, andthe destruction/creation processes (operators or other mathematicaltechniques used to implement them) will affect the information contentfrom the point of view of the observer’s knowledge (“transfer of spacialinterface”). This (pre)concept should be compatible with the quantumlaws of black-holes etc.

In a way, the approach replaces the usual quest for a unified groupG to represent the internal symmetries of fields, functionals on space-time:

Field : “Space− T ime′′ → Rep(G)

with a search for the right implementation of a “space-time” incorpo-rating the symmetries directly into its structure using duality.

We must warn the reader that, in a top-down design approach, thespecification of the theory will start from general principles towards aprecise conceptual interface, explained in a high-level language. Theactual implementation is almost totally ignored in the first part (User’sManual), except when an existing specific mathematical theory mayshade more light regarding the ideas had in mind.

The more technical description towards a possible implementationof the present ideas is deferred to the second part, The Reference Man-ual.

Some “clues” which seem to be important, yet prone for specula-tions and alternative approaches are deferred to the Annex A.

Overall is seems that the main point is that “information/ entropy”is the missing dual aspect of the traditional approach (energy-matteretc.). In some sense a “fast implementation” approach consists in “dou-bling String Theory” to incorporate the entropy/ information current(think of the Yes/No-corolla as ST’s pair of pants), allowing for dualitybetween external (topological distinct RS) and internal DOFS (vertexoperators), by implementing insertion / elimination operations (QDRwith duality). We believe that ST is not “so ill-formulated” ([Woit],

22 INTRODUCTION

p.18), but rather not conceptually understood in the past. In order todo so, physicists and mathematicians alike should revisit the concep-tual aspects first, relaxing the old “strong culture where everything yousay has to be very precise and you have to be able to rigorously proveit.” (loc. cit. p.19). And ... let Computer Science join the efforts 2!

4. Acknowledgments

Bibliography. It rather reflects the author’s (more recent) readingprefferences, than providing the optimal or primary sources. In severalplaces the exposition and new ideas (in the current presentation form,but some persistently haunting the author since the “glass age” 3) wereprompted by the reading of a specific article/book; the citation wasmeant to acknowledge it. When the author’s idea was “confirmed” bythe reading, the further plead for the idea was “left as a burden” tothe corresponding author/source.

Dedications. The book is dedicated to my family. It was madepossible by my teachers, too many to be mentioned here.

But it’s too soon to look back! The final picture is not clear, butthe work to be done is; let’s get it started! (see the Annex B).

About the author. With a Mathematics, Computer Science and(self taught) Physics background 4, his work “off shell” often containsa (superposition) mixture of “What if” besides some “pure” ideas, notbeing under a “technicalities constraint” (“on shell”). Ultimately welearn from mistakes too; and sometimes lough at somebody else’s mis-takes (and cry!) ... it’s Arts and Sciences after all!5

2For computational purposes “upgrade the technology” (Rieman surfaces).3“Voi chi entrate qui, lasciate ogni esperanza”, but beware of author’s humor!4Not quite the list from [‘t Hooft]!5’tis too!

Part 1

User’s Manual

6

An important stage in the conception of the present project was re-alizing that a further unification is needed, expressed as usual by a newmissing Unifying Principle. Bellow we reproduce the description of themain project of VIReQuEST [VIRequest-UP], with minor changes,as it was conceived at the inauguration of the Virtual Institute for Re-search in Quantum Entropy and Space-Time B. It is the source of theDWT Program, as described in the present version, stating the “hints”and also “hopes” that such a principle is still out-there.

6Not too many formulas in this part ...

CHAPTER 1

The Search for a New Unifying Principle

Major breakthroughs are based on a change in perspective due tounifying principles.

I will try to backup this statement and revisit some folklore funda-mental questions and theoretical difficulties (”paradoxes”) which in theauthor’s opinion should be solved as a result of a unification steamingfrom a new principle. At this stage (proposal level) we are able to “list”what seem to be the major pieces of a puzzle: a theory including thebenefits of, and built with the technology of the present quantum the-ories (Quantum Mechanics/Quantum Field Theory, Quantum Gravityetc.) while releasing the conceptual “tension” of the “measurementparadox”. In my opinion, we do not always need express contradic-tions between experiment and theory (nowadays what theory predicts,say string theory, lies “safely” outside the experimental range ...). Theunderstanding (“revelation”) may come from a new way of looking atthe same “technical tools” (e.g. Special Relativity - see §2.1, etc.).

1. Models, models, models!

Recall that we model reality and we do not know what reality IS;many books have been written on the subject, so I will only mentiona few relevant names: Kant, Mach etc. and revisit a few points (ques-tions), briefly:

1.1. What do we mean when we ask “What is “time”?”?Implicitly we refer to a concept within a theory (framework/contextetc.) which usually belongs to a specific community or person’s knowl-edge (KDBMS :- ), linked (pointed to) via a tag like Newton, Einstein,Heisenberg, Feynman etc.. Or, when asking “What is an Electron?”,the answer ... depends on “hidden variables”; “Electron” may referto the corresponding particle in Lorentz’s theory, or the de Broglie’swave, Dirac’s spinor etc. (bad scenario: the context changes with theopponent’s tentatives to avoid our “theoretical hits” ...). Even worsestill, it can be quite misleading when “explaining” quantum mechanics,and in the same statement making use of the term “electron” to refer

25

26 1. THE SEARCH FOR A NEW UNIFYING PRINCIPLE

to the quantum description and then to ponder in classical terms aboutit ...

In this sense, there are many meanings one can call “time” (or“space” etc.) within various theories; so one has to be careful aboutthe implied context.

1.2. “Is it a particle or a wave?” The “electron” for instance,is very well modeled as a particle by a few theories, when it comes to acertain range of experiments, yet there is a need for other theories mod-eling the “electron” as a wave because of another class of experiments... Overall quantum theory has a unified explanation for “all” experi-ments (of a certain kind) and the Complementarity Principle may bethought of as a “Two classical charts atlas of Quantum Mechanics” ...

1.3. Interpretations of quantum mechanics. Why do we needto “interpret the result” of a quantum mechanics computation in clas-sical terms?

Classical mechanics is contained within quantum mechanics ([LL],p.12), and it is not just a “limit” (Correspondence principle). Indeed,the measurement process involves a “Q-probe” (quantum probe: mi-crosystem, elementary particle etc.) interacting with the measuring ap-paratus (usually a macrosystem) and the result of the experiment itselfis modeled, or at least used by the experimental physicist (or processedby some software!), in classical terms (we acknowledge macroscopicalevents: dots on a screen, beeps in a counter, bobbles in a chamber etc.).Even a Stern-Gerlach experiment (i.e. involving “internal states”) in-volves the interaction of a Q-probe (the electron) with a magnetic field(macrosystem!) AND a detector (beeps on 2 counters, providing theinput to a classical gate/computation). So, in a way Quantum Me-chanics is a phenomenological theory! (beyond the Kantian statementof the type “we only model phenomena ...”).

1.4. The main “lesson”. from above is that there are implicitchannels of information which are present, yet probably not correctly(or completely) modeled within the corresponding theory! The roleof observer in classical physics is that of “user”. A crucial objectiveis to have a unique description independent of observer (covariance;classical heritage). This is no longer tenable in quantum mechanics:“results” depend not only on “what” we observe, but also on “how”,which in turn depends on what do you intend to do with “the result”.Nevertheless we are still looking for a “standard” in these proliferationof “encoder-decoder” business.

2. THREE ... REVOLUTIONS: RETURNING TO PRINCIPLES! 27

A unifying point of view, as a “slogan”, if one has in mind the uni-fication alluded to above, is that “All is quantum computing” (see also“Feynman processor” [Milburn] etc.), i.e. any interaction, whethersystem-system (Einstein ≈ “I like (!) to think I don’t have to look atthe Moon for it to exist”), system-observer (quantum phenomenon),observer-observer (genuine communication!) are of the “same kind”(Which kind? that is the question!;-)

2. Three ... revolutions: returning to principles!

Let us consider Newton’s simplifying picture of Kepler’s Laws as astart for scientific modeling of (mechanical) phenomena. 1

2.1. First revolution: Special Relativity - gave a new look at thetechnical tools already available at that time: Minkowski space, Lorentzcontraction, conformal invariance of Maxwell’s equations etc. Yet theconceptual break-through consists in “understanding” the meaning (andbuilding various connections with other concepts - networking ...): theunification of space and time (technically already done by Lorentz andPoincare - see [V], p.25 - but ... “What is it that we are doing?” wasprobably the lurking question of the day). The unification was derivedin an “axiomatic” manner from the principle c = constant (correspond-ing to a constant Lorentz metric, or rather its conformal class ...). Aprobably more important principle is the equivalence between mass andenergy:

Principle I : E = mc2.

A “simple equation” yet with huge implications (Nagasaki, Hiroshima:-( ).

2.2. The second revolution: General Relativity or Quan-tum Mechanics? In the author’s opinion, QM is The Revolution,changing the way physics is done (see 1.4). General Relativity is a“jewel” amongst mathematical-physics theories, again starting from aprinciple, the equivalence between accelerations, gravitational or not(or masses: inertial or gravitational):

Principle II : mg = ma

General Relativity “upgrades” the Newtonian geometro-dynamic de-scription “force of some kind=centripetal force”:

Force = Mass ×Acceleration

1Or ... is it “a culminating point of the scientific revolution of the seventeenthcentury”? [Katz], p.425


to a pure geometric description (space and time were already merged)“matter tensor ∼ geometry tensor”:

Matter Tensor = κ Einstein Tensor

which, beyond the new “technical tools” (semi-Riemannian spaces,Ricci curvature etc.) amounts to passing from a description of dy-namics as “curved motion in flat (universal) space” to “flat motion(geodesics) in curved space(-time)” (again as a figure of speech ...). Inother words, taking a phenomenon (gravitational force) from the LHSof Newton’s principle and incorporating it into the RHS as Einstein’stensor (essentially the average curvature; κ denotes the gravitationalconstant). The “trick” proliferated: then came Kaluza-Klein, attempt-ing the same with electro-magnetism. It did not work as well, since“internal degrees of freedom” could not be well accommodated as ex-ternal degrees of freedom (i.e. dimensions of space-time). The alterna-tive was to build degrees of freedom outside the “obvious” ones, leadingto Gauge Theory etc. Meanwhile the technology advanced and stringtheory is capable of such feats, introducing “real” dimensions (for agrand total of 11? 21? ... cheaper by the dozen!). Some of them, ofcourse, need to be “hidden” from every-day “access” by compactifica-tion, declaring them small enough (just another model for space-time...).

In the “phenomenological camp” the opposite tendency may benoticed (in the spirit of quantum mechanics; see 1.3): let the degreesof freedom (and states) be “internal” (abstract) ... and Vertex OperatorAlgebras appeared! (or maybe to tame The Monster? :-)

So, where is the third?

2.3. Space-Time: Is “motion” possible? We do not need Zeno’sparadox (see [Katz], p.56) to claim that motion is not possible ([LL],p.14) 2. Of course, we have to specify in which theory; in quantummechanics, of course, since otherwise classical mechanics deals greatwith motion/continuous evolution/dynamics (Poisson manifolds etc.),and we’ve learned not to talk about what reality IS, but only mod-estly (?) about our (scientific community at large?) best model aboutit/Him/She (one preconceived model for each specific “glasses” wechoose to wear ...) .

In quantum mechanics there are “states” and “transitions”, as ina sort of a “complexified” Markov process, where, amazingly, the pos-sibility of having a result in two ways may cancel each other’s contri-bution (“indecision”), rather then build up the probability! (to model

2Zeno’s Arrow paradox seems to urge for Lorentzian contraction at least.


mathematically this feature, we choose superposition + interference,implemented as a linear theory over complex numbers).

The incompatibility between knowing the position AND momentumat the same time, for the same direction (Heisenberg’s uncertainty prin-ciple), conceptually refutes classical trajectories altogether (but stillrefers to classical concepts!). 3

If we insist in adopting QM to investigate the “motion process” andstill have a “classical understanding” of what the electron “does” in atwo slit experiment we have to conclude that “IT” goes through bothholes simultaneously! (The “best approximating” classical statementfor the quantum occasion ...). So, “Is motion possible?” Well ...

2.4. What is an “Event”? The differences in approach (Classi-cal Mechanics, GR, QM/QFT) start with the concept of “event” (see[Ion00]).

For Newton the “event” is a “particle” (existence) “somewhere intime”; these three concepts, existence, space and time, are “absolute”,i.e. independent of observer and of each other.

For Einstein, “existence” is still “absolute”, although the “event”occurs in space-time (still “absolute”, even after the advent of GR).After the QM lesson, we should agree that we model correlations: Ainteracting with B produces a C (say electron in a magnetic field yield-ing a beep on the up or a down counter; and the observer? ...). Thereis a missing factor here to be explained later on, appearing in a parallelbetween quantum and classical computation - see [VIRequest-CS]).

To implement “correlations” one needs to define the “states” and“transitions” (e.g. using categories: objects and morphisms). Thereis usually a “time-ordering” issue here: states first, then transitions... This may be thought of as developing the theory starting from the“free case” (inertial reference frames/ free theory in scattering methodetc.) and then adding “interactions” (all frames/scattering matrix etc).It is essentially the old Newton’s goal (and Descartes’ methodology)of representing functions (or theories) as power series or theories asother series: indexed by Feynman graphs, Riemann surfaces etc., i.e.building the “big processor” out of “micro components”), and calledperturbation theory (yet this is not the whole story - see Introduction§2).

3From a constructive point of view, one can always rewrite a C++ object-oriented program in a modern version of FORTRAN, or obtain QM as an emergenttheory [A], or based on classical keywords [BM]etc.


2.5. Quantum Field Theory. In QFT we have a continuum ofdegrees of freedom (the values of the field) only because we stronglybelieve in a given continuous space-time. Roughly, QFT is an “up-grade” of QM as a (complexified) Markov Process, where the completegraph represented (a matrix indeed), is replaced with a class of graphsand the complex numbers as coefficients are replaced with operators(propagator) (a figure of speech again ... and again ...)

Then came Feynman (bringing QFT to the masses, thank you! :-)- see [Zee], p.41) and introduced what we will call The AutomatonPicture: states and transitions, whether these are paths in space-time(external DOF) or transitions in internal space (IDOF).

Since we ultimately look for transition amplitudes of an interac-tion in the context of a framework based on the free case (classical inessence: we know how many particles go in, and what comes out, inclassical terms ...), sum up the amplitudes for all possible “scenarios”(correlation function as a sum over Feynman diagrams). This is a ba-sis in the “transition space” (space of all “paths”). In some sense, theanalog of a partition of unity is the partition function.

The “problem” is, that if we believe “motion” is possible in a con-tinuous space-time, then we end up with too many “paths”, divergentintegrals etc. Physicists have learned quickly how not to step in quicksend, while mathematicians had a hard time building the bridge overthe “swamp of infinities” (constant/variable, infrared/ultraviolet, im-portant/negligible etc. :-)).

2.6. External/Internal Degrees of Freedom: The AutomatonPicture. The natural way to “solve” the problem of too many paths(and infinities?) (let’s just cut the Gordian knot already!) is to realizethat all we need is a category of “paths” and an action allowing to builda representation of this “Feynman category” with suited coefficientscorresponding to the internal degrees of freedom (phenomenology) hadin mind (hopefully towards a GUT, encompassing all of them, butgravity).

By now it appears that gravity is an organizational principle withinthe space-time description (GR), rather than an exchange interaction.Trying to push the beautiful particle-field picture from scalar and vec-tor fields to spin 2 tensor fields (... obstinate ?) could be the “take abigger hammer” (string theory ... M-theory?) approach to “crash thenutshell”, approach which looked so repellent to some (Grothendiecketc.). It worked with Fermat’s Theorem, right? But, “What’s takingso long?” (see [Are-we-nearly-there-yet?]).


... or rather try to implement gravity as a pairing between theFeynman category, which captures the causality (there is NO universaltime at the micro scale, so that we have to deal with time orderedproducts / OPEs etc.) and the “coefficient category” which capturesthe macro-behavior/hidden details (see §1.3) in an adjunction whichtrades additional external degrees of freedom (e.g. applying the ho-mology differential, i.e. insertion of an edge [Ion03]) for additionalinternal degrees of freedom (technical details should not clutter thepicture at this point!).

This should be done, perhaps, in conjunction with a model forthe information flow (see §1.4), since there are several macrosystemsinvolved, and an experiment, like a quantum computation, involvesclassical read/write operations subject to classical logic/laws ... (see[VIRequest-CS]).

No matter what the specific implementation will be (e.g. usinggraphs, networks, categories etc.), it will capture the idea of automa-ton (states and transitions; e.g. cellular automata in [Wolfram] or[Cells are circuits, too!] etc.), implementation written in one’s fa-vorite object-and-relations oriented high level language.

2.7. Is there a “time”, after all? Indeed “time” is THE deli-cate concept; or rather a plethora of interconnected concepts! We alllike to ponder on the fundamental questions, especially as adolescentsor young researchers trying to find new ways ... (see “Time’s Up, Ein-stein”, by Josh McHugh, Wired 06/2005, p.122). It was the analysisof what time is, that led Einstein to a clear picture unifying Newton’suniversal space with his universal time. Even at that stage, one couldponder on a hidden assumption Einstein implicitly made: transitivityof synchronization (“the early years”: challenging everything ...). Itmay indeed fail in GR, if there is no local time (non-integrable or-thogonal distribution to a Killing vector field etc. - see [O’Neill]),and instead of spending $200,000 on a “Michelson-Morley experiment”trying to reintroduce the “ether” (! see “Catching the cosmic wind”by Marcus Chow, New Scientist, 2 April 2005, p.30 ... or rather not)one might rather test the above mentioned possibility, which definitelyholds true at some level of accuracy (the “ergodic principle”: if it isnot forbidden by the laws of physics, if will happen ... eventually): theproblem is “In what conditions?” ...

So, better investing more into theoretical research (rather thenbuilding bigger muscles for physicists to smash everything), and maybefind out that the “ether” is ... the “Higgs field” 3 ... or not!


But since we aim at a deeper model (beyond SM or ST), where“events” are “pure correlations”, such issues are secondary. One lessonlearned from Special Relativity is that there is a causal cone; events canbe spatial separated (no causal correlation possible - not talking aboutentanglement yet ...), or if causally correlated, than they must be time-separated. Yes, a “proper time”, is a different concept (“continuity ofexistence”), rather playing the role of a parameter (implementationdependent); as opposed to “laboratory time” in quantum mechanicsetc.

So, what we need is a “Causal Structure” and that is precisely whata Feynman Category provides!

Feynman Category =⇒ Causal Structure.

In bombastic terms:1) Event: NO! Correlation: Yes!2) Time: NO! Causality: Yes!Of course, we do not request a total “removal” of time from the

“old” theories (we still wake up at 7, take the kids to school at 9 etc.);what’s been said is well said. We request its limitations be acknowl-edged.

If a causality structure is given, then to benefit from the presentand past theories one has to deal with embedding it in a classical 4dmanifold (d=11, 21?), as some “background space”; or at least, afterrepresenting it in one way or another (e.g. decorating punctures on Rie-mann surfaces with operators, VOAs etc.) one has to come up with anOperator Product Expansion (OPE) as a much more complicated issuethat the usual 1-parameter group of unitary transformations capturingthe dynamical evolution, or “time flow”.

What is left of the idea of 4-coordinates as a “... starting point ofthe mathematical treatment” ([V],p.24)? First of all, one should post-pone the “mathematical treatment” until the “design” of the theory atconceptual level is complete (or at least satisfactory; “implementationspecifications” of the physics model), then let the (“implementation”)specialists implement the theory ... (another story! we would not havehad QED a few decades ago, right? It had to be done fast, no time(...) to wait for mathematicians to be pleased with a “rigorous, i.e.mathematical, implementation”).

What I am advocating for is a “device independent, portable in-terface” between math and physics models (author-independent imple-mentations, user friendly :-)) ...

Then there are some holistic questions ... There are 3-pairs ofnon-commuting observables representing external degrees of freedom

3. A “NEW” PRINCIPLE 33

(q1,p1, etc.). Why 3-d? why 3 generations? etc.; these could be “sep-arability questions”, telling theories apart, but we feel there is more toit than it meets the eye ... (see Ch.9 §3).

At the pragmatic level of VOAs, the substitute for Lorentz/Poincaretransformations (or conformal invariance) is having a built in Virasoroalgebra, which in a sense is the simplest graded Lie algebra ... Whatis its deeper message? (Sure, diffeomorphisms of the circle: quantumphase?; a way to remove the “continuum” from the quantum complexphase? ...).

3. A “New” Principle

Returning to General Relativity, perhaps the most important con-sequence, beyond expansion of universe and Hubble’s constant, is theconcept of black hole. The unification of GR and quantum theorywas initiated by S. Hawking as an extension of GR incorporating theblack hole radiation. Since then, three laws have been identified (see[Smolin], p.92).

The first law relates temperature, as a measure of energy per DOF,with acceleration as a measure of the interaction (Newton’s sense):

Unruh′s Law : Temperature /~ = Acceleration/c.

It expresses a principle, therefore the simplest (physicist favorite) wayis enough: linear relation. Together with Einstein’s equivalence prin-ciple, it suggests that there is an energy distribution for the 2-pointgravitational correlation (in our quantum discrete picture).

The second law:

Bekstein′s Law : ~ Entropy =1

κArea/(8π),

relates Entropy, as a quantity of information needed to completelyspecify a state (the “quantum memory size”) and Area, which in adiscrete (geometric) model should be thought of as a measure of thepossible In/Out interactions (“quantum channel capacity”?). Beyondthe “global statement”, adequate for stating an equivalence principle,there should be here a “local/discrete” Stokes Theorem at work ... (?)

It is reassuring to find that Lee Smolin mentions implicitly sucha “would-be” principle: “one pixel corresponds to four Plank areas”([Smolin], p.90), although it could rather be “one interaction qbitcorresponds to four Plank areas” ... (?)

Later on (p.102), he derives some conceptual implications whichare evaluated as not admissible, IF there is no theory to back them up(we have learned allot from the old story: “Euclid’s Parallels”, axiom


or not? Let’s derive the “unbelievable consequences” first, then decidehow to build the theory).

Finally the third law relating temperature and mass, but in anopposite way as the first law, is:

Hawking′s Law : Temperature = k/Mass,

or alternatively:Mass = kβ

(with an eye on the entropy: Boltzmann’s correspondence etc.).

It refers to the radiation capability of a black hole (‘‘density ofI/O-interactions”?), rather then its energy distribution per DOF.

The situation is reminiscent of Newton’s position when simplifyingKepler’s laws ... so let’s look for a “new” unifying principle! (we dohave a “situation” here, right?).

3.1. “Mind versus matter”. Recall that E = mc2 (Principle I),in a sense, unified energy and matter.

Quantizing energy (bound states):

Principle III : Energy = hFrequency (= ~ω)

should correspond to quantizing information in some sense (What aboutunbounded states?).

The new Unifying Equivalence (Super)Principle will be labeled “Mindversus Matter” to convey its scope. It states a correspondence betweenenergy and information, both quantized/discrete:

New Super-Principle IV: qbit↔ ~ (S(qbit) = ~)

aiming (so far) to unify the “observer” and “observed” of quantumphysics, explaining the “measurement paradox”, and why not, provid-ing an interface between the “safe” science and the other “believe-it-or-not” areas of investigation (conscience etc. - You name it :-)).

The idea is that a transfer/fluctuation of a unit of energy shouldcorrespond to a quantum bit of information (“Precisely in what sense?”... a very good question indeed!). An additional DOF (E/I) (internal,i.e. type of particle, or external (!), i.e. space-time “location”) changesthe partition function describing the distribution of amplitudes of prob-abilities in a way similar to a black hole “leaking” a qbit of information.The theory should naturally incorporate the black hole laws in the con-text of a General Relativity relocated from its natural habitat (man-ifolds with a metric/Lagrangian) to the realm of Feynman Processes(representations of Feynman Categories: string/M-theory rephrasedand background free/mass generation upgraded). ... It is still missing,but in conception! (see also Ch.8 §3)

3. A “NEW” PRINCIPLE 35

3.2. Are black holes prototypical? Again it is reassuring thatthe idea of the above Unifying Super-Principle, in one form or another,is present in the remarkable book [Smolin] (p.101): “There is some-thing incomplete about a law which asserts a balance or an exchangebetween two very dissimilar things.”. Paying too much attention to itsdraw backs is not always a good idea (p.102). Yes, if one would justclaim E = mc2, would not be enough ... But again, a theory startswith an idea, a new principle (1% of work), and then one designs thetheory top-down (99% of the work - Edison).

So, Lee Smolin is talking about a balance between “atoms” and“geometry”, but only in gauge theories there is a clear cut distinctionbetween external DOF (space-time) and internal DOF (implementingthe type of particle as a representation of a gauge group and then“marrying” them as a principle bundle etc.).

Moreover, a distinction between “atoms” and “geometry” leadsback to an “absolute space-time” point of view. This is no longertrue in a Feynman Theory (FPI adapted to Feynman Processes as rep-resentations of Feynman Categories) where an insertion of a new graphshould be thought of as “adding geometry” (and also as a change ofscale!) but then, under the functorial adjunction, in this way newinternal DOFs are introduced.

Now my “bet” regarding the two profound questions from [Smolin],p.102, is:

(A) Yes, there is an “atomic structure” of the geometry of space-time (i.e. leads to a better model - see 1 - e.g. PROPs or LQG’s “grainsof space-time”), and our Unifying Super-Principle generalizes in a sensethe idea behind Bekstein’s Law. Indeed, in a discrete Feynman Cate-gory model, “area” corresponds to the number of interactions (or somemore sophisticated concept depending on the author-dependent imple-mentation, yet “counting” is the safest way to proceed - combinatorics,what else?), which from Unruh’s law, “carry” a certain energy; roughlyspeaking a “space-time event” A→ B has a double role of both inter-action channel (s/t or ... neither!) and information channel.

B) Yes, the Digital World Theory incorporating the theory of in-formation (Shannon, quantum computing etc.) on top of a FeynmanTheory, will have as natural consequences the black hole radiation laws(... and much, much more :-) ).

How to switch from black holes, thought of as “prototypical” whenit comes to “global” quantum aspects, to the general case, say in termsof Feynman graphs? In a brainstorming way:

Horizon → Hidden causally ∼= No interactions→


→ Non − connected Feyman Graphs ...(?).

Then: “black hole radiation (interactions) appear” → “inserting non-connected FG in another FG” ... !? (not implementation ready :-))

It is too soon for technicalities :-)

4. Conclusions

So, to stress even more the main “evolutionary steps” of the key fun-damental concepts (“repetitio est ...” sometimes annoying, right? :-)):Newton Space Time Particle xor wave N.A.Einstein Space-Time Particle xor wave ObserverHeisenberg Space Time Particle/Wave & ObserverDirac Space-Time Particle/Wave & ObserverFeynman Path Integral QuantizationM.Ph.-Folklore Representations of Feynman CategoriesThe DWT 1.0 Representations of Extended Causal Structures

Here “Extended Causal Structures” refers to the incorporation ofthe concepts of entropy and information processing [VIRequest-CS]in order to unify the classical interactions “particle-particle” and “particle-observer” modeled by Quantum Theory with “observer-observer”, i.e.genuine communications (for symmetry reasons at least, but hopingthat it would lead to a better understanding of “reality”, for exampleof the measurement paradox).

How to put together all the above “design constraints” in a coherenttheory, is another story ...

Then came The Institute (idea) :-) 4. It is meant to be the Web Pro-cess aiming to stimulate the upbringing of the Digital World Theory(“The Quantum Matrix” representing The Hologram) by taking ad-vantage of the Butterfly effect (in a teacup or in our local inflationarybubble of universe ...?).

4See Annex B - VIReQuEST.

CHAPTER 2

Miscellaneous: a “warm-up”

The ideas and comments included bellow as a start up, will berevisited later in more detail; the present chapter is only a warm-up ...

1. On the measurement “paradox”

Where the deterministic description becomes probabilistic?The state of a quantum system (QS) represents our knowledge

(model) of the system and its evolution is modeled in a determinis-tic way as being governed by Schrodinger’s equation. On the otherhand it should come as no surprise that our knowledge of the systemchanges “abruptly” when a measurement is performed. What changesis our description of the system.

Ultimately we only model reality and our models are not the realityitself (phenomena versus reality - “a la” Kant). But the change of thestate due to interactions between subsystems should also be thoughtof as “relative measurements”.

Deterministic or not?Hidden variables which are not local can be implemented to provide

a classical interface to quantum mechanics which gives us “peace ofmind” (we’re in control - reality evolves in a predictable way). This isjust a way to disguise the same consequences, which are probabilisticin nature.

We are used to translations from one language into another (andthankful, as it disseminates knowledge), but some languages have afuture ...

2. Modeling a complex process

We prefer to talk about “processes” rather than of systems, to em-phasize the role of the evolution/transition in the context of the Au-tomaton Picture (Path Model).

2.1. The duality: internal - external. A process captures theduality between a system (existence) and its evolution (change).

37

38 2. MISCELLANEOUS: A “WARM-UP”

This in turn reflects on the two inseparable components of a modernapproach to modeling reality: software and hardware! One can alwaystrade one for the other!

To “fix a scale” amounts to splitting between I/E DOFs, corre-sponding to a chosen “modeling resolution” etc. A change of scaleoperation appears in various contexts: partition of states in the micro-macro description (µM -correspondence), collapsing EDOFs and aver-aging over IDOFs (colored subgraphs) etc.

Once a scale is fixed, one has to model the components of the sys-tem (e.g. attaching vertex operators, associating a macro observable onstate space cells etc.) and model the interactions between components:the type of interaction (e.g. class of graphs for a φ3-theory) as a “per-turbation” of a “free theory” (e.g. Feynman rules once propagators areselected).

2.2. Structural modeling (SM). Modeling, as a (human) pro-cess for reflecting a part of reality into an artifical construct, is a cyclicprocess when including the experimental feedback (a learning processin a broad sense!).

Structural modeling adds emphasis on adding structure (say alge-braic) to the usual quantitative/numerical capabilities any scientificmodel should have. The structure consists in relations between ob-jects/concepts/variables, leading naturally to Path Models (see [DelM]or [SL] etc.).

Some of its key features, although almost universal, appear in dif-ferent (specific) guises in various “places”.

A path model for instance, with multiple independent observedvalues and multiple dependent observed values ([SL] p.4), is essentiallyan IO-computing process.

Confirmatory factor models (loc. cit.) consisting of observed vari-ables that are hypothesized to measure latent variables, are essentiallymicro-macro correspondences of the Boltzmann type.

The observed variables may be thought of as “readable”, i.e. globalvariables as opposed to latent variables ([SL]), or “hidden observables”in QM, which are “internal variables”, or “local variables” which neednot be directly accessible (although for testing purposes one can tryto read/print them; except when quantum computing/interaction isconcerned, eavesdropping/observation will alter the state etc.).

2.3. Why Path Models? The use of Path Models is attributedto Sewell Write (1918), a biologist ([SL]). Path analysis is used to testrelationships.

3. WHAT IS THE INFORMATION FLOW? 39

Now QM teaches us that there is nothing else but correlations, sodesigning Path Models is rightfully termed causal modeling. The re-searcher, in an early stage of the design process, specifies a model “apriori” based on theoretical considerations. This model specification,involving the available relevant theory, research and information, to-wards the developing of a theoretical model, is a critical part of SM(loc. cit.).

3. What is the Information Flow?

The Einstein-Podolski-Rosen paradox (EPR-paradox) exhibits theinconsistency of QM under adjunction of classical assumptions/ inter-pretations: non-locality as an “action at distance” (classical view offorce/field etc. producing a change at a spatially 1 separated distantpoint). The role of information is not considered, since it plays no rolein the classical physics: it’s “all” about “motion” and the number ofparticles (NOP) does not change!

In the teleportation setup the crucial concept is that of informa-tion flow. The quantum model (within the DWT at least) generalizesthe concept of causality into the concept of correlation which incor-porates correlations due to information exchange which might not belocal 2. One then distinguishes a t-correlation (time correlation = classi-cal causality) and s-correlation, referring to/modeling space-correlatedsubsystems, which from the observer’s external (macro) point of vieware distributed in “space”. The need for this dichotomy steems fromlimiting the quantum computation as being sequential or parallel (inan outsider’s description).

Now, although from the observer’s point of view the NOP doeschanges (creation of particle-antiparticle pairs), there is something thatdoes not change: “information is conserved”, the number of qbits isconserved, and appears as a “motion” up and down relative to the ex-ternal global time of (or description made by) the observer. This is theinformation flow. The information flow carries s and t-correlations!

The root of the incompatibility between the basic assumptions ofclassical mechanics and (relativistic) quantum mechanics, and exposedby the EPR-paradox as a non-locality, is the non-constancy of theNOPs relative to a classical external description, usually based on alocal or global “time”.

1Declaring “spatial”, we mean “not causal”; the paradox prone conditions arehalf-set!

2Not even local to the system under study and involving the observer’sintervention.


I.e., in the S−O interaction, “straightening” the s/t-correlations ofthe S − output that O perceives as Input, therefore having “to decodethe message” into “spatial” or “temporal” (process the info as sequen-tial or parallel; ultimately label consecutively blocks of symbols forwhich the order is indiscernible), the S-computation (the informationflow) is not always an “s/t-grid”. And on of the basic Redemeister’smoves, straitening an S into an I does not work in the world of quan-tum measurements (Measuring the observables A, then B then A, isnot equivalent to measuring A twice Ch. 6 §1).

The non-constancy of the NOPs, in the spirit of the General Rel-ativity, means that “Space-Time” “evolves” (is not just “geometry”)and since in the spirit of QM the evolution is not deterministic (if ac-counting for measurements, or at least in the spirit of DWT), then“Space-Time” cannot be “fixed” as in GR (“pure geometry”) or as inString Theory (unable to survive without a background space mani-fold). Therefore, in the spirit of DWT at least, “Space-Time” with itsIDOFs is the “vacuum”, which is a resolution of the point/qbit. The“singularity resolution”, or call it the “experimental scale”, dependson the questions (input) O provides to S, and as in any computation,classical or quantum, requires various amounts of memory being used...

4. The “understanding” of QM

There is an extensive literature on the physicists’ ways of “under-standing” quantum mechanics. A first key issue is the meaning itself ofthe term “understanding”; it refers to the compatibility between QMand our classical model used to perceive reality, for example assessingthe position of a (macro)pointer of a macro-apparatus etc. To properlystudy this relation, one has to model and study the quantum-2-classical(Q-C) computing interface (e.g. in our suggested ABE-model of mea-surement).

The three ways presented in [Stapp] (Bohm’s interpretation, branch-ing universes, Heisenberg’s model) attempt to deal with the “quantumjump” (collapse of wave function etc.) in the context of the “macro-scopic possibilities previously generated by the deterministic laws ofmotion.” (loc. cit., p.19). Indeed, limiting to modeling motion will notallow escaping the deterministic realm, even that of QM at the levelof Schrodinger’s equation. The (more) complete picture has to includethe dynamics of information/entropy of the S −O-system (process).

6. CLASSICAL OR QUANTUM MECHANICS? ... WHAT’S THE DIFFERENCE ANYWAY!41

5. Is it all in the brain?

Ultimately there is nothing special about O in the above “dualsystem” (S-O), except (we need to emphasize) that the description is“order sensitive”: model(S − O) 6= model(O − S), for a given pair Sand O. In the broad picture any of the S and O could be “machine”or “human”.

When it comes to the “measurement paradox”, the O-componentof the above S → O-system (process) will achieve a conversion fromquantum computation (communication) to classical computation (com-munication), usually associated with an I/O-operation. It does notmatter if O is a human or a macro-apparatus, “macro” meaning thatit “behaves” (i.e. it is accurately modeled) as a classical system.

Usually O is pictured as a human staring at the pointer of an in-strument. The “conversion” Q → C was already in place, done bythe apparatus. The point is, our eyes are not “good enough” to per-ceive (read) a quantum communication (computation), and this is whywe are provided with a conscience performing classical computations.But for difficult decision-related tasks, we probably are endow with asub conscience performing quantum computations (nice try, right? :-).And we do have trouble “reading the results”, i.e. what our “deepinner self” inspires us to do. The “crudest conversion” seems to occurwhen we wake up; the nonsense we remember of our dreams is reminis-cent of a measurement of a superposition of states/thoughts (quantumcomputation results, perhaps).

It is conceivable, continuing the above speculation, that we willfind the way to by-pass the classical interface when the merger be-tween human and (quantum) machine will be achieved: The Human2.0 [Human 2.0]. Then the new possibilities are hard to imagine ...(Any ideas, Mr. Spielberg?).

6. Classical or quantum mechanics?... what’s the difference anyway!

The so called correspondence rule (Dirac quantization) is impres-sive, but perhaps a bit misleading ...

To understand the “difference” between CM and QM one has toovercome the difference in “language” used to encode them, by bringingthem on a common ground (see also 4.4.1).

Note that CM, say in Newton’s formulation, is an “affine theory”.Galileo’s relativity group which allows to realize space as a homoge-neous space, represents not only independence of coordinate systems,but due to translations it represents the “free theory” encoded in the


second order differential equation of motion. The external forces (orassociated potentials) appear as perturbations, providing a frameworkfor modeling interactions.

Now quantum mechanics appears from the start as an effective the-ory: states and transitions implemented as a linear theory, i.e. homo-geneous. There is a zero vector, but which is not a state; in fact statesare “rays” (lines), so that its true geometry is projective and when rep-resenting observables one alternatively has a projective representationon the original space of states, and the need for a central extension torelate the two representations is clear etc..

To compare CM and QM one has to translate CM say into a ho-mogeneous theory first. The simplest way is by adding one ficticiousdimension to space and consider the embedding of Newtonian space asR3 × 1. Then Galileo group is realized as a group of linear trans-formation on R4 (see [AI], Ch.3, p.152). Now the Newtonian spaceappears as a projective space (almost: without the North pole!) andthat the “larger space” is a compactification of the Newtonian space!One can proceed to implement the theory in operator form movingfrom the classical observed values, say energy, as scalars to an eigen-value form, since the “states” (position etc.) are now “rays” in R4.Observables will be operators etc.

In this “quantum-like” formulation CM is a linear theory and canbe “better” compared with QM.

The first major and crucial difference is the lack of superposition,while the second is the absence of interference.

To admit “superposition” for CM would mean “delocalization”: asystem which is a “particle” could be at two places at one time, i.e.the system would be distributed in space, yet the parts would notbe independent, but s-correlated! Of course, this would be a steptowards explaining the two-slit experiment, since now a particle could“split” and its two s-correlated parts would go through the two distinctopenings in the wall (not s-correlated) and then “recombine” on themeasurement plate ...

A moment’s thought reveals that this is the right way to enhancethe power of our models regarding reality. One should not “encode”all our knowledge at the level of “states” and postulate a classical anddeterministic dynamic law, but should be more cautious and distributethe possibilities between states and the transitions which are supposedto model the “ways” a cause “determines” and effect (Path Models).

7. TO THINK OR NOT TO THINK CLASSICALLY? 43

A law of dynamics is still required on top of this, as in the prototypi-cal approach of Feynman’s regarding high energy quantum phenomena(FPI).

Even better, one should allow for “partial knowledge” (information)about the system (mixed states) and make the transition to a statisticdescription first, before comparing with quantum mechanics. This step(“difference”) is essentially the adjunction sets-vector spaces togetherwith a 0, 1 to [0, 1] change of logic’s coefficients (probabilities).

To accept within CM the second crucial difference, “interference”(say by complexifying everything) is much harder/unusual/paradoxical.It seems to enable what is left to be able to explain the fringes in theabove experiment, which may be thought of as the generic (prototypi-cal) quantum phenomenon.

Then, where are Heisenberg’s uncertainty relations in this trans-gression from classical to “quantum”? Would it be enough to “add” acompactified ghost-like dimension: an S1 complex phase? (R3×1, i).

7. To think or not to think classically?

The classical meaning of these assumptions is revealed by lookingat their dual role in the context of interaction-communication duplexinterpretation.

Classical information (communication) consists of strings either 1(x)or 0 (Z2); same with classical mechanics (interaction), which is basedon functions (∀x∃!y), which leads to deterministic models etc.

Quantum information (COM) deals with strings of qbits consistingof a superposition (complex linear combination) of 1 and 0 )(“just”change the “coefficients”: CZ2!). This transforms the classical “ex-clusive choice” into an “offer” of possibilities (of evolution/correlationbetween cause and effect), but also allows a “fine tuning” mechanicsto ballance positive and negative contributions, “pros and cons”, intothe model: a multitude of paths/ways a cause may produce an effect.Viewed in this way, it is a natural upgrade of the bold “deterministicway” of classical science (“we can understand everything, in princi-ple”), to a more humble, realistic point of view: “still “dubito, ergo ...”modeling”.

A priori we don’t really know (or we are not so sure), how event Aled to B, but we have some hints ... “this way OR that way” (NOT the“God-like confidence”: this XOR that). We wish “tertium non datur”,but ... the simple/naive picture about reality is history by now ...

Then, how should “We” think, if our models and apparatuses arequantum (deep down)? When we read the quantum information we


project onto a measurement basis [BZ] because ... (?) or we reallydon’t have to!? (n.a.) Indeed, why should we “reduce”/collapse/ etc.the quantum description? Well, obviously because we can’t process QI(not well anyway), since we are not equipped with quantum computingcapabilities/algorithms/logic (yet), since ordinarily we wouldn’t benefitfrom it ... Or do we?

The fact that the system is in a state (message symbol) of 1xor0 isa “property” of the system, but having the system in a state of 1or0 isnot [BZ], because we don’t like them. It seems that “property” reallymeans “classical property”. This reflects the fact that in an S → Ointeraction/communication, if S is modeled as a quantum system, yetO as a classical system, then and S − property is not an O− property(how else?)...

Ultimately, the question is when O should “think in a quantumway” and when not! We should get used to Q-think since the quantumworld (from nano-technology to astrophysics and beyond, homeopathy,cold fusion, cryptology etc.) reveals untapped possibilities reminiscentof miracles (“How is that possible?”, i.e. conceivable, since otherwiseperfectly “real”!).

Well, a good (small) step for mankind would be to “upgrade” thecurriculum ... (probably I shouldn’t go there!). Teaching 400 yearsold physics and 2000 years old math in school [Smolin-3], when theknowledge accumulates exponentially, is leaving everybody behind!

Solutions? Plenty! For example, if you would like to learn physics,go to ’t Hooft’s web site 3 and ... teach yourself! (see also [‘t Hooft]).

On the other hand, why hurry so much? “Do we really have to?”(“Langsam, aber sicher!”).

8. Add a pinch of ... Einstein relativity!

Now the “genuine” relativity, Einstein relativity that is, can bethought of as another “projectivisation effect”.

Again add a ghost-like dimension to space and describe Galileanmotion in the resulting projective space: relativity seems to emerge(Minkowski space-time) ... (?).

Some say that this “concoction”, that is mixing quantum mechanicsand relativity in a coherent way, yields Quantum Field Theory! Itwould be instructive to see a “preview” of this “prediction” in such asimplified way (theoretical “gedanken experiment”) ...

3http://www.phys.uu.nl/thooft/theorist.html How to become a good physicist?

Part 2

Reference Manual

CHAPTER 3

The Principles of DWT

The mathematical implementation of DWT is based on representa-tions of Feynman Causal Structures (FCS), called Feynman Processes.The name should suggest that FPI approach to QFT may be thoughtof as a complexified and enriched version of Markov processes. QFT, asa 2nd quantization, is in a sense a “micro theory”, when compared toQuantum Mechanics as an effective and macro-theory. Alternatively,QM is a (0+1)-QFT [Zee], i.e. “space-time” is collapsed to a point. AFeynman Causal Structure plays the role of a resolution of a point.

The above alluded to representation is rather a pairing, allowing toincorporate IDOFs and enabling the I/E-duality.

FCS are essentially PROPs with additional structure, generalizingQFT (algebras over the Feynman PROP), CFT and VOAS (algebrasover the Segal PROP) and TQFTs viewed as algebras over cobordismcategories etc.

1. Why Path Models?

The reasons steem from the main corollaries:- Leads to Automaton Picture (universal computing model etc.)- Enables both “local” and “global” (i.e. non-local) features of the

theory- Extends QM: implemented as matrices over complex numbers, it

is a Feynman theory in disguise (complexified Markov process; but notenriched!).

- Why the IO-computer model? because, as Feynman said it [Fey1],modeling a quantum system starts by splitting the universe in twoparts (system and non-system) and modeling the interface between thesystem and the rest of the universe (external influences). We add tothis suggestive image the third component, the observer (“modeler”),which tries to control and make sense of the In’s and Out’s. It leads tothe ABE-model of measurement.

2. The Causal Resolution

Modeling reality is management of DOFs (computational resources).

47

48 3. THE PRINCIPLES OF DWT

The “perturbative approach” delineates the “free” from the “inter-actions”, at each stage in the resolution.

The “point” is the system overall, without any parts distinguished,i.e. the theory at the corresponding “scale” is effective. Various “res-olutions” distinguish subsystems (parts of the parts of the parts etc.)modeled as “points”, as opposed to interactions which are modeled as(multiple) arrows (small category, rather then graph, but to keep theterminology simple, we will still use the term graph).

Mathematically, an example of a causal resolution of a system isa double complex Gn,m [FI1], bigraded by the number of internal andexternal vertices of the source and target of the interaction graphs. Theaxiomatic version includes the other usual examples.

The (asymptotically) Free Theory amounts to the external struc-ture (external points - we will avoid the term “legs”) while the internal“virtual” points with the internal arrows correspond to (model) theinteraction(s).

In the Lagrangian FPI picture, splitting the Lagrangian L = Lfree+Lint amounts, via Wicks Theorem to giving the class of graphs and thepropagator as the inverse to the quadratic part, once the representationis given via an action and the Feynman rules.

The free classical theory amounts to Galileo’ relativity (symmetriesdetermining a homogeneous space).

The role of graphs, besides being the building blocks of transitionprocesses:

Γ−Γ→ Γ+

is to encode the In/Out data corresponding to preparation apparatusand observation (measurement) apparatus (again, after representation,i.e. “coloring” the vertices with operators etc.)

In this way “space-time” is the “vacuum” from which pairs of par-ticle and anti-particles pop in and out. Our need to “fill the space”with say R3 between particles should be satisfied by retaining its sym-metries (Galileo, Poincare etc.), since usually Space-Time comes as ahomogeneous space. Then one can “convert” the EDOF per parti-cle (point/object etc.) into IDOFs, towards a unification of symme-try groups (internal and external). At this stage it is productive tosimplify the picture and assume that there is a universal IDOF perpoint, namely a qbit C × C, with its symmetry group U(2). Thinkingin this way allows for a direct connection with the CS-interpretation.On the other hand, due to the “coincidences” of structure and rela-tions between quaternions and the other key groups of transformations(SL2, SU(2), SL(1, 1),Diff(S1) etc.) there are various ways (“enough

3. THE FUNDAMENTAL PRINCIPLES OF DWT 49

resources”) to connect with the traditional picture (gauge groups, con-formal symmetries, Poincare/Lorentz symmetry).

3. The Fundamental Principles of DWT

The 1st Fundamental Principle of DWT was explained above §2:a System is modeled as a Feynman Process, i.e. a representation of aCausal Resolution of DOFs (QDR).

As suggested by the lows of radiation of black holes Ch.1, there is anew unifying principle going beyond Einstein’s equivalence between en-ergy and matter (see Ch.6 §6.2). This principle related to the missing“4th Law” ([Davis1]) implies in a concrete way the equivalence be-tween space-time and matter/energy, or from the alternative point ofview, the equivalence between information and energy (matter/space-time etc.).

More precisely, the 2nd Fundamental Principle of DWT states theduality between external DOFs, determined by the QDR and internalDOFs, given by a functorial representation (as a Feynman Process):

Feynman process ∼= duality(causal structure,matter&energy).

The familiar EDOFs allow to describe the external dynamics of (anobserved) system, i.e. its motion. IDOFs allow to describe the internaldynamics, i.e. its evolution in internal state-space.

Although duality blurs the distinction between particles and fields,i.e. particles, modeled as (labeled) vertices and interactions modeledas relations between vertices (Feynman rules and all that), the externaldynamics primarily concerns the dynamics of particles/vertices whileinternal dynamics concerns the dynamics of internal states and typesof particles.

Now, it is natural to assume that the external dynamics is controlledby the Action Principle (energy/Hamiltonian dependent), while theinternal dynamics is controlled by the Entropy Principle.

The 3rd FP of DWT states the balance between energy E andquantum entropy I:

F + TI = E = mc2,

where F is the Helmholtz free energy (see partition function §4), Eis the relativistic energy, T is the internal temperature and I is thequantum information charge/entropy (see Ch.6).

Such a conservation law should be obtained from a symmetry prin-ciple via the Noether Theorem.

50 3. THE PRINCIPLES OF DWT

A quantitative development of the above ideas will be deferred tillafter reexamining the classical and quantum information flow and en-tropy (Ch.5 and Ch. 6).

CHAPTER 4

Causality: Information versus Time flow

The dynamics is usually associated with “time”, and modeled as a“time flow” (1-parameter, continuous or discrete, semi-group of trans-formations etc.).

The DWT realizes the importance of the concept of “informationflow”. Roughly put, the causality models interactions, and while thereis no “foliation” allowing to define a “global time, there is a current ofquantum information “flowing” in a quantum process (quantum com-munication). The observer in the SO-model (system - observer) im-plements a “global time” as a labeling of the information acquired(interaction experienced in a S-S-interaction). It is also a “flow”: theflow of symbols/states is the IO-information flow at the level of thecorresponding interface (S-O, O-S etc.).

Due to our similarity in information processing (roughly the samehardware!), humans (etc.) agree on a “simultaneity” facilitated by thehigh speed of light, when averaging over quantum fluctuations whichwere irrelevant for us in the past (i.e. our previous “communications”!).

A quantum fluctuation, say a creation of a particle-antiparticle pair,viewed as a creation of additional internal DOFs, in the DWT is consid-ered equally a creation of “space-time” (there is nothing “in between”particles, besides the potential creation of additional I/EDOFs!).

Classical information and Shannon entropy can be investigated interms of decision trees. Keeping in mind that a Feynman process is anenriched (“multi-paths”) complexified (changing coefficients from realto complex numbers) Markov process (focus on states and transitions),enlarge the class of trees as “models” to the class of graphs (networks,of course) and replace probabilities associated to edges (paths) withamplitudes of probability, to enable creation and destruction of quan-tum information (qbits).

Then the Shannon’s entropy which at the level of trees is a residueof a more powerful invariant (to be explained bellow) “upgrades” tovon Newmann entropy which is still a residue of the correspondingquantum invariant upgrading the classical invariant.

51

52 4. CAUSALITY: INFORMATION VERSUS TIME FLOW

Recall that allowing graphs as mode general models, introducesloops (feedback), which provide the quantum corrections to the classicalapproximation/picture.

In the diagrams that represent quantum processes (contributions tothe resolution of a quantum process), quantum information (entropy)“flows”, as intuitively “felt” for example in [Coecke1].

The definitions and exemplifications implementing the above “projectdescription” will be given as marginal comments on [BZ].

The categorical implementation based on monoidal/braided cate-gories with duality is deferred to the Reference Manual.

CHAPTER 5

Modeling classical information flow

Classical information (logic) deals with elementary symbols consist-ing of either 1 XOR 0 (exclusive “or”). Same with classical mechanics:for any x there is unique y, i.e. functions. This leads to deterministicmodels (no “creation” operation/coproducts) governed by the classicalmessages Z2N (sequences of 0 and 1).

Quantum information needs elementary symbols which are complexlinear combinations of 1 AND 0 at the same “time”, called qbits. Thisallows to create dichotomies; it’s an “offer of possibilities”, a choicegiven, leading to an escape from fate!. Quantum mechanically it al-lows a truly “dubito ergo cogito”, since it allows for multiple contribu-tions (ways), i.e. paths, for “one cause” to produce an effect. For anobject oriented (categorical) mind, such a correlation is described as aHom(cause, effect) in categorical language; the physics interpretationis clear: the set/class of paths, as Feynman was thinking.

What is truly new is that some alternative ways may diminish afuture possibility (destructive interference).

1. Measurement: a “paradox”, or what?

Now will this help understand the measuring paradox? Mathemat-ically, measuring (an R-operation - see Penrose), is a “projection into ameasurement basis” [BZ]. With the interaction-communication paral-lel in mind, to “read out” the result of the quantum communication (theinteraction/ correlation cause-effect) we need to “classically decode”!Why? may be because we do not want (actually “need”) to processquantum information, or we use devices incapable to do so (bulky clas-sical apparatuses; but if a polarized filter is followed by an other one,and another one etc. in a sequence of measurements, this would nothappen ... except: what’s the use for us? we were not “eavesdropping”on what the result was, and we are really curious with our classicalcurious mind!). We and our macro-apparatuses are equipped only withclassical computing algorithms and logic, be it IO-level only or not.When we claim that “properties” (and we mean classical properties)

53

54 5. MODELING CLASSICAL INFORMATION FLOW

do not exist prior to th measurement we refer to the comparison of thesystem’s behavior with classical M-observables of our choice.

Of course there is no 0 or 1 M-values prior to measurement, yetthe quantum property of the system (part of the holographic world,yet reasonably accurately modeled as a Hom(cause, effect)) is there(what is “is” ... well, that’s not my story!), whether we read about itor not!

It is a “hidden property”, since it is not classically measured (noclassical IO) by us. The same projection, truncation etc., occurs whena QS → CO interaction/communication takes place, involving a quan-tum system QS and a classical observer CO (our conscience or appa-ratus).

To model and predict the outcome (behavior) of such a “combo”one may chose the ABE-eavesdropping process, probably leading to alogic which is an extension of the (irreducible) quantum and totallyreducible classical logic (see Ch.7), as it was suggested elsewhere (§3,Diagram 1).

What can we do about it? ... this is the question! With ourclassical hardware (body) we indeed are confined to classical software,for the day-time and everyday activities at least; but I believe that theintegration of acquired information is done otherwise, more efficiently...

2. Shannon or Von Newmann?

Now, to use Shannon information to measure quantum informationwould be surprising, as it was not designed to do so! Yet the ideas,some implicit, true, are deep enough to be used without change; allwe need to do is to upgrade the implementation: “change coefficients”and change the trees (forests) into ... (more general) graphs! This isquite natural, since quantum phenomena can be often well modeled as“loop corrections” (i.e. don’t forget the feedback, even if it is back intime! ... that is, relative to our communication’s time!)

Then a correspondence emerges relating quantum entropy and clas-sical entropy, represented by Shannon’s and Von Newmann’s mathe-matical implementations:

Q− info

reduction // C − info

vN − Entropy

? // S − Entropy.

3. SHANNON’S MEASURE OF INFORMATION 55

Now in both cases entropy is an invariant of the µM -correspondence(M-parts or µ-states; e.g. Boltzmann correspondence), which is de-scribed usually in terms of partitions, but which can be modeled by(decision) trees or more general graphs (see also §3.3):

ρ− states

vN−Entropy

// µ − states

S−entropy

< M >ρ?// < M >µ (M − cells)

This will disclose a deeper underlying invariant, we call InformationCharge, possibly related with Kolmogorov-Sinai entropy as a limitingcase, for which the usual “global entropy” appears as a residue sensitiveto the external structure of the tree/graph (see §6.1):H(Res(tree)) = |tr(QI(tree))|2, SvN (graph) = tr(QI(graph)). (?)

The quantum information charge flows through the network as a cur-rent subject to the usual Noether formalism: it is due to symmetriesof the Lagrangian controlling the dynamics of the network, expressedin the language of category theory as braided categories with duality.

To be more precise, we need to “creatively” review Shannon’s ap-proach to information theory, with an eye on [BZ], turned critical forliveliness, definitely not to offend in any way!

3. Shannon’s measure of information

Since a comparison “quantum - classical” is always useful, we willconsider both quantum and classical descriptions at times.

3.1. What is a (good model of a) quantum system”? As-sume a quantum system of “simple type” (H) (the Hamiltonian formsa complete set of observables) is modeled by the (prepared) state ψrevealing Ei as a classical measurement of the quantum (classical) ob-servable H (E), energy, with a probability distribution π = pi. Thenpi = |λi|2, where λi are the probability amplitudes relative to a mea-surement basis ψi (i.e. we know how to filter/refine/partition the sys-tem ...) ψ =

∑

i λiψi. The expectation value of E is < E >ψ=<ψ|H|ψ >.

The probability distribution can be alternatively introduced notusing amplitudes as above, but also using a partition function Z(β)depending on a global parameter β (the “temperature”), under theassumption of a “normal distribution” law (see §4).


3.2. What is a (good model of a ) classical system? Nowthe Boltzmann correspondence for a classical system, in view of thelinearization of classical physics (Ch.2 §6), can be interpreted as follows.The classical system is in a superposition ψ =

∑

i niψi with bits ni ∈BZ2 as coefficients.

The classical urn can be used as a typical example. The urn isthe (classical) system with N parts (subsystems), the color is the M-observable, with m = Spec(M − observable) possible colors which de-termines a partition of the state space Ω (mN states): P = Ωi with|Ωi| = ni, where i ∈ [m] = 1...m. Now shake the urn (i.e. applyan unknown evolution, decoupling the system and the observer by atransfer of entropy: decoherence).

Then “listen” to (interact with) the system: extract the balls one byone. In fact the model changes (NOP not constant), but if accountingfor the balls extracted (message received in this S−O-communication),then NOP is conserved in the larger Maxwellian box (see Maxwell’sdemon) with N − k balls on one side and k on the other of the one-way permeable screen. To understand the information implications,we should think of the balls as information carriers (symbols), andthe sequence of colors as a message received by the observer. It is aclassical information source, but with memory, characterized by theprobability distribution π = pii=1..m of occurrence of the “emitted”symbols (states).

3.3. Decision trees. A statistical approach, like the one above,inevitably uses averages and the limit of large numbers. We are lookingto bridge the gap with “particle physics”, and therefore think abouta measurement as an exchange of a “infon” carrying the “informationcharge” of I qbits.

As a provisional stage, think of the associated partition function ofthe Urn as a linear combination of labeled binary trees:

Z(d) =∑

t tree

ct/|Aut(t)| t,

where d is the maximum depth (“number of questions”), for exampled = NH, where H is the Shannon entropy:

H = −∑

pilogpi.

The leaves (terminals) represent length N sequences. The number oftypical sequences, i.e. those realizing an actual (experimental) proba-bility distribution “close” to the given one π, is:

W ≈ 2NH .


¿From the point of view of QM, a decision tree as above, with verticescorresponding to questions and descendent edges labeled with Y/N,represents a sequence of measurements having the questions as observ-ables with outcomes Y/N (spin-like). The Shannon entropy is sensitiveto the external structure of the tree, not at its particular structure:

H = H(Res(t)),

where Res(t) is called the “residue” of the tree, representing the exter-nal structure of the tree, i.e. the corolla resulting from collapsing allinterior vertices and edges.

To relate with the Feynman formalism, the leaves (final events/block messages) may be identified with paths joining the root (“source”)and particular outcome (“target”) (see §6.0.5).

A more general framework has to allow all trees (classical), not justD-ary trees (“Brownian motion in an D-dimensional lattice).

Shannon entropy (information) represents the minimum number ofquestions to determine any such particular path (sequence), i.e. theinformation of the longest path, in other words the info contained inthe “directions from root to leaf”.

Intuitively speaking, if the path is the cause and the particularsequence is the effect, then there is unique cause determining the effect.In the quantum world one should allow loops, and mimic the pathsanalysis (see what follows) with amplitudes; Von Newman entropy mustbe used (obtained) in place of Shannon entropy.

The Shannon info density is the info per “jump” or transition insuch a “Brownian path”. Think of the “space” as a quiver, similarto Markov chains. If there is a source without memory, it is called aBernoulli scheme.

In the quantum case, the quantum event is the message received,and depends on the type of measurement performed (i.e. what ques-tions were asked). The answers, i.e. the particular message received,can be predicted with a certain probability, depending on the observer’sprior knowledge about the system. Recall that, in the Hom model, ameasurement consists of preparation, interaction and observation:

System→ Observer : Hom(cause, effect).

If the “cause” itself is an interaction, it leads to the ABE-model:The “Level 1” of structure consists of the SO-interaction/communication,

as above. The “Level 2” represents a classical “eavesdropping” on thequantum process from Level 1.

We will revisit Shannon’s theory of classical information and try torefine it in terms of information flow in graphs.


The Measurement Process

Preparation Observation

Encoder

Alice

C−int. ))SSSSSSSSSSSSSSSSSS

Q−Interaction // // DecoderBob

C−int.uukkkkkkkkkkkkkkkkkk

Eavesdropping

Eve

Experimenter

(1)

Figure 1. The ABE-model of the measurement process

3.4. Shannon’s entropy: a framework. A simple frameworkto model an interaction/communication (system/information source)is a µM-correspondence (e.g. Boltzmann, etc.). It consists of a statespace Ω (states/symbols), together with a partition P = Ωii∈I , suchthat the indices are in a (1:1?) correspondence with the values of anM-observable, say the energy E:

IΩ

E

##FFFFFF

FFF

P // Range.

There are various levels of mathematical sophistication implementingthe idea: foliations of the state space (e.g. Hamiltonian actions etc.),vector bundles, groupoids and moduli spaces etc. We will focus onthe underlying idea and work on the above toy model: the relation(quotient space) determined by a (measurable) function defined on afinite set (measure space relative to the counting measure):

Ωpr→ “Spec(E)′′.

The statistical approach modeling “our independence” regarding thesystem (we do not control it!), starts from assuming a probability dis-tribution π = pii∈I representing our best estimate regarding the mea-surement outcomes (symbols received).


The goal is to measure the additional information needed to charac-terize the events, i.e. to specify the “address” of elements of Ω (memoryneeded to store the address).

Shannon was looking for an invariant depending on the probabilitydistribution π:

H(p1, ..., pn), n = |I|,which in turn may be assumed to be correlated to the partition P (e.g.normal distributions etc.).

3.4.1. It’s all about ... colorful decisions! Our goal is to introducea finer invariant on decision trees which will produce Shannon’s en-tropy under the equivalence relation determined by the residue, i.e.the external structure of the colored tree:

H(Res(tree)) = Q(tree).

A colored tree is labeled by subsets in the state space, and the edgescary probabilities determined by the values on their vertices:

e = Ωi → Ωf , p(e) = |Ωf |/|Ωi|, S(e) = − ln p(e).

The invariant Q is finer since here the “Feynman rule” involves a trivialpropagator: ω = 1 (see §6.0.5).

Extend the probability function defined initially on edges, functo-rially, on paths:

p(v1 → v2 → ...→ vk) = Πp(vi → vi+1).

Similarly, S has an additive extension,The residue of a colored path in a colored tree is

Res(γ : Ωi → Ωf) = (Ωi,Ωf ),

i.e. the colored external structure of the colored tree. Then p(γ) =p(Res(γ)).

The alternative notation:

∂t = Res(t)

will be used, as a hint to the role of boundary of the external structure,when the graph is interpreted as a cobordism 1.

At this point we would like to coin a fancy formula, trivially satisfiedhere, as a beacon for later developments of a quantum version of thetheory. If r is the root of a tree labeled with the total state space and t

1This role will be exploited in connection with holographic theories Ch.10 andblack hole entropy Ch.9 §4.


a terminal leaf labeled with the corresponding “favorable events”, thenthe associated probability is:

p(t) =

∫

γ∈Hom(r,t)

e−S(γ) Dγ.

3.4.2. Shannon’s axioms for classical entropy. The entropyH(p1, ..., pn),defined as a function of the probability distribution, is assumed to sat-isfy the following requirements [Sh] (see also [BZ], p.3):

1) It is continuous in pi;2) (Normalization) If π is the equiprobable distribution, i.e. pi =

1/n, then H is an increasing function of n;3) If the partition of the state space is refined at the ith outcome, i.e.

Ωi = A∪B is a disjoint union, with associated probabilities a+ b = pi,then

H(p1, ..., a, b, ..., pn) = H(p1, ..., pn) + piH(a/pn, b/pn). (2)

Shannon shows [Sh] that only

H(p1, ..., pn) = −∑

pi ln pi

satisfies the above axioms.At this point we elaborate a different approach based on the idea of

“scaling of DOFs”, or the “quantum dot resolution” (QDR), i.e. relat-ing the operation of collapsing/insertion of subgraphs (DOFs) with theinformation loss/gain. Mathematically speaking this bring up front thecoalgebra structure present in QFT renormalization, as introduced byKreimer and developed together with Connes [Kreimer, CK1, CK2].The relation with the resolution point of view comes from [Ion03,Ion04-1], which ties Kreimer’s coalgebra structure with Kontsevichgraph homology [K92] (see also [FI1]).

3.4.3. An invariant of trees. In the above example, with n = 2 andi = 1 for simplicity, consider the corresponding colored trees t (left)and t′ (right):

Ω

~~~~~~

~~~~

@@@

@@@@

@ Ω

AAA

AAAA

A

~~

Ω1 Ω2 Ω1

~~~~~~

~~~~

AAA

AAAA

Ω2

A B


and let γ be the subtree of t′ with root Ω1. Then collapsing γ yields t.This is expressed by saying that

γ → t′ → t, t = t′/γ

is an extension of graphs and t is a quotient of t′.Then the above relation can be “restated” (anticipating a bit) as

the Euler-Poincare mapping property:

H(Res(t′)) = H(Res(t)) +H(Res(γ)), (3)

where now H should be replaced with a relative entropy:

H(p1...pn|p),∑

pi = p

such that, if p = 1 then H(p1...pn|p) = H(p1...pn) (see 6).Let’s define a function of colored trees Q(t) and require that it

satisfies two axioms:1) It is an Euler-Poincare mapping, i.e. given an extention as above:

Q(t′) = Q(γ) +Q(t)

2) Satisfies the normalization condition:

Q(corrola) = p · (−∑

pi ln pi).

Our colored trees are assumed to satisfy a conservation condition.The set labeling a parent is the disjoint union of sets on its children(descendents), i.e. no “entropy production” at this stage.

As a consequence of our simple “Feynman rule”, the “probabilityflow” is conserved, i.e. at each branching, the sum of probabilities ofthe output edges equals the probability of the input edge.

An Euler-Poincare map as above is an abstract version of a “dimen-sion function”; other examples: Euler characteristic from geometry etc.It is perhaps the “thermodynamic weight” foreseen by Birkoff and vonNewmann (see §3).

Then the divergence of the information flow yields Shannon’s rela-tive entropy H via a divergence theorem (Stokes/duality again 2).

Hologram Theorem 3.1. 3

H(∂t) = Q(t)

The above properties/axioms come from the correspondence be-tween the probability distribution and partition function §4, under theFPI interpretation §6.0.5.

2H = dQ - see the path algebra derivation interpretation of the entropy §7.2.23See Ch.10


In order to clarify our point of view to the reader (and to ourselves),we will take a detour to statistics and thermodynamics, in order to linkthem to the information theory approach to entropy.

4. Probabilities and Partition Functions

A common framework is provided by the µM -correspondence, i.e.the multiplicity of micro states (partition) corresponding to a particularobservation ([Entropy], p.255), classical or not. One may bear inmind the urn example (see bellow), for the classical case, or loc. cit.([Entropy], p.255).

Imagine a system modeled as consisting of N non-interacting sub-systems (“particles”), exhibiting n possible levels of energy (the M-observable), for a total of nN possible M-states (macro states). In theCS dual language, the M-states of the particles are “symbols” distin-guished by the observer at this “resolution”).

4.1. Micro and macro states. The following combinatorial dis-cussion has a dual role: studying one system composed of N non-interacting subsystems with state space Ω, which we will refer to asthe “space-like” point of view (prefered in this section), or studying Ncopies of one system, i.e. an ensemble, referred as the “time-like” pointof view (see 5.3.1).

Let Ω denote the possible micro states (µ-states) of one particle,usually associated both with external and internal DOFs.

The M-observable E : Ω → R determines the possible internalmicro states E = Eii∈I of one particle. The M-observable E foliatesthe µ-state space of a particle, determining a partition P = Ωii∈I =Ω/E. We will call Ω/E the internal micro-states (indexed by I) or,in view of our usually “big apparatuses”, the macro-state space of theparticle (M -states).

Remark 4.1. Indeed, we tend to “project” our apparatuses’s out-comes (our “artificial eyes and ears” etc.) as internal DOFs (e.g. spinetc.), while the “remaining relations”, the so called “motion”, be at-tributed to “space-time” as modeling the external DOFs.

The members of the partition denote the corresponding internalmacro-states (M -states), and the state space is an “extension” (bun-dle/fibration etc.) of internal macro-states by external micro states(addresses: partly known and partly unknown in a measurement):

Ωi → ΩE→ E ∼= I.

We will say that the particle is of type (Ω, E) (or just E).

4. PROBABILITIES AND PARTITION FUNCTIONS 63

Now energy is a special M -observable: it “discriminates” statesaccording to their probability of occurrence, leading to the so calledBoltzmann’s distributions in correspondence to partition functions (see4).

Definition 4.1. An M-observable is a Hamiltonian for a µM-correspondence iff the associated partition is compatible with the equiprob-able distribution partition.

In other words, the observable factors through the probability den-sity. Then one can reduce the analysis to the analysis of the modulispace Ω/E together with the associated probability density π.

A system of N-particles has as state space Hom([N ],Ω, the µ-statespace of the system.

States fall into “categories”, represented by the type of the partitionunder the M -observation E:

Hom([N ],R)

Hom([N ],Ω)

E∗

66mmmmmmmmmmmm∃!

col// Hom([N ], I).

E∗

OO

N/col// N − partitions.

A µ-state µ is mapped to col(µ) representing the outcome of the E mea-surement of the n-th particle (“color”). An M -state is the associatedpartition of N :

M = N/col(µ) = mii∈I ,say (m1, ...,mn) in the example from [Entropy], §12.4, p.255, whereI = [n]. Of course, the “completeness condition” holds

∑

mi = N and:

|Hom([N ],Ω/E)| = nN , |Ω/E| = n.

The number of µ-states µ of an M -state M is:

WN(M) = |(N/col col)−1(M)|.In the above example (I = [n]):

WN(m) = N !/m!, m! = m1!...mn!.

We will say that E foliates the state space into cells (addresses)ΩN/E, which correspond to macroscopical states (M-states). To the M-observable E we may associate the decision tree (corolla) (Ω, (Ω1, ...,Ωn)),where the leaves correspond to the possible measurements E = Eii∈I .

The expectation value of E is:

< E >=∑

niEi =

∫

[N ]

M∗(µ).


The large numbers assumption relates the combinatorial descriptionwith the “limiting description” in terms of probabilities distributions(see 5.3.1):

M − states : π = limN→∞

([N ]/M∗(µ))/N.

relating the combinatorial probabilities ni/N , a projective space pointassociated to the combinatorial M-state (see 4.4.1), and the probabilitydistribution π = (p1, ..., pn), as a continuous limit M-state, which isalso a projective space point, due to the normalization

∑

pi = 1.

4.2. The Law of Large Numbers. The correspondence betweenthe combinatorial description of the micro-macro states (Boltzmann)correspondence and the stochastic description in terms of a probabilitydistribution is essentially the content of the Law of Large Numbers.

The probability of a particle to be in the i-th cell, i.e. its µ-stateto belong to Ωi is pci = mi/N .

Then the (combinatorial) probability distribution associated to theM -state M of an N -ensemble of particles of (Ω, E) is πN = (pc1, ..., p

cn)

(I = [n] for readability).Now if considering systems with more and more particles of the

given type in a sequence of macro states M(N) such that the associatedprobabilities (projective points) converge (assuming the large numbersprobabilities exist):

π = limNπN ,

then, in the above example, with the Stirling approximation used:

limNWN (NπN)1N = eH(limpiN ),

where H(π) denotes the Shannon entropy.So, with e = limn(1 + 1/n)n in mind

limNW (N)1N = eH lim

N

will be interpreted in terms of the classical information charge functionon trees §6.1.

4.3. Shannon entropy and typical configurations/sequences.Now assume that the macroscopical knowledge about this “modulispace” Ω/E is given via the associated probability distribution π = pi,where pi = mi/N , defining the so called typical category (“generic cat-egory”).

The probability of such a category is:

p(m) = WN (m)/nN .

4. PROBABILITIES AND PARTITION FUNCTIONS 65

Then Shannon entropy is the large numbers limit (via Stirling’s for-mula):

limN lnWN (m)/N = −∑

i

pi ln pi.

It represents the amount of information per particle necessary to iden-tify the “address” (category) of a state of the system.(?)

4.4. Partition function. The “continuous limit” (probability dis-tribution) may be encoded in the partition function of the system.

Let < E >=∑

piEi be the expectation value of E, thought of as aconstraint.

Define the partition function associated to E ([Fey1]):

Z(β) =∑

e−βEi.

Then there is β such that

pi = e−βEi/Z(β).

They maximize Shannon’s entropy H(π).Indeed a Lagrange multipliers method applied to the optimization

problem with two constraints:

max H(π) = −∑

pi ln pi,∑

piEi = E,∑

pi = 1,

yields:

∂piH = βpi + γ => pi = e−γ+βE,

and therefore defining:

Z =∑

e−βEi => γ = lnZ, pi = e−βEi/Z,

with β determined by the energy constraint.Then

H = βE + γ, γ = lnZ,

or alternatively, after defining the Helmholtz free energy by Z = e−βF ,obtaining the Gibbs entropy ([Entropy], p.256; H(π) = S(E)):

H = −βF + βE.

The equivalent relation TH + F = E will lead to one of the Funda-mental Principles of DWT (Ch.3).

Now, why “free energy” and “reduced energy”? because we arein a projective space situation relating a “free theory” with a reducedtheory, as we will explain next.


4.4.1. Probabilities and real projective space. Probability distribu-tions are points of a projective space due to the normalization (con-straint) condition

∑

pi = 1. At this point a comparison with quantummechanics is instructive; remember the Markov-2-Feynman “upgrade”:complexify and enrich the theory, no “free-2-reduced theory reduction”,since both are theories defined in projective space!

CM/Statsitics QM

States π = (p1, ..., pn), ψ = (c1, ..., cn),

π =∑

piei ψ =∑

ciψi

Normalization ||π||1 = 1 ||ψ||2 = 1

The Bohr correspondence rule:

pi = |ci|2

is the link between the classical (“real” as in R) and quantum pictures(“complex”, although in fact real!).

In QM the state is a vector ψ which can be represented as a linearcombination of pure states (measurement basis) with coefficients ampli-tudes of probability, i.e. according to Bohr’s statistical interpretation,complex numbers whose modulus are the probabilities of measuringthe corresponding outcome (relative to an M-observable). The quan-tum states are in fact rays in the projective space of the correspondingHilbert space, represented (ambiguously modulo a phase) by a vectorof L2-norm 1: ||ψ||2 = 1.

So the probability distribution π = (p1, ..., pn) is the classical analogof a state, except it is normalized relative to the L1-norm:

||π||1 =∑

|pi| = 1.

If one considers arbitrary basis elements of the corresponding ray inRn

+, then its norm is the partition function:

Z = (Z1, ...,Zn),Z = ||Z||1, pi = Zi/Z;

of course, still under the assumption that probabilities (or Zis) aredetermined by the energy levels (see 4.1):

Zi = e−βEi.

5. THE THERMODYNAMICS LEGACY: A HIDDEN MESSAGE? 67

In some sense (leading to the 3rd FP-DWT Ch.3), the internal dy-namics, i.e. the evolution of the state π, is governed by the MaximumEntropy Principle (max H(π)) 4.

The free energy is the energy counterpart in the “free theory” inthe Rn

+ space. (?)

Remark 4.2. In view of the above discussion §4.4.1, the differencebetween classical (statistics) and quantum mechanics is that the formeris an L1-norm real projective space theory (“analysis-like”), while thelatter is an L2-complex projective space theory (“geometry”). But the“hart” of QM/QFT is the annihilation-creation not only of particlesand anti-particles, but also of “possibilities”/histories (superpositionwith interference).

4.5. Taking absolute values and limits: old habits! Why isstatistics a (internal) dynamics in R+? As if a Z2-graded version (Y/Nwith probabilities, i.e. “double the bit” ...) might be the missing link(intermediary stage) between the classical picture (logic) and quantumpicture (logic) ... Is this related to considering a projective representa-tions (via central extensions) and linking the “projective space” picturewith the free theory?

Also Shannon entropy is a classical limit of the combinatorial de-scription (counting) of configurations (permutations/symmetries, Youngtableaux, braiding, spin and statistics etc.). Stirling’s approximationlinks the two:

lnn! ≈ n ln n− n+ 1,

providing an asymptotic conceptual correspondence, as in quantiza-tion prescriptions (e.g. Dirac quantization etc.). What are the “trueconcepts”, after the “old habit” of taking limits is abandoned?

5. The thermodynamics legacy: a hidden message?

To better understand the role of entropy in physics, besides con-necting with information theory, it is mandatory to review what ther-modynamics has to say about it.

The following “recall” has a pronounced subjective flavor, in viewof the Quantum Dot Resolution philosophy of DWT.

An informal thermodynamics primer may be found at [ThD] (whohas time for books anymore ...). “If we restrict ourselves to crystals...” (p.2), lattices or our discrete QDR, the main keywords suggest a

4What about the “external dynamics”? (FPI & localization ...)


certain hierarchy:

Entropy

ttiiiiiiiiiiiiiiiii

**TTTTTTTTTTTTTTTT

Internal (Reduced)

**UUUUUUUUUUUUUUUUUExternal (Free)

ttjjjjjjjjjjjjjjjj

Energy & Enthalpy

This cannot be a coincidence; it must be the duality at work (internal/ external) together with the dual description projective / affine space.

5.1. Internal Energy U . We will rephrase [ThD] p.2.5.1.1. What internal energy is not. Neglecting external energy (e.g.

gravitational potential) due to interactions, i.e. due to the “visiblestructure” at a given “resolution”, and discarding internal energy which“never changes”, i.e. it is constant during our modeling process (noevolution, e.g. “mass”?), what is left are “vibrations ...”, i.e. thedynamics of some internal DOFs.

Again thinking of “energy” as an M-observable “foliating” the µ-state space into equally probable events, we understand the temperatureT as referring to the external contribution to DOFs, i.e. the numberof vertices of the graphs (via Boltzmann’s constant), while the “totalinternal energy” being due to the pairing with the “colors/labels” as-sociated with vertices, i.e. providing the types of subsystems via theirsymmetries: internal DOFs:

U = 1/2 T × EDOF × IDOF.

A thermodynamic macro-state is usually given by “the number of atomsN , the pressure p and the temperature T” (loc. cit. p.3).

We will speculate on “pressure” p later on (5.2.1). Now, what isthe dynamics?

5.1.2. Is it heat “virtual” work? A change may be due to heat trans-fer Q or “mechanical” work W , or changing the number of subsystemsN ! Assume for the moment N is constant, to focus on heat transfer.The First Law of Thermodynamics states:

dU = dQ− dW.


As a working hypothesis one may try to “explain” heat transfer via“work”, more precisely as an interaction process:

Subsystem (a)dQ //

dU

''OOOOOOOOOOOSubsystem (b)

−dW

dU77ooooooooooo

i.e. “reduce” the heat transfer to a “virtual work”:1) Convert internal energy dU into work “at (1)”;2) Transfer work −dW “from (a) to (b)” via interaction;3) Convert work −dW into internal energy “at (b)”.

5.1.3. Vacuum causal bubbles. A possible quantitative implementa-tion is suggested next. Consider the two subsystems S1 and S2 “point-like”, i.e. no structure is externally visible at this stage, so the con-sidered energy is internal. Insert a subgraph γ1 for S1, i.e. “resolvethe singularity” at (a) as a “virtual” generation of structure (vacuumcausal bubble). In step 2 above, redistribute the structure, by changingthe clustering of DOFs, say into γ2 containing the point representingS2, so that the remaining (only) point will represent S1.

Then collapse γ1 to obtain the “same” configuration, at the level ofnumber of points at least.

What is the actual “math code” for the above, is yet to be seen,but the simultaneous presence of collapsing and inserting graphs withan identity result at the level of external structure is reminiscent of achain homotopy:

ds+ sd = id,

where d is a graph homology differential (collapsing an edge) while scorresponds to insertion of an edge ...

5.2. Enthalpy and free energy. We will try to address somepossible lines of thought, keeping in mind the quantum jumps pictureCh.8 §1.2 and the radiation laws of black holes which tend to “redefine”the surface area and volume (§4).

Remark 5.1. If graphs model Space-Time, what is the “dimension”of Space-Time? The full structure is a 3-category (think 3-simplices)with a Poincare duality with “reduces” the picture to graphs plus or-der information and duality. At the geometry level, Hodge duality isinvolved (see also Ch.9 §3.

5.2.1. Pressure and volume. The pressure and volume as macro-scopical classical observables must be reconsidered in the context of adiscrete causal structure. The classical mechanical picture is clear and


quantization will carry it to the quantum realm, but still in the frame-work of a continuum space-time; what happens if a granular space-time(QDR) is used instead?

“Any mechanical work must change the volume”, since “something”must move! Is it due to a change in position or momentum, since in thediscrete picture v is not dx/dt (i.e. use Poincare-Cartan form ratherthen a Lagrangian). Either:

work

''OOOOOOOOOOOO// ∆ position

∆ momentum

dynamics

66mmmmmmmmmmmmm

or work as a change corresponding to some action, is momentum timesthe position change.

If pressure p, as a measure of interaction per unit of surface, isconstant:

dW = pdV, p = constant.

5.2.2. Enthalpy and free energy. The state function measuring theenergy needed to “form a substance” is called enthalpy [ThD]:

H = U + pV.

It measures the “heat of formation”, whether it is a chemical reactionor a particle interaction.

2H2

AAA

AAAA

Ae+

>>>

>>>>

>

• // 2H2O • γ //

O2

>>

e−

??

To find study equilibrium states, a minimum principle will involve boththe enthalpy and entropy, since the change in entropy occurs parallelto the change in enthalpy.

Remark 5.2. This “balance” between energy and entropy, competi-tion for energy between internal energy and external information needs.It will later lead to an new equivalence principle between informationand energy §6.2, allowing to account for a conversion between internaland external DOFs.

To study the equilibrium states of purely mechanical systems, i.e.composed of non-interacting subsystems, the appropriate state functionto minimize is enthalpy.


To study the equilibrium of thermodynamic systems, i.e. manyinteracting subsystems, one should minimize the free enthalpy, alsocalled Gibbs energy:

G = H − TS (G = U + pV − TS).

Remark 5.3. The last formula should be compared with the ac-tion term representing the change in the momentum potential in theHamilton-Jacobi formulation [Gosson], p.12:

Φ(r, t) = Φ(r′, t′) +

∫

γ

pdq −Hdt.

It suggests that the above thermodynamical potential is an internalversion of the external momentum potential, in this alternative (field-like) formulation of Hamiltonian dynamics.

The pV term (better d(pV ) = pdV + dpV ) is related to the ex-ternal description of the system (geometry and dynamics of EDOFs),while TS term (better TdS) accounts for the changes in the internaldescription.

Remark 5.4. The similarity with pdq−Hdt reflects upon “time” asa global and macroscopical description (symbols = labels) of “internalchanges”, which are not “motion” (external changes).

The entropy S of reversible thermodynamic processes is related toheat transfer Q and temperature T :

dS = dQ/T.

Then Gibbs energy is essentially:

G = U +W −Q,

i.e. an account for various energy contributions: internal and external(In/Out).

For constant volume and variable pressure, we learn that the ap-propriate state function to minimize is the free energy, called alsoHelmholtz energy:

F = U − TS, (F = G − pV ).

5.2.3. They’re just state functions after all. These, U,H,G,F , areall state functions, also called thermodynamic potentials and only oneis needed for a complete description from the point of view of ther-modynamics. This implies that the probability distribution, viewedas a complete state of the system, is determined by the correspondingfoliation by any of these “energy functions” (see §4.1).


Remark 5.5. When “relaxing” the constraint p ∼ dq/dt we prob-ably should add an additional state function to account for the dualitybetween internal and external DOFs. A tempting way to do so is topair them as a complex number, probably related to the “upgrade” fromprobabilities (identified by an energy M-observable) to amplitudes ofprobabilities (identified by two state functions).

5.2.4. The energy conservation: a homotopy? Since pV term (orpdV, dp V etc.) represents work, the Gibbs energy leads to a balanceequation:

E −W = U − TS.

Now the macroscopic equilibrium point of view, i.e. “thermostatics”,needs only one state function for a complete description. In otherwords, a complete family of independent M-observables consists of onlyone state function, i.e. a function of the type “energy” which separatesthe fibers of the microstate space relative to probability distributions(classical stochastic states).

The duality between internal and external DOFs will be representedby a balance between energy and entropy. This will involve two “a pri-ori” independent state functions, U responsible for controlling IDOFsand TS controlling EDOFs; the work term W = pV should be thoughtof as an external source, while Q = TS represents an internal source:

E − U = d(W −Q) = pdV − TdS.

5.2.5. To minimize or to maximize, this is the question. To findthe equilibrium M-states, i.e. to solve the “Thermostatics Problem”(thermo - since one state function suffices, and statics because, well,no dynamics is involved), minimize the free energy F = U − TS ormaximize entropy S ...

To calculate the canonical probability distribution π (M-state: thestochastic state or classical information source) corresponding to a statefunction E (some energy M-observable), one can use either (1) thecombinatorial picture counting the micro-states per M-state 4.1, or (2)give the partition function which will determine the M-state via theMaximum Entropy Principle (see §4). The large numbers assumptionstates that the second is the continuum limit of the first (see 4.1).

Although any state function describes the equilibrium of a thermo-dynamic system (the Thermostatics Problem), for crystals, the mostconvenient is the free enthalpy [ThD], p.6.

Clearly there are too many “If ... Then ...”, i.e. a case-by-caseanalysis when dealing with complex systems, which is ... expected ina way from such “expert systems” :-) Still there should be a unifying


point of view (framework/tool kit etc.) with a more “friendly userinterface” ...

For example, if energy is constant (constraint) maximize entropy(partition function correspondence). If energy varies “slowly” it meanswe are studying the “dynamics” (or kinematic?) of equilibrium states.Outside of the equilibrium is “hic sunt tiggers” land ... (see [Entropy]).Taking into account I/EDOFs in duality brings some hope though ...

5.3. Entropy - statistical considerations. Broadly speakingthe attribute (prefix) “Macro” entails forgetting labels and averagingquantities (“statistics”) while “micro” refers to individual quantities,carefully labeled (“addresses” etc.)

5.3.1. Ensembles. Given a (type) of system, an ensemble is a col-lection of copies of the system. If the system is an urn, say with N balls(internal structure - subsystems with EDOfs ignored but with IDOFsmodeling colors), then a collection of ν such urns or rather a repeatedinvestigation of one such system, idealized, is an ensemble. Even adice, with no explicit internal structure (N = 1 subsystems, yet with anon-trivial internal space with a measurement basis with 6 elements),can be “cloned” in a gedanken experiment, or rather through severaltimes and studied under the scientific assumption that the behavior isautonomous (astrology has a different point of view, of course).

The alternative “copies” or “sequences of “time chains” ω1, ω2...(Markov chains etc.) invites the computer science interpretation ofstates as symbols and ν-ensembles as messages of length ν. This dualitymay be sketched as a diagram:

Combinatorics of One System Source of Information

ν − Ensemble(Message of length N)

T ime chains .

Then the statistic entropy in the combinatorial picture corresponds tothe (classical/quantum) channel’s capacity.

There are two possible large numbers limits. One is the em con-tinuum structure limit limN→∞, with the purpose of “simplifying” thecombinatorial “chaos” a human mind can’t handle (avoids?), startingwith Newton’s endeavors in mechanics and brought to an end by thequantum theory. It is useful (mandatory?) when attempting to model“space” (EDOFs etc.) by continuum structures (manifolds etc.).

The 2nd limit is the large numbers limit limν→∞ which is again use-ful in approximating the original discrete structure of time, consisting


of communications exchanging quanta of information, in order to “sim-plify” the description towards a continuum model for “time” (rescalinga long ν sequence to [0, 1] is a sort of a “zoom out” to smooth outirregularities and the fine grain structure).

The relations between the various “boundary theories”, includingthe “classical corner” limN limν, will not be discussed here.

¿From the three types of ensembles, micro canonical (isolated sys-tem: fixed energy and number of particles), canonical ensemble (incontact with a heat bath at fixed temperature with a fixed number ofparticles) and grand canonical ensemble (allowing for exchange of heatand number of particles), only the last one is general enough to beconsidered in a common framework for energy and information.

As a technical point, considering ν “copies” of a system with statespace Ω, or rather states ωi ∈ Ω, i ∈ [ν] again corresponds toHom([ν],Ω),which may be interpreted as a “trajectory” in state space. If the sys-tem has N subsystems (“space-like dimension”) then the “space-timetrajectory” is an element of Hom([ν] × [N ] → Ω). If the time-order isirrelevant, then a sequence of states such that νω systems of the ensem-ble are in the state ω is a (homological) chain in C0(Ω) = ∑ω∈Ω νωwith finite support of “volume” ν.

5.3.2. Statistical entropy. The statistical definition of entropy is viaBoltzmann’s equation

S = klnW

where W represents the number of µ-states per fixed M -state. It canbe derived either using the continuum limit, as above N → ∞ or thelarge numbers limit as in [Stat1]. The entropy of an ensemble

∑

ω νωis

Ων = ν!/∏

νi!, lnΩν ≈∑

−ν∑

pi ln pi = νH(π),

where pi = νi/ν, and the Shannon’s entropy (per symbol) is obtainedvia Stirling’s approximation.

Conceptually we are in a F = ma situation, where S belongs to theidealized framework (information source) while W is a precise theoret-ical concept (counting states) corresponding to a particular / actualtype of system.

Notably S itself is not a state function, but TS is, term which entersthe energy balance.

5.4. What do you say, Mr. Feynman? ¿From Helmholtz freeenergy F we may calculate everything else ([Fey1], [Stat1] 4.3), viathe relation with the partition function:

Z = e−F/(kBT ).


5.4.1. Energy: dark or white? The corresponding energy balanceE = F + TS reflects the duality between the “external dynamics”,associated with info/order etc., which we will call “white energy” incontrast with the terms affecting the “internal dynamics” of “not vis-ible” (internal/averaged etc.) DOFs we speculate may provide the“correction terms” in a quantum version of General Relativity, as analternative to the “missing matter” from the energy-momentum side.

5.4.2. The partition function - revisited. So what is the partitionfunction just a “normalization constant” or a function? Neither; in thecontext of QFT it is a generating function, but here we will turn offinteraction for now:

Z(E;β) =∑

j∈Ω

e−βEj .

It depends on the chosen state function, say energy E, and temperaturekBT = 1/β (patience please ...).

As explained above 4.4.1, the probability equipartition assumptiongives the correspondence between energy and Boltzmann’s distributionvia constraint entropy maximization:

(Eii∈I, Z)max H(π)

∑

piEi=<E>∑

pi=1

// (π = pii∈I , β)

The converse is direct; given π and β, Ei = −1/β ln pi.Therefore, if viewing the energy as fibrating the state space Ω (the

“extension/bundle” picture), then the partition function is a way toredistribute the “energy gaps”:

ΩE→ Spec(E)

e−β(·)

→ R,

where Z = (Z1, ..., Zm) (say |I| = m) is interpreted as a coordinatefunction on the M-state space Ω/E ∼= Spec(E), and the normaliza-tion constant is ||Z||1 (see 4.4.1). So temperature, or preferably β, isessentially an infinitesimal generator ...

Now the correspondence between the projective space descriptionin terms of probabilities π and the “free theory” in terms of E is:

(Z : Spec(E) → R) 7→ (π, β).

Then the relevant state function is not entropy, but rather TS in someconvenient coordinates:

Q(Z) = Q(Spec(E), β) = H(π)/β.


Alternative coordinates Xi = − ln pi = β(Ei − F ) or Yi = Xi/β =Ei − F (departure from the free energy) should also be considered:

Q(Y, β) =∑

i

Yie−βYi.

If Boltzmann’s probability distribution is obtained maximizing entropyas a function defined on the projective space of the M -state space ofthe system or ensemble (type of info source):

H : P 1Spec(E)R → R

under constant energy as a constraint, then in general, what should bemaximized in the free energy picture? Free energy is the only qualifiedcandidate!

5.4.3. What is energy after all? Clearly the probability descriptionlives in the projective space; but what about energy? It is alwayspositive, and failures of this requirement are dreaded and quickly fixed(on shell at least).

But being positive is a symptom of projectivity (the quantity is“infinitesimal” already, can’t take a ln any more).

So, we should reexamine relativity in an attempt to trace back thisfeature ...

5.4.4. The other observables: piece of cake. The other thermody-namic observables can be derived from the partition function [Stat1]:

< E >= − 1

Z

∂Z

β, cV =

∂ < E >

T, etc.

Furthermore [Fey1], p.7:

H = −∂F∂T

, etc.

or [Entropy] p.256:

H(π) = lnZ(β) + βE, dH = βdE.

We should remark that entropy has contributions from interactionsSf (formation entropy) as well as from configurations Sc = H (en-tropy of mixing) §5, yet here (interactions off) only Hc is “visible”.Formation entropy corresponds to “colored configurations” (internalDOFs) while entropy of mixing corresponds to labeled configurations(“position-momentum” etc.)

The relation with Poincare-Cartan form is intriguing:

PCF ∼ pV − TH ∼ −Sf − Sc?

6. FEYNMAN PATH INTEGRALS AND ENTROPY: GRAPHS INVARIANTS! 77

towards a similarity (speculation: nothing new, right?) between therole of Planck’s constant (energy quanta) and Boltzmann’s constant(information quanta!?).

6. Feynman path integrals and entropy: graphs invariants!

Although we are in the realm of “tertium non datur” (probabilitiesdon’t destructively interact, trees etc.), the sum over possibilities idea,which is the mile-stone of Feynman’s approach naturally applies to ...decision trees.

One may think of a of a sequence of answers to a sequence of ques-tions represented by a decision tree as outcomes of measurements, orcomputations, in an algorithm: going from “Start” to “Stop” via var-ious branches. For example, if the root represents the question (mea-surement of) A, then depending on the subsequent questions, B inde-pendent on the A result, or B or C depending on the A-outcome, wewill have different labeled graphs (or just trees):

A

>>>

>>>>

> A

>>>

>>>>

>

A(BB) : B B A(BC) : B C

The probability distribution - partition function correspondencemay be extended to such graphs, yielding none other then the FeynmanPath Integral!

6.0.5. FPI and probabilities. If a probability distribution π (sto-chastic M -state), which may be viewed as a colored corolla, corre-sponds to the partition function describing the statistics of the system§4, then a colored decision graph corresponds to the amplitude viaFeynman rule under the FPI formalism:


BoltzmannCorrespondence

ProbabilityDistribution

PartitionFunction

π Z

FPI

Feynman diagram Feynman rule Green function

Γ W (Γ).

In a sense Feynman Path Integral formalism generalizes Boltz-mann’s correspondence between probability distributions and partitionfunctions.

The colored corolla is the transition function of the (minimal) Start-Stop Automaton (STA):

•

~~

AAA

AAAA

A

• ... •(4)

If the Input is the probability p then the Output is one of the proba-bilities p · pi.

Recall that under the Boltzmann correspondence pi = Zi/Z wherethe partition function Z = (Z1, ..., Zn), Zi = e−βEi corresponds to anM -state (E, β) (or rather M -observable E).

Proposition 6.1. The expectation value of the energy determinesthe entropic state function:

H/β =< E > .

Proof.

< E >=∑

i

piEi =∑

pi(− ln pi)/β = H(π)/β.


Theorem 6.1. The Boltzmann correspondence extends from corol-las to Feynman Path Integral (sum) on graphs, relating the entropy Hand expectation value of energy < E >:

HT =< E > .

Indeed, the expectation value of the “algorithm” (the decision graph),is:

< M >=∑

γ:r→i

p(γ)Mi =∑

p(γ) ln e−S(γ) = −∑

p(γ) ln p(γ)/S?

Mi is the “info contribution” (action) flowing through the path γ. Foran elementary edge (channel) of probability p, the flow is:

W (• p→ •) = − ln p/β, p = e−βW ,

where β corresponds to the information capacity of the chanel (propa-gator in the QFT framework).

If sources are present, then a balance equation holds between theInput Probability pIn (information) and Output Probability (informa-tion):

pOut = pIn +W (• p→ •),in complete analogy with the “momentum flow” determined by thegenerating function (action) (see [Gosson]):

Q(r′, t′) = Q(r, t) +W (r, t; r′, t′).

So the role of β, the inverse of the thermodynamic temperature, “lo-calizes” in the general framework (QFT and quantum information) tobe related with the channel information capacity.

The difference between the classical and quantum theory at thispoint is at a flip of a switch away: turn i on (and loops/feedback; morein Ch.6).

But before that, we will elaborate on the content of the theorem onthe tree from the example of Section 3.4.3.

6.1. Entropy and Information Charge/Potential. The cor-respondence between entropy and Feynman Path Integral is a naturalgeneralization of the Boltzmann correspondence between a probabilitydistribution and the associated partition function, as explained above6.0.5. But the role of “temperature” was understated, so we will reviewthe “enhanced picture”.

The probability distribution is a state belonging to the projectivespace. In order to compare it with the “energy picture” (partition


function) one needs an additional parameter: β, playing the role of theinverse of temperature in thermodynamic.

Projective Space

BoltzmannCorrespondence

// Path Space Formalism

(β, π) Z(β) : Spec(E) → R

β, 1

pi

1

Ei

pi Zi = eβEi

Decission graphs

Res

OO

QDR

FPI

OO

(β,H(π))H(E)·T (E)=<E>

// < E >=∑

iEie−βEi.

A normalization of the above correspondence means assuming 0 freeenergy F = 0, i.e. normalizing the partition function Z = ||Z||1 = 1(one can always take advantage of the gauge transformations Ei →Ei + C in the RHS). Then pi = Zi and Ei = −(ln pi)/β yields theinverse of the correspondence associating to a partition function, underfixed energy expectation value, the probabilities maximizing Shannonentropy,

π·E =< E >= H(π)·T, max limπH(π),

∑

i

pi = 1,∑

i

piEi =< E > .

6.1.1. The “right invariant” goes by the right rule! We claim thatH(π) is not the right invariant for the combinatorial picture, but HT ,the state function from thermodynamics, is. But this will change therule labeling decision trees, to a genuine “relative” version: do not as-sociate probabilities to edges, rather associate number of states νi tovertices! For example, if ν = |Ω| is the number (measure) of the µ-statespace and νi = |Ωi| is the number of micro-states corresponding to themacro-state Ωi, then LHS is the probability edge of the corolla repre-senting the µ − M -correspondence, with the probabilities pi = ν/νilabeling (coloring) the edges, while the RHS is the configuration edge(address branching instruction) of the same corolla, where the number


of configurations ν(i) (or some other EP-map: number of qbits, quan-tum dimension of a modular category / representation of a quantumgroup etc.) labels (is applied to) the vertices:

• pi→ •, ν• → νi•, pi = νi · ν−1.

The “other data” (vertex probabilities or edge probabilities) can bereadily computed in either formalism.

The probability formalism is a relative one (moduli space / pro-jective space), therefore not appropriate to a “Noether informationcurrent” picture, as the particle formalism (“particle=state”; forgetIDOFs for now).

To relate TH and H, we will stare a few moments to the Equation3, in the special case of a previous example 3.4.3, in order to relatethem:

TH(

ν

~~

AAA

AAAA

A

ν1 ν2

) = ν H(

1ν1/ν

ν2/ν

???

????

?

• •)

= −ν1 ln ν1 − ν2 ln ν2 + ν ln ν.

since conservation of number of states yields ν = ν1 + ν2. So TH isadditive (EP-map or a “functor” in the categorical picture: action orenergy) due to Stokes Theorem in disguise:

TH(ν(ν1, ν2)) = H(ν1, ν2) −H(ν).

In other words:

H =∂(TH)

ν,

which, as we shall see, makes me wonder whether ν plays the role ofβ!(?)

Now the ν’s are the appropriate candidates for an Euler-Poincaremapping due to the conservation of number of states (and informationcurrent etc).

Indeed, equation 3 now reads:

TH(Γ) = TH(Γ/γ) + TH(γ).

But how to derive in an axiomatic manner H(ν) = −ν lnν? In factthis is the “wrong formula”! With Stirling’s approximation in mind,the illumination comes: the right (hidden) quantity is ν ln ν − ν, sincedue to the conservation equation:

−∑

i

(νi ln νi − νi) + (ν ln ν − ν) = −∑

i

νi ln νi − ν ln ν !


Definition 6.1. The information potential of a labeled vertex is:

Q(W ) =

∫ W

0

lnxdx = [x lnx− x]W0 = W lnW −W.

The Stirling approximation was “stretched” a bit, to avoid the un-pleasant constant term of 1 (although we abhor limits, we used it; hmm...?).

Now extend Q additively (on the “tensor algebra”):

Q(ν1, ..., νn) =∑

Q(νi).

¿From the definition it follows that:

Q(ν) =

∫ n

1

∫ ν(i)

1

ln(x)dxdi

(just one more formula to think about). Then TH is the EP-mapsensitive to the residue of the tree (graph):

TH = dQ : TH(ν → ν ′) = Q(ν) −Q(ν ′).

Speculating on the relation with temperature and heat transfer is tempt-ing, but it will be postponed (not too much!); (n)or the interpretationof Q as a generating function of a flow ...

6.1.2. Invariants: discrete versus continuum. So, not the entropyH is the key, but rather the “informational energy” TH with its defin-ing “potential” Q(ν), which measures the information charge stored by“adding the ν DOFs”; since also

dQ

dν= ln ν

TH may be thought of as a measure of the information flow (“gradient”/ differential, what’s the difference when there is a metric :-)) andthe well known thermodynamic relation between entropy and heat isobtained:

H =dQ

T.

The continuous invariant Q, corresponding to the large numbers limit,is just the infinitesimal measure of the internal symmetry of the Quan-tum Dot measured by the discrete invariant:

C(ν) =∑

ln νi! = ln |Aut(P)|,

where an internal symmetry of the partition of Ω is a permutation ofthe elements of Ω preserving the partition.


They are related via Stirling’s approximation:

C(ν) ≈ Q(ν) (N

∑

0

≈∫ N

0

). (5)

6.2. Concluding speculations. This is not entirely surprising(but pleasant indeed!), since the Feynman path Integral formalism wasoriginally invented as a way to tame the combinatorics of the pertur-bation approach in QFT while the work horse of high energy physics,gauge groups and symmetry, with its representation theory, invokes theYoung tableaux machinery for grinding permutations etc.

So what are temperature and entropy? Inseparable dual (micro-macro) aspects of information theory regarding a system.

The relation with the levels of energy:

Ei = −T ln pi

suggests to think of Si = − ln pi as a information capacity (flux). In-deed, since pi = νi − ν:

Si = S(ν → νi) = ln ν − ln νi, Ei = TSi

and therefore:

Q(ν) =

∫ ν

0

lnxdx =

∫ ν

0

S(• → 1)d•,

which is yet another relation open for speculations ... Think of

I(ν) = − ln ν = S(ν → 1)

as the information charge corresponding to the information potentialQ. Then

I = βE (6)

is the missing Unifying Principle, relating energy and information (Ch.3).

But then, since temperature determines the energy per DOF:

< E >= kBT,

the relation with entropy suggests a “quantization condition” must bepresent; is it

1 qbit = β < E >?

Is it related to a PCF / Maslov index formalism? (see 3.3).There are many other tempting questions regarding this “puzzle”;

we will record them for later use ... Is the QDR finite? Is kB relatedto a quanta of information? Is β a “coupling constant”? etc.


7. Relative entropy and information channels

Conditional probabilities are related to relative entropy and se-quences of measurements. Classical logic is clearly inadequate to han-dle the corresponding tree representations of measurements (as decisiontrees), since it is sensitive to the external structure only.

Consider for example two binary-outcome measurements A and B,with associated probabilities p(a0), p(a1) and p(b0), p(b1). Then “refin-ing” the outcomes of A with the measurement of B, or B first then Ayields two distinct decision trees, denoted tAB (shown bellow) and tBA:

A

zzzz

zzzz

!!DDDD

DDDD

B

xxxx

xxxx

x

B

##GGG

GGGGG

G

a0b0 a0b1 a1b0 a1b1.

The measurements are independent if the joint probability is the prod-uct of probabilities; then the conditional probability is defined by:

p(ai ∧ bj) = p(bj|ai)p(ai).This relation implicitly refers to a path A → ai → (aibj) of the tABdecision tree.

Now tAB 6= tBA and a finer invariant will distinguish between thetwo. An example to be considered is the measurement of spin/polarization§6.

The “probability” is the classical information flow in decision trees(without spatial correlation: S-correlation), while the amplitudes ofprobability represent the quantum information flow in graphs, whereS-correlation is possible in causal structure, and responsible for phe-nomena like entanglement and teleportation etc.

Back to decision trees, their external structure is the classical shadowof the quantum non-commutative reality:

Res(tAB) = Res(tBA).

The Shannon entropy H and the information potential Q are classical,being unable to distinguish the internal structure. A quantization isnecessary, beyond the extension ofH (or TH) from trees to graphs withloops colored by complex amplitudes. The general picture is a quantumcomputing with objects on vertices and unitary operators on edges, asa categorification of the classical picture where one counts DOFs on

7. RELATIVE ENTROPY AND INFORMATION CHANNELS 85

vertices ν = |Ω| and applies a boolean calculus to the correspondingprobabilities (roughly speaking).

As a guiding idea, the quantum entropy is essentially Von New-mann’s entropy applied to quantum computing, which in turn is anupgrade of quantum mechanics, with its Hilbert spaces and unitaryoperators replaced by (braided) monoidal categories with duality (e.g.type II-factors [S]; [Ion01]; [Coecke2] etc.), leading to representa-tions of PROPs (QFT, CFT), representations of cobordism categories(TQFTs, HQFTs) and finally to representations of causal structures(Feynman process).

7.1. Heisenberg commutation relations. We believe that, withinthe above conceptual framework, the information flow counterpart ofHeisenberg Canonical Commutation Relations is a “normalization ver-sion” of:

tBA − tBA 6= 0.

Now quantum measurements can be more intuitively modeled as repre-sentations of trees, including the “mother of all quantum experiments”,the double slit experiment, which should allow the derivation of theHeisenberg CCR.

7.2. What is relative entropy. We will proceed to analyzingequation (5) from [BZ], p.4 (n = 2 will suffice):

H(p(a0), p(a1b0), p(a1b1)) = H(p(a0), p(a1))+p(a1)H(p(b0|a1), p(b1|a1)).

Conservation of states implies:

p(a1) = p(a1b0) + p(a1b1),

while independence of A and B (“commutation assumption”), implies:

p(a1bi) = p(a1)p(bi|a1).

Now this defines p(ai|bj) in the case of the joint probability distribu-tions. But what is p(ai|bj), really?

7.2.1. Transition cobordisms. A measurement of A followed by ameasurement of B is a graph cobordism between the outcomes of Aand the outcomes of B, called the In/Out boundary points. In factA plays the role of a source or preparation stage, while B is the ac-tual measurement, or target of the measurement process (interactionor communication).

In the above joint case scenario, the graph is a bipartite graph withinputs ai and outputs bj. The edges aibj are labeled by the conditionalprobabilities tij = p(bj |ai), representing the entries of the transitionmatrix T (B|A), of this Markov process.


We will denote with Hom(ai, bj) the set of paths (or the vectorspace with the corresponding basis). It corresponds to a component ofthe graph, called a communication channel. The number of channelsis the degree of A: dim(A).

A micro-macro correspondence associated to an observable A withprobability distribution π is the Markov process 1 → A, with transitionprobabilities

T (1, ai) = p(ai) = p(ai|1).The associated colored graph is the corolla 4.

Definition 7.1. An IO-process is the colored graph cobordism <B|T |A > represented by the following composition:

1 → AT→ B → 1.

Anticipating complexification,

B∗ ∼= (B → 1), ∼= (1 → B).

Of course T (bj, 1) = 1 for all j.7.2.2. Entropy as a derivation. Note that the function H(p) =

−p ln p is a derivation:

H(pq) = −pq ln(pq) = H(p)q + pH(q).

This property allows for the following constructive definition of theentropy.

Definition 7.2. The entropy of the Markov process T = T (B|A)is the matrix H with the (i, j) entry the entropy of the (ai, bj) commu-nication channel:

H : T → R, Hij =∑

γ∈Hom(ai,bj)

H(γ),

where H(γ) is the value of the extension of H(p→) = −p ln p as a deriva-

tion on the path algebra of T (quiver).The total entropy of the Markov process is

H(T ) = ||H||,the norm of the matrix of channel entropies.

The entropy of an elementary measurement A (macro-state etc.) is

H(A) = H(1 → A) = −∑

i

pi ln pi = H(1 → A→ 1) = H(A∗A).


The entropy is determined by the information potential:

H = Id+ I, H(p→) = p+ I(p), I(p) = −

∫ p

0

ln zdz.

As a corollary, written in terms of Q, the entropy of a measurement(macro-state) A is:

Q(∑

i

pi) −∑

i

Q(pi) = 1 −∑

(−pi ln pi + pi) = H(A).

7.2.3. Entropy as a functor. Recall that matrices may be viewed asrepresentations of quivers, as done above in the case of entropy. It willinherit the main property of matrices: H is a functor on the categoryof Markov processes, viewed as composable graph cobordism:

H(T1 T2) = H(T1) H(T2).

7.2.4. Relative entropy.

Definition 7.3. The relative entropy of the Markov process T :A→ B with input data A, is the average entropy:

H(B|A) = ||H(T )|A > || =∑

i,j

p(ai)Hij.

Then we have the following property (compare [BZ], p.4).

Proposition 7.1.

H(< B|T |A >) = H(A)+H(B|A) = H(1 → A)+H(A→ B)+H(B → 1).

For example, with n = 2 and factoring

Γ = A (Id⊗B), Γ = (p1, p2) (1, (q1, q2)),

the above property reproduces the equation (4) from [BZ]:

H(Γ) = H(A) + p1H(1) + p2H(B) = H(A) +H(Id⊗B|A).

7.2.5. Entropy in the Energy Picture. The correspondence betweenprobabilities and energy levels via partition function allows in particu-lar to go back and forth from an interpretation in terms of informationtheory (“group level”?) to an interpretation in terms of energy (infini-tesimal level).

Changing variables z = e−βE yields:

Q(pi→) =

∫ pi

0

= −β2

∫ ∞

Ei

Ee−βE = −βP (Ei),

revealing a connection with an underlying spectral measure.


7.3. Overview: Z2/R/C/H-physics. The formalism introducedwithin the framework of Markov processes, is viewed as a real Feyn-man path integral approach (probability theory/Markov chains - “R-FPI). The relations with Classical Mechanics in its particle-wave dualformulations (Lagrangian or Hamilton-Jacobi) and with Quantum Me-chanics/QFT as a complexified version of Markov chains, are open forspeculations. Classical mechanics focuses on external DOFs via PCFpdx −Hdt, while statistical mechanics and thermodynamics focus oninternal (classical) DOFs via −Edt ∼ βE, allowing for a channel ca-pacity versus local time analogy.

The entropy H is an infinitesimal character on the edge coloredpath algebra (Probability Picture) while TH is a (character?) on thevertex colored path algebra. The correspondence between the discrete(combinatorial or micro-state space) and continuum (partition functionor Energy Picture) frameworks amounts to the approximation

∑ ∼∫

under the correspondence νi = Zi.Proceeding from classical deterministic mechanics, with its True/False-

deterministic approach, via statistical mechanics with a sort of “fuzzylogic” towards quantum mechanics, is marked by a change of coeffi-cients; from Z2 to probabilities R and to amplitude of probabilities C.There are repeated attempts to further proceed and use quaternionsH to incorporate the 3D-space, so elusive to a rational explanation for“why 3D?” (see also Ch.9 §3.1). A “good sign” is that a qbit is not justa complexification of a bit, but rather its double ... is it H = C⊕C theright number system? It is non-commutative, but manageable, with alot of potential regarding the symmetries due to “coincidences” remi-niscent of the fact that “vector physics” is just the algebra of imaginaryquaternions.

One of the distinguishing features of classical physics and informa-tion theory is the presence of a global time (“clock”) corresponding tothe existence of a unique factorization of transition processes allowingto separate “parallel” and “sequential” (at least locally). In the quan-tum world such a factorization may be “artificial”/non-canonical, or atleast correspond to additional data, as the need and role for OperatorProduct Expansions advocates: in general there is no global or evenlocal time in an arbitrary causal process. Or may be a local “timecut” exists locally at a vertex: In=Past and Out=Future ... Anyways,what “flows” is information (Information Picture) or energy (the En-ergy Picture).


So, there are two sides of the coin, the Information Picture and theEnergy Picture, related via information capacity / time:

< H >= β < E > .

Ultimately, the mere mechanical motion we call “time” is no match forthe “time we experience”. But then “time”, as information/causalityetc, flows again ... “E pur si muove!

CHAPTER 6

Modeling quantum information flow

Chapter 5 dealt with the classical theory, although the new ideasare not dependent on the classical context. The difference between theclassical and quantum theory at this point is a flip of a switch away:turn i on with a Wick rotation (imaginary temperature!?) and ... allowfor feedback (loops).

1. Constructivism: the critique of the critique ...?

As a preliminary “program” in developing the quantum analogs, wewill comment on the critique from [BZ] of Shannon’s entropy formal-ism.

The Shannon postulates should be understood as “hiding” the PathModel approach, and therefore they are not inapplicable to QM, butneed “upgrading”.

As any “inspired generalizations”, it should be conceptually sim-ple: sum over histories remains, taking care of the “visible structure”(graph-cobordism), while the trace (“internal residue”) should takecare of the internal counterpart of the possible hidden histories.

In general the B outcomes bj may depend on the A outcomes, sothe joint experiment is not the general case; it already assumes a sortof independence of B and A.

In the simples possible case, a quantum jump, measuring positionor momentum should be modeled as txp 6= tpx, leading to Heisenberg’sCCR. Similarly for the double slit experiment, with its alternative: leftor right hole?

With quantum measurements as operator versions of transitionsA→ B (7.2.1), and “uncertainty” (or information gain) related to vonNewmann entropy S, the Heisenberg CCR should follow as

S(t[A,B]) = ~.

The interpretation of QM will look natural in terms of decision trees,extending the lapidary justification: “It just works this way!”.

91

92 6. MODELING QUANTUM INFORMATION FLOW

Adding interactions requires including graphs with loops (feedback).So the upgraded QM-Model is a representation of graphs; from Heisen-berg’s matrices to Feynman diagrams (from QM to QFT).

Res(tA) < − > Spec(A).

But, still ... “QM, where is thy heart?”; probably the superpositionwith destruction of information (not just products but also coproductstoo).

And ... “QM, where is thy soul?”; probably the inevitable non-commutativity of information gain, as a consequence of “conservation”:if you acquire (“take”) information, you drain (change/perturb) thesystem (see the ABE-model of measurement Figure 1 §3).

Therefore Information gain does depend on the order of acquisition!Also, in the quantum realm, beware of inequalities, since they hidebrutal truncations; look for the missing terms (“neutrinos”).

The dual pictures (Information / Energy) shed different keywordson the same IO-process 1 → A→ B → 1: measurement or interaction.

Source→filter

encoder

transitioncommunication→

screendecoder → Target.

Then, some of the so called “paradoxes” regarding our knowledgeof th system’s properties (“real” or not “real” etc.), for example com-paring different successions of measurements ABA 6= AA, should beanalyzed in view of the relative quantum entropy and information flow:

Teleportation Diagram

• •

• •

OO

•

OO

•

??~~~~~~~ •

__@@@@@@@

??~~~~~~~ •

OO

•

OO

•

OO

It is no longer surprising that a B intervening measurement may dras-tically change the “function” of the gate, leading from “idle check”AA to teleportation! (see e.g. [Kaufmann], [Coecke3] etc.). Thefundamental concepts involved are: S and T -correlation (parallel andsequential computations), which cannot be reduced to a local space-time description while denying “spooky actions” ...

The two examples of [BZ] should be reanalyzed in this light (§3).

2. QUANTUM ENTROPY 93

2. Quantum entropy

The ideas introduced in Chapter 5 will be implemented in the quan-tum context as a change of coefficients from real to complex. TheBohr interpretation is the projection from one formalism to the other:p = |ψ|.

2.1. Quantum transitions. Now we will turn internal DOFs on.Outcomes ai of an M-observable will become eigenvalues of the

associated operator with the corresponding measurement basis. We willoverload the notation, without distinguishing between ai the eigenvalueand |ai > the eigenvector.

The quantum analog of a transition A → B is a representation ofa graph-cobordism, called a VO-graph, where the vertices are labeledby states and edges by operators:

aiUij→ bj, p = | < ai|Uij|bj > .

The definition hints towards vertex operator algebras, as algebras overa PROP (functor from the PROP to a modular category or so).

2.2. Mixed and pure states. The relation betweenA = (1 → A)and mixed states ρ =

∑

i piPψ will be investigated later on.

2.3. Von Newmann entropy. The analog of the classical en-tropy H, extending Shannon’s entropy to the category of graph cobor-disms, should be an extension of Von Newmann entropy:

S(ρ) = tr(ρ ln ρ),

leading to FPI:

S(1 → AU→ B → 1) =

∑

γ=∏

e

S(γ)?

2.4. Non-commutativity of measurements. In the classicalcase (see 7) H(tAB) = H(tBA), where tAB = tA ⊗ tB etc., under theassumption that A is independent of B, i.e. p(bj |ai) = p(bj|1), whichis equivalent to the condition that the joint probability is independentof order:

p(1 → ai → bj) = p(aibj) = p(ai)p(bj) = p(bjai) = p(1 → bj → ai).

Now in general S(tAB) 6= S(tBA).As an exemplification, consider the polarization gedanken measure-

ment experiment from [BZ], p.7.First, one should not consider just A, a measurement of the polar-

ization in the direction of the z-axis, followed by B the measurement

94 6. MODELING QUANTUM INFORMATION FLOW

of the polarization in the direction of the x-axis, but rather the 2-Dvector space V =< sin θMx + cos θMz > and a transition from V intoitself:

VU→ V.

As a guide in comparing with the classical information, in the quantumworld, the analog of the correspondence between probability (p = | <bj|U |ai > |) and energy (p = e−iE/Z), is the correspondence betweenstates ψ (amplitudes) and phase.

3. Two good examples

Instead of the whole theory, “we” will analyze the “two good ex-amples” 1 from loc. cit..

3.1. Example 1. (left to the reader).

3.2. Example 2. (left to the reader).

4. Quantum information

The “total” info content of a quantum system is ... the systemitself! What we need, I think, is a measure of this quantum interact-ing/communication, i.e. a measure of the quantum information sent(quantum information capacity etc.).

1Was it Gelfand? ...

CHAPTER 7

Classical and Quantum logic

The above mathematical representation of information flow hassome interesting connections with the quantum logic introduced byBirkhoff and von Newmann quite a while ago [BN].

1. Propositional calculus

Since then it became quite clear (loc. cit. §4 p.825) that “The cen-tral idea is that physical quantities are related, ...” and it seems strangeit took so long until Feynman interpretation gave the “categorical fla-vor” to quantum physics, although the appropriate language (categorytheory) was in place at the time (of Birkhoff and von Newmann). AlsoMarkov’s work ... (etc.).

The mark of classical mathematics was still too strong an influ-ence to “break away” with tradition; even Birkhoff and von Newmannfocused on the correspondence between “observation-spaces” and “sub-sets of phase-space”, i.e. at the level of objects (loc. cit. §6, p.826).Not the objects alone, but together with the relations should be setinto a (functorial”) correspondence, i.e. data and programs or on theother side states and correlations.

The (A) “subsets in phase space” should be thought of as pro-grams and data, i.e. quantum computation (I/O and processing) orquantum interaction at the level of mathematical models, while (B)“experimental propositions” would rather refer to measurements con-sisting of preparation (I: “source” of input states/particles/qbits etc.)and observation (O: “target”/recording apparatus etc.), with a quan-tum process/interaction between the two. A “physical quantity” is thistriple; e.g. “spin” needs source and axis and Stern-Gerlach magnet etc.

2. The lattice: creation and annihilation!

The propositions of quantum mechanics were related with the struc-ture of a lattice. It is tempting to speculate (at this early stage) thatthe idea is a “premonition” (“hides”?) of a stronger claim, that thelanguage/logic reflects the representation of a causal structure as ex-plained before, i.e. of a Feynman process. In other words, “Does QM

95

96 7. CLASSICAL AND QUANTUM LOGIC

represent the (quantum) communication process?”, i.e. leading to rep-resentations of (decision) trees, PoSets (lattices), graphs etc.?

It is striking that the trademark of QFT, i.e. creation and annihila-tion operators, formally correspond to the structure of lattice (creationand annihilation of DOFs, what else is there?).

It is also indicative of an important connection (math path model -information interpretation/formal language/automata) that “classical”distributivity fails in quantum logic (when measurements correspond-ing to M -observables) do not commute, but a substitute exists: themodular identity (p.832). A Path Model interpretation will be consid-ered elsewhere (§5.1).

The classical distributive property “... is the characteristic propertyof set-combination.” (field of sets is a Boolean algebra) (p.831). But“sets” are not realistic models; reality is holographic (!) and we can’treally tell if “something” (object, event? etc.) is in isolation (“part ofa set”) ...

Somehow “set theory” and “holographic world” are deeply incom-patible. Categories with objects and relations (we don’t need the phi-losophy of set theory to build it) is a much more flexible and high levellanguage; therefore better suited to describe our models.

3. The modular identity and ... entropy!

Returning to the modular identity, it is “a consequence of the as-sumption of that there exists a numerical dimension-function d(a)”:

D1 : Ifa > b => d(a) > d(b),

D2 : d(a) + d(b) = d(a ∩ b) + d(a ∪ b).

The “hint” is clear: “D1-D2 partially describe the formal properties ofprobability” and the modular identity “is closely related to the exis-tence of “a priori thermo-dynamic weight of states.” (p. 832-833).

Birkhoff and von Nwmann were thinking (probably), about entropy!Indeed, a “dimension-function” is a special case of an Euler-Poincare

map on extensions, similar to the above finer invariant (the “charge”of the quantum information flow/current). Its residue is the (clas-sical/quantum) entropy H(lattice/tree/forest/graph) (colored withprobabilities or amplitudes), depending only on the external structure(terminals and roots).

5. CONCLUSIONS 97

4. Relation to projective geometry

Now any lattice of finite dimensions are products of projective ge-ometries and a finite Boolean algebra. This looks like a nice “mar-riage” between classical and quantum mechanics, providing the neces-sary framework to explain the measurement “paradox”.

The transition from a “quantum” to a “classical” description wouldcorrespond to a “truncation” (projection) dropping the “projectivepart”. It is at least a mathematical candidate to implement the “quantum-2-classical interface”.

The correspondence between projective geometries and skew-fieldssuggests a connection with the moduli space of non-commutative dif-ferential forms (left action of F on T •(F ) etc.). The relation with the“homogenization” process of classical mechanics (from “affine”/free ge-ometry to “linear/“reduced” geometry) will be investigated elsewhere.The role of the Riemann sphere, viewed as a bifield will be explainedelsewhere. For now, let’s note that the qbit has a certain redundancy,and its projective space is the bifield (Riemann sphere), as the basicI/O quantum memory unit.

Complementarity, implemented in loc. cit. p.835 as an involution,will be interpreted as an antipode (of a sphere or Hopf algebra).

5. Conclusions

How many ”logics” are there? We learn from [BN] (p. 836) that“... one can construct many different models for a propositional cal-culus in quantum mechanics, which cannot be differentiated by knowcriteria.” at that time, at least. If these “quantum logics” are derivedfrom Feynman processes, i.e. representations of Feynman causal struc-tures, then one may try to classify them using homological algebra (see[Ion04-1] etc.).

In any case one should not move to the continuous-dimensionalcase and stay within the graded-objects realm (keywords: skew fields,projective geometry and Grassmanians ... differential forms and non-commutative differential forms etc.).

While statements of quantum mechanics correspond to a projec-tive geometry (see Ch.2 §6), classical dynamics descriptions constitutea Boolean algebra. A “classical explanation” of quantum mechanics“facts” should belong to an extension of one logic by the other.

Now the classical (commutative) case decomposes into 1-dimensionalindependent constituents (characters), while quantum mechanics has a“greater logical coherence [BN], p.836, leading to the impossibility ofmeasuring different quantities independently. As explained elsewhere,

98 7. CLASSICAL AND QUANTUM LOGIC

the µM -correspondence is now implemented by decision graphs, ex-tending the usual finer joint partitions due to independent (commuting)M-observables. The coherence of quantum mechanics, mathematicallyimplemented by Heisenberg CCRs, appears at the level of physical in-terpretation as “uncertainty” or at the level of computer science levelas different entropy of the information flow graphs tAB and tBA (see§2.4).

This interpretation allows to address the two questions in [BN],§18, p.837.

5.1. What is the experimental meaning? In a sense the dis-tributive property of classical lattices corresponds to probabilities (real/classical information/ entropy: R), while the modular identity of quan-tum lattices corresponds to amplitudes of probability (complex/quantuminformation/entropy).

5.1.1. Creation or annihilation? As a corollary quantum informa-tion (qbits) can be branched and merged (quantum gates for graphs,partition of unity for trees etc.), which may appear to an observer as“created and destroyed” (meet and join lattice operations), while clas-sical information can only be destroyed (entropy increased - SecondLaw of Thermodynamics).

The fundamental role of creation and annihilation operators (bial-gebra structure) becomes more and more clear ...

5.1.2. 2nd Law finally tamed! Also the “limits” of the 2nd Lawprovide relief from a “thermodynamical death”. to the contrary, theQuantum 2nd Law of Quantum Information allows for diversity andcomplexity to emerge from “chaos”. Then the “arrow of time andentropy” becomes a subjective matter of the more complex observer“looking down” to the less complex system ...

5.2. What about “tertium”: “datur”, or not? At a moretechnical level, the most objectionable classical logic assumption is[BN], p.837:

a′ ∪ b = 1 =⇒ a ⊂ b,

where a′ denotes the complement of a. The dual relation is:

a ∩ b′ = 0 (False) =⇒ a ⊂ b,

enabling the proof by contradiction (“tertium non datur”!).With hind site we view this relation as hinting to a split decompo-

sition (direct sum). In contrast, in the quantum world, the ‘factoriza-tion” has an “overlap”, being analog to the atlas of trivialization of acomplex line bundle ([CK2]). In the algebraic-geometric language the“points” of the algebra of observables/states is not a split extension, but

5. CONCLUSIONS 99

it consists of two subalgebras. Correspondingly, the quantization al-ways involves a central extension before the “deformation step” (globaldeformation extending the infinitesimal deformation).

So, “tertium datur” after all! It’s “black or white or ... gray”(Strictly speaking, “reality” is complex: God is playing ... qbits!).

CHAPTER 8

Quantum dots and bits

Now what to do with a resolution of degrees of freedom, as a newcomer in the plethora of mathematical-physics models? We have to tieanchor it in both classical and quantum physics (see §3), but first, itneeds a name; how about: “the Quantum Dot Resolution” (QDR)?

The term quantum dot originates from nanoscale physics [QD], re-ferring to tiny spatial regions of size 100-200 nm (loc. cit. p.12) bigenough to be considered “macroscopic objects” (“artificial atoms”),yet small enough to exhibit quantum effects (only understood and de-scribed using quantum mechanics).

1. Quantum dots: theory and practice

The quantum dots may be thought of as corresponding to externalDOFs “colored” by qbits in a representation of a the causal structure(“space-time”) given by a resolution of a “point”. A quantitative corre-spondence between the experimental quantum dot and the theoreticalone would be worth pursuing.

1.1. What are position and momentum? The usual conjugate(dual) concepts of position and momentum exhibit a conceptual incom-patibility even at a classical level: the existence of a position denies ina sense the possibility of motion. The mathematics (derivative as alimit) poses of course no problem, yet it clearly is misleading in thelight of the quantum phenomena.

So back to an old question: “How is motion possible?” or rather“How to model motion, really?”.

The analysis of information gain during a classical measurement ofposition and momentum as a joint measurement, reveals (or claims)that the a choice of the order of measurement, i.e. position first x(t)and momentum next p(t + ∆t) or the other way around, should beirrelevant.

But the instantaneous momentum (theoretical concept, after all)can be approximated (what else is out there?) as a pair of positionmeasurements, x(t) and x(t+ ∆t) (see §3.2.4). It is a classical space-time correlation.

101

102 8. QUANTUM DOTS AND BITS

To approach a Path Model description (since there is no absoluteevents nor space-time [Ion00] etc., except correlations), we propose amatrix model based on the concept of quantum jump.

1.2. Quantum jumps. It should be thought of a “quantum space-time correlation”, in analogy with the above description. Mathemat-ically it can be modeled as follows, making use of an ambient man-ifold for representation purposes (the intrinsic description will entailshortly).

For a fixed map r : Γ0 →M in C(M) representing an embedding ofa causal event into the ambient manifold M , say Riemannian (classicalstate-space), define the “conjugate variable” p : Γ1 → TM as the ma-trix of tangent vectors at the ith vertex in the direction of the (unique)geodesics to the jth vertex:

pij = Ti(ri → rj), “rj − r′′i ,

if there is a link between i and j.If Γ is an edge, then it should be thought of as a “quantum jump”, so

we move away from the classical “local picture” (“manifolds”) towardsa “quantum picture” (Path Model) based on transition “functions”(quivers, Markov process, FPI etc.).

Then the analog of momentum as a “conjugate variable” (dual?)should be “delocalize”, but when the points are “close” we should re-cover the (or a) tangent vector. In this picture the “dimension of thetangent space” varies with k(r) the connectivity degree of the quantumdot, but we hope that the “effective dimension” (in some sense) will beat most the dimension of M ([FI2]; see Ch.10).

Indeed, Γ allows to implement, in a sense, a stronger duality be-tween the classical Hamiltonian/Lagrangian “conjugate variables” (laterwe will analyze the role of the Lagrangian), due to the correlation be-tween Γ0 and Γ1, i.e. between “free theory” and “interactions” (Lfreeand Lint).

For simplicity we will use the notation X = (r, p) : Γ → TMwhenever convenient. Note that r and p are “coupled”, but in thegeneral case, for example when considering the Poincare-Cartan formon the tangent bundle extending the Lagrangian form (see [AI], p.283),they are not.

Now the (generalized) Feynman rule (the “amplitude”) may be de-fined:

< FΓ(ω),X >=∏

ωri(∏

pij),

where ωri =< ω, r >i= ωi(ri).

1. QUANTUM DOTS: THEORY AND PRACTICE 103

The∏

j pij is (like?) a k-volume form at the corresponding quantumdot.

The big product (over i) is a form on the configuration space of Γwith coefficients in (M,η).

Now integrate over the big configuration space to get the (general-ized) Feynman integral

KΓ(ω) =

∫

C(M)

< FΓ(ω),X > DX

as a pairing between “big forms” (“amplitudes”) and “big chains”(“paths”) with kernel ω (“multi-propagator”) for a given graph (in-trinsic “interaction pattern”; Wilson chamber observable!).

The relation with renormalization as a change of variables corre-sponding to averaging over unimportant variables ([QD], p.3, “... elim-ination of uninteresting degrees of freedom”, i.e. collapsing subgraphs(n.a.) etc.) and involving a transfer of structure process is consideredin [FI2].

The information content/entropy of the underlying graph is relatedto the total quantum information gain of the process (quantum com-munication), having a classical component determined by Shannon en-tropy and a quantum component determined by Von Newmann entropy(see Ch. 6).

1.3. Position and momentum incompatibility. Returning tothe roots, Heisenberg uncertainty relation applied to position and mo-mentum expresses (also) the conceptual inconsistency between “be-ing in one place” and “moving” (see also “How is motion possible?”).In the transition picture of quantum jumps the concept of “motion”(change/transition etc.) is central, yet incorporating the concept of“position” (state) (again the duality object / morphism etc.):

xp→ y.

But “existence” actually refers to knowledge, so “position change” willbe interpreted as an “information flow” from x to y. At the quantumlevel, it is modeled by a transition amplitude (complex) from x to y,which reflects into a classical probability of transition from x to y.The probability density of the position (or wave function) changes,let us say in a classical (limit) way from certain at x ((1, x) → (0, y)configuration) to certain at y ((0, x) → (1, y)).

An intuitive picture is provided by the analogy with an electric cir-cuit, where the edges of the Feynman graph (various types of particles)


are the elements of the quantum circuit, the quantum information (uni-tary operators) flows through the elements and the classical observablesare associated with the In/Out measurements. This is just pictoriallyexplaining the quantum interaction/computing duality: a quantum sys-tem is a quantum computer (not too programmable, though) etc.

The main addition to the old Copenhagen interpretation of the wavefunction as the amplitude of probability, is its role of information flow.In the context of a discrete model (DOFs) as representing space-timeitself (“all there is”) this leads to quantum information charge and cur-rent etc. The mathematics is similar to the Hamilton-Jacoby theorydescribing Hamiltonian mechanics as a dynamics of the momentumamplitude p = ∇rΦ [Gosson], explaining the natural classical inter-pretation of Quantum Mechanics in the Bohmian formulation.

The interpretation of the “anomalous” term in the radial part ofthe Schrodinger’s equation will be related with the (classical) Shannonentropy.

Reinterpreting position and momentum is just the first step in amassive change: “categorify classical mechanics” §3, in order to fa-cilitate the transition to quantum physics (Feynman processes) in itsrelation with statistical mechanics (Markov chains and all that), as aguide to the representation theory of Feynman processes.

2. Reversibility: information flow and time reversal

Andreev Billiards [QD], p.131, constitute an interesting challengeto explain in terms of information flow backwards relative to the ex-ternal global time. An electron propagating in a metal trying to entera superconducting mirror, is forced to retrace its path, as if time isreversed! It is called the Andreev reflection.

The first lesson we learn is that one has to model particle-antiparticlepairs for completeness, even if a “local” experiment, not of EPR-typeinvolving entanglement, concerns just the particle ([QD], p.134).

2.1. Category theory and information flow. Monoidal cate-gories with duality are the perfect framework for modeling informationflow ([Ion01]; [Coecke2]). Hilbert spaces and unitary operators rep-resent just the global/bulky approach, where there is just one (simple)object, the Hilbert space and duality, as a pairing between H andits dual H∗, is provided by the inner product. Unfortunately they“hide” the underlying algebraic structures (simplex/quivers and theirrepresentation) allowing for a natural implementation of the automatonpicture: Markov chains and Feynman processes.

3. CATEGORIFYING CLASSICAL MECHANICS 105

In particular, the annihilation operation implemented by dualityin a category with duality can be interpreted as a “time mirror”; it“bounces” the information flow back in time (Ch. 6), the global, ob-server’s time that is. Is there a connection with Andreev’s billiards([QD], p.131)? It would be surprising not to! The author suspectsthat the quantum dot resolution, once the theory matures, will pro-vide a nice mathematical framework to model such “strange quantumphenomena” (Dirac’s sea of particles and the “vacuum” etc.).

3. Categorifying classical mechanics

To facilitate and improve the “quantization process” a categoricaldescription of classical mechanics would be useful as a half-step acrossthe corresponding “conceptual gap”.

3.1. Phase spaces: classical and quantum. A starting pointis to model the “phase space” as a category, merging “existence” and

“motion”, i.e. position and momentum, into a “transition” xp→ x′.

Classically there is a configuration first M and the classical phase spaceof “potential motions” is Ω = T ∗M .

This is a local /set theoretical point of view (set first then func-tions). The categorical point of view merges the two inseparable as-pects of change, which classically were abstracted into the clear-cutdichotomy position-momentum (absolute existence and instantaneouschange!?).

The quantum phase-space is thought of as being modeled as QΩ =Hom(Category,Manifold), 1 in view of the classical-quantum connec-tion. The “category of models” C consists of various types of causalstructures. For example: configurations of graphs in a given space-time, or if disregarding interactions, just the space of configurations ofN -particles Hom([N ],M).

An element of the quantum state space is a representation of anedge, signifying a quantum transition in external “space-time”, i.e.

γ(e) = x1p→ x′, with γ ∈ Hom(e,M) (forget the internal DOFs for

now). This is a “bit” of a quantum trajectory (external q-bit!?), tohave an analog term for the classical trajectory.

3.2. Classical limits. The correspondence between the quantumbits of trajectory (quantum jumps γ as elements (functors) of the quan-tum spate-space QΩ) and classical bits of trajectory (points of the state

1To avoid loosing the big picture, a correspondingly high-level language / no-tation is used.


space Ω) is plainly:

QC : QΩ → Ω, Hom(C,M) 3 (xp→ x′) 7→ (

x+ x′

2, p) ∈ T ∗M.

The relation with the usual quantization and classical limit, at thiscrude level of description, is still worth investigating; maybe this clas-sical correspondence is related with the “ ... embarrassing “middlerules” in the “Feynman type” approximations.” [Gosson], p.148.

3.2.1. ... and generating functions. What is the relation between(quantum) propagators and generating functions W (x, p;x′, p′)? (seefor instance [Gosson], “Generating functions” p.133). There shouldbe a quantum propagator, i.e. the classical categorical analog of thegenerating function, under the correspondence:

QW (xp→ x′) = W (x, p;x′, p′), p = p′,

provided the initial momentum equals the final momentum (e.g. [Gosson],p.?).

In general, the following rule seems “natural”:

W (x, p;x′, p′) = W (x(p+p′)/2→ x′).

3.2.2. ... and actions. Although there is a fine distinction betweengenerating functions and actions (see [Gosson], p.141), at this stagewe may think of the generating function W as an action:

W (x, p;x′, p′) =

∫

γ

pdx −Hdt,

where PCF = pdx −Hdt is the Poincare-Cartan form.In the free particle case, assuming M = Rn (affine space):

W = m/2(x− x′)2/(t− t′) = m(x′ − x

t′ − t)2 × (t′ − t).

It is of the form Kinetic Energy × time.In a relativistic picture, a quantum jump should have the form

qj = (x, t)(p,E)→ (x′, t′). Its (infinitesimal) “relativistic work” is the

Poincare-Cartan form:

W (qj) =< (p,E), (∆r,∆t) >∼ pdx − Edt = PCF.

Therefore the generating action is the action of the relativistic quantumtrajectory:

W (γ) =

∫

γ

p∆x− E∆t,

3. CATEGORIFYING CLASSICAL MECHANICS 107

where γ represents a sequence of quantum jumps (transitions) and thediscrete PCF is:

Categorical Poincare Cartan Form (qj) = p∆x− E∆t.

Is then the action positive only

W (xp→ x′) = pdx − Edt ≥ 0?

Allowing particles and anti particles (particles traveling back in time)seems to complicate the picture ...

3.2.3. ... and Heisenberg relations. Can Heisenberg uncertainty re-lations in the categorical picture be obtained from a quantization of theaction?

A quantization of the action could be the requirement that “Apropagator propagates a multiple of ~” (external momentum; no bo-son/fermion issue here):

W (xp→ x′) ≥ ~. (7)

Such a requirement could be interpreted as a quantization of space-timedynamics (see §3.3).

3.2.4. What about measurements? Returning to measurements, whatdoes it mean to “observe the position or the momentum” in the cate-gorical picture? Observables should rather be functors than functions!To understand the conundrum, a commutative diagram might help:

QΩQC−limit //

Q−(X,P )

Ω

(X,P )−measurement

PC1Project/Average

//R

2.

Either an averaging process of the probability distribution (if QX orQP are valued in a probabilistic projective space) should compensatetaking the classical limit.

At this point a simpler picture emerges in the “extreme” cases, inorder to brainstorm a possible approach. Measuring precisely the posi-tion of a quantum jump (∆x = 0), could be interpreted as a functorial

operation onp→ x

p′→, towards a ∆p = |p′ − p| 6= 0, while measuring

the momentum accurately, i.e. ∆p = 0, suggests an operation on xpx′,

and therefore ∆x = |x − x′|. Now a lower bound on the quantumpropagator could lead to ∆x∆p ≥ ~ ...


3.3. Exercises. So Phase Space was transformed from the classi-cal T ∗M into a Feynman category C (causal structure), but the maintask is still ahead (“geometric quantization”).

What is the analog of a Lagrangian manifold (see [Gosson], §4.6)?Define a Poincare-Cartan Functor on the Phase Category (causal

process) as an analog of the classical PCF form. Is it related to theTH invariant (§6.1)? Define the phase of a Lagrangian subcategory asa pairing associated to the PCF. Is it related to Q?

IsQ, the information potential/charge (Definition 6.1), a generatingfunction for the information flow?

Q(A) = |Aut(A)|, S(A→ 1) = lnQ(A), TH(γ) =

∫

γ

S(γ(t))dt?

Is there a theory of the Maslov index leading to a quantization ofinformation flow? See [Gosson], p.169 and compare the above formulaswith:

m(γ) =

∫

γ

d(ln |W |), m(γ ? γ ′) = m(γ) +m(γ ′).

The classical minimum capacity/action principle [Gosson], p.173, onenergy shells < E >= constant establishes a relation with the Maslovindex:

PCF (γ) = iπm(γ),

coming from a quantization of the action on loops (see Equation 7):

PCF (loop) = ~.

If a quanta of information exists (the qbit, right?) representing anelementary bit of space-time (causal structure), then, in a would bediscrete model of quantum gravity (see Ch. 9), is it related with thecosmological constant?

CHAPTER 9

Quantum gravity and information

Gravity is special. This is agreed, but from usually different pointsof view. It is a phenomenon which resisted the many attempts ofunification with the quantum description of the “other forces”...

But, is it a “force”? 1

I never liked having to solve a problem by “this or that method”;it limits you options of success. Continuing the tradition of field-particle duality, modeling forces as an exchange of a boson betweentwo fermions, is such an imposed method to “solve the problem ofQuantum Gravity.

But General Relativity showed that gravity is more then a force;it cannot be separated from the fundamental concepts of space-time-matter-energy. Then why should gravity be a “force”, like the others?If so, what is “space-time” then and where does the mass (energy) comefrom? There is nothing left to put in their place, except any physicist’prefered vacuum, e.g. string theory’s background space.

No, do not waste your last interaction in this way; then the Mind-Matter Interface (“Final Frontier”) may be left unsolved!

The opinion that gravity is an “organizational principle” alreadyemerged and gains credentials ([Penrose], etc.).

Since matter (IDOFs: mass or energy) determines space-time (ED-OFs) producing the “illusion” of gravity (GR), we will try to “produce”this illusion as an entropic (informational) effect, making use of thetradeoff between energy and information (I/E DOFs), as embodied inthe main DWT principles (Ch. 3).

1. Artificial Intelligent Geometry

Artificial intelligence requires adaptability, the signature of life ratherthen dead matter. So why not “Geometry” (E/I DOFs and their dy-namics), if it aims to model Mind (Information) beyond the traditionalMatter (Energy)?

The first step is to have “adaptable models”/variable geometry.

1Or is it I.T.? :-)

109

110 9. QUANTUM GRAVITY AND INFORMATION

1.1. Monte-Carlo simulations. As an example of a large classesof models using discrete space-time within the Automaton Picture(Path Model) we mention the Ising model [DJ], Ch.5. The latticeindexes the subsystems (N -particles), the spin is the simplest case ofinternal DOF (qbit) and the Hamiltonian/Energy is the M-observable.The lattice is fixed initially, but in other models (QG coupled with mat-ter), “adaptive methods” are used which update the matter system byupdating the triangulation (loc.cit. p.254).

Without going into details, we will just point out that the cluster-ing technique used in the Metropolis updating conceptually is a zoom-in/zoom-out of DOFs, suited for an implementation as a QDR, withits insertion/elimination operations (subgraph = cluster, flipping thespin on a cluster amounts to flip the single qbit address in the quotientlabeled graph etc.). This project will be called The Hierarchic IsingModel.

Updating geometry by changing the triangulation is not convenientto be done using the “classical” (topological) Pachner moves (p.255),but inserting edges in the framework of graph homology (homologicalmoves).

It was proposed by the author long-ago that the “correct” approach(i.e. efficient) is to abandon manifolds [Ion00] and to work with theirdiscrete incarnations, triangulations and simplicial sets, prone to anabstract (functorial) reformulation. The immediate benefit of beingreadily “understood” by computers in view of computer simulations isobvious in the Monte-Carlo simulation studies.

General relativity banished the concept of fixed geometry, correlat-ing the metric and the distribution of matter. Yet another step shouldbe undertaken removing the fixed topology, and even the “continuoustopology” altogether. We of course, advocate the Resolution Picture(QDR).

In this discrete framework GR should be implemented via a Hilbert-Einstein metric for a link-distance (and triangle-distance?) but withthe inclusion of an entropy term TH, together with a coupling betweenthe matter fields and the information current (Project Quantum Grav-itational Entropy). A convenient laboratory would be the HierarchicIsing Model with its “detailed balance” related to entropy/informationflow (edge insertions = entropy production etc.). Updating geometry(EDOFs) and matter fields (IDOFs) should be a dual process.

2. COMMENTS ON LOOP QUANTUM GRAVITY 111

2. Comments on Loop Quantum Gravity

For an introduction to LQG see [Rovelli-1] (also [Rovelli-2]), or[Smolin-2] for a brief and leisurely overview.

In our opinion LQG is a useful theoretical lab for discrete mathe-matical models of space-time. Unfortunately, it manifestly disregardsthe other interactions, which is against the spirit of GR, since “mat-ter creates space-time”, with its “illusion”: gravity. Claiming that thespin networks resemble Feynman diagrams, but have noting to do withfundamental interactions is a way-out of considering internal DOFs,besides the basic spins (q-bits). Of course, once the basic languageis developed, a higher language interface could be built on top of it... But why wait, and not acknowledge that vertices of graphs (spinnetworks) are “real matter DOFs”, as in the QDR approach? It isquestionable if there is a need for “potential space-points”, i.e. “vacan-cies” as space-time vacuum states (lowest energy levels) to be filled oroccupied with matter (higher levels). But this looks as a “refinement”of the QDR approach when considering in a more precise manner theinternal DOFs etc.

One of its virtue, that it derives the fact that space-time is discrete,is just a confirmation that one should change the conceptual basis ofthe theory and model causal structures as discrete structures; infinite,but (a resolution) of “finite type”, to rephrase what Greeks were sayingall-along (sort of Aristotle’s “potential infinity”). For this, a categori-fication is in order 3, abandoning for good the “manifold approach tospace-time”.

The missing ingredients are: the lack of scaling capabilities, noflexible internal DOFs and therefore not a framework for the dualitybetween external and internal DOFs.

The existence of a fixed “grain of space” is reminiscent of a Λ trun-cation of ultra-violet infinities in renormalization theory; or of a fixed“numerical method” ... On principial (philosophical) grounds, it is notadvisable, even if it is “derived” from the initial continuum formalism.That space-time is a two step construction, spin networks then foamsrepresenting the evolution of a spin network, is not general enoughto cope with the general case of causal structures, where informationflows, but there is no local “space × time” product structure; unlessone views the foam as a causal structure and the In/Out spin-networksas models of source and target of the interaction, in the spirit of DWT(of course :-).

It should not be hard to transfer the technical tools into the richerQDR framework (AI-geometry and E/I-DOF duality with PCS-duplex


interpretation), with the main goal of deriving the black hole radiationlaws (“Project LQG++”) as an overall consistency check (i.e. subsum-ing QM, GR and thermodynamics).

3. Is “reality” 1-D?

It is hard to explain why there are 3 generations of fundamental par-ticle, or why is space 3D (11, 21 etc.) etc. But this last question wouldbe settled if the “space” (external local DOFs) would be (modeled) 1D,while “time” would be modeled as a causal structure (subsystems ininteraction); you would not ask “Why 1D?”, would you?

3.1. Are quaternions the new coefficient-material? This last“move” away from classical, consisting of treating R3 (or SU(2) / spin)as a local internal symmetry rather then a global “illusion”, is con-sistent with the categorification of physics and the trade between I/EDOFs. “Just” represent the causal structure attaching to vertices qbitsmodeled as quaternions (tensor algebra; related to twistor program?).The last, or rather latest, change of coefficients from C to H §7.3 shoulddo the trick!

Then, how to relate such a structure with “normal physics”? Again“changing coefficients ”; if what we are talking about before is “QFTas derived functors a Feynman process on a causal structure over thequaternions” [Ion04-1], then take “cohomology with coefficients” in agenuine space-time manifold as your favorite background (Calabi-Yau?2 [FI2]); if the causal structure is the punctured Riemann surfaces(Siegel) PROP, then ... string theory should emerge. In other words,most of the technicalities are there; all we need is to take a step back(pull back) to reach for the underlying ideas forced upon us by good-practice, then leap forward, beyond our preconceived ideas ...

In order to explore the quaternionic picture, understanding thedeeper meaning of a Wick rotation is important; are we in a semi-Riemannian setup R3,1 or the more algebraic framework provided byquaternions (skew-field, Clifford algebras-quantization etc.)?

If the conformal theory is rigid, how rigid is “quaternionic analysis”(SO(3, 1) ∼= SU(2) ⊕ SU(2) and Virasoro algebra)?

¿From the Feynman dimensional analysis point of view, is The The-ory (say QG) renormalizable with a 1D-space-time?

With external and internal DOFs on an equal footing, do we stillhave a “no go theorem” against unifying Poincare group with a GUTgauge group?

26 dim + 2 the string = 2× 4 ... C(G; T ∗H)

3. IS “REALITY” 1-D? 113

Will the “Super DWT” explain the three generations of particles?

3.2. All good things come as triples; why? Indeed, besidesSpace-Time being 3D, there are 3 generations (families) of particles, 3quarks, 3 flavors, 3 colors etc. (not to speak of other “trinities” ...).

The author’s believe (or optimistic attitude) is that these are coinci-dences, until one creative mind manages to link them within a totallysurprising way. Then the expectations are to derive the facts fromeven a smaller number of fundamental assumptions, according to theaxiomatic ideal, from Greeks to Hilbert and beyond. On the otherhand knowledge comes in data-bases, and is better accommodated ina hierarchic-relational structure. We should not be afraid of loops or“circles” (unless really vicious); we should perfect our ways to exploitthem (better KDBMSs).

For example, in the Standard Model there are still a lot of particles(that’s OK: DB-types), but also a lot of processes used to derive theirproperties: Lagrangian terms (fields and interactions), renormalization,mixture angles etc. (OK again: it’s a procedural DBMS).

Now the “family problem” is related with the “neutrino problem”(see [SciAm/SE]), solved by a model (oscillation of neutrino types)which cannot be accommodated within the Standard Model. And thekey issue is mass again, as different levels of excitations of the sametype of particles, leading to 3 generations.

So, what is mass? See [SciAm/SE], p.33. Is is explained via aLagrangian mechanism introducing a new special field, the Higgs field.The interaction of particles with the Higgs field accounts for the massof the really fundamental particles, beyond the mass of protons andneutrons due to internal kinetic energy of quarks.

One of the “special”, unique feature of the Higgs field is that itslowest energy level is nonzero: “the universe is permeated by a nonzeroHiggs field [SciAm/SE], p.36. It does not take much to start speculat-ing whether these Higgs Lagrangian terms in one theory (SM) play therole of the “grain of space-time” in another (LQG), and might be assim-ilated with vacancies in the a version of QDR of DWT where “emptyEDOFs” (potential) should be considered (vertices labeled by “Higgs”internal DOFs) with which the “normal folks” (electrons etc.) could“switch places” (vertices acting as vacancies). It is a way to introducea “background space”, yet of a different kind compared with the usualvacuum state implemented in the context of on a Space-Time manifold(or sigma model etc.). Recall that the distinction between free theoryand interaction might not be so clearcut in certain Hamiltonians (La-grangians), but once made, corresponds in QFT for example (FPI), to


the “choice” of the class of graphs and propagators via Wick Theorem,i.e. how to pair EDOfs and IDOFs ... In the QDR of DWT, due to theduality between E and I DOFs, this clearcut distinction does no longerapply; what are the benefits? It is too soon to tell, but in principle thetheory should be much more flexible to accommodate new proceduresto derive previously “unexplained phenomena”.

3.3. Matter or antimatter? Speaking of which there is one moreunexplained “mystery”, biggest of them all ([SciAm/SE], p 31.5):“Why is the universe made of matter rather then antimatter?” (notthat it matters :-), but it’s intriguing ...).

An easy way out is to invoke an anthropocentric argument. Tosimplify the “picture”, let’s assume we are dealing with just one tye ofparticle/antiparticle.

If anti particles are particles traveling back in time (Feynman),and since each particle with its antiparticle is just a Feynman knot(loop) in the universe, all there is around us is a big link (entangledknots; can be quite a big mess). What we experience locally is a braid(tangle etc.), which can be completed, in our imagination at least,to just one link. Then the question “Why is the braid (“the mess”)“concentrated” in our local universe?” can be dealt with invoking a“stratified complexity principle” (can’t define that; it’s just a “picture”,remember?) to conclude that humans (the biggest “mess” ever :-))must belong to that part of the universe ...

Technically inclined people would like to invoke charge-parity vio-lation (loc. cit.). The present state of the art cannot entirely explainthe amount of matter around us. Then may be entropy can help: dueto CPT invariance, CP-violation can be traded for T violation, exceptphysics (classical and even quantum), is traditionally T-invariant (onlyhighly complex theoretical mechanisms can account for a local time-reversal violations ... ?). But what about “entropy” (information flowetc.; if we introduce a twist - indeed!- in the categorical framework:left duality not the same as right duality); will it help? Probably yes,in conjunction with gravity, as an organizational principle perhaps ...So, after all, particles alike ... cluster alike; the more “time” passes(or rather the degree of complexity increases), the larger the clustersof matter particles and the larger the clusters of antimatter particles.Then, do we really need a “high” CP-violation? Is at the present timethe clustering at the level of cluster of galaxies? Etc., i.e. it seemsthere is plenty of room (space) to turn our theories one way or another... but it should be a simple and beautiful turn! (Occam, Einstein, etc.“are watching”)

4. THE LAWS OF BLACK HOLE RADIATION - REVISITED 115

4. The Laws of Black Hole Radiation - revisited

With the “ideological progress” in the present version in mind, wewill review the implications at the level of the program [Ion00], sincethe birth of the present project [VIRequest-UP].

4.1. Entropy “resides on the surface”. Recall that the entropyis a derivation Ch.5 §7.2.2 and it is sensitive to the external structureof a graph (causal structure, thought of as one possible space-time).

This is consistent with the fact that “Black hole entropy resideson the surface; therefore, maybe the degrees of freedom reside on thesurface; ...” [G] p.53. Indeed the “external interactions” are the “legs”of a graph inserted (or collapsed subgraph). As mentioned elsewhere,there is no “fixed” structure, since it depends on the current scale.The conversion between internal and external DOFs implemented viainsertion and elimination of EDOFs (and correspondingly averaging ofIDOFs) should be investigated in connection with the concept of blackhole horizon and its radiation laws.

4.2. QFT and black hole thermodynamics. It was suggestedearlier that black holes (BH) are “prototypical” when it comes to theunderstanding of the quantum bits and dots of reality. Recall that(current theories) characterize BH by their mass, angular momentumand charge and “the laws of black hole mechanics literally are theordinary laws of thermodynamics applied to a system containing ablack hole.” [W], p.9.

In a sense a BH is a “unit of spacetime”, so it may be modeledas ONE EDOF: the DOF resolution (QDR) “ends” there; a BH is a“simple subsystem” or rather a BH is a “collapsed subgraph/system”(indeed collapsed!): γ ⊂ Γ → Γ/γ, with the point the subgraph γcollapses thought of a the BH 3. The collapsed DOFs to a sink (allinteractions are incoming) plays the role of a BH.

Now Hawking radiation (creation of a pair particle and anti-particle)is naturally modeled as an edge insertion (graph homology differential[K92, FI1]), splitting the one EDOF (the BH) into ( a superpositionof) two DOFs and a correlation. To enable a “local time” (“since cre-ation”), probably it is better to replace the edge with a “coproduct-like”V (3 EDOFs etc.).

In any case, the “loss of quantum coherence” is a natural featurein the perspective of [LB]. Although the BH IDOF might have beena pure state before the interaction (insertion), relative to “Eve’s time”

3The continuum with the “empty space” for a nice and easy mathematicaltreatment seems like a waste of ... modeling ingredients!


(outside time description of the transitions etc.), the split of BH intoBH’ and Pair yields still a pure state of the whole system BH’ andPair, although each subsystem, BH or the Pair, are represented by amixture.(?)

In [LB], one of the main points is that, in general, two pure statesmix after interaction; although the system is still in a pure state, thetwo subsystems are each a mixture. Indeed, the interaction of the twosubsystems produces an “exchange of information”. Traces of this in-formation exchange are left after the interaction, as “memories” of theinteraction experienced. As advocated throughout the book, the dis-tinction between “system” S and “observer” O is relative; it is merelyan “orientation issue” S → O, yet the nature of S or O is not an issue(“alive or dead” just changes the information processing capability -in the extended sense: matter/energy/information). In a “system-to-system” interaction S → S (classical setup; although it should applyto “Alice → Bob” communication equally), there is an “exchange ofwave functions” if you please ... The exchange is permanent in thedescription of the observer (Eve), who’s responsible of the distinctionbetween the two systems in the first place (“subjective”? yes; there isno other way! (see Ch.10 §3).

The subjectiveness of descriptions is related to the hierarchy of themodel and to the fact that there is no “universal time”; the need for aresolution approach appears more clearly, including a certain “stratifi-cation of space”. “Spacetime” is no longer a bundle (or product Space× Time), since correlations are not globally sequential XOR parallel;at a certain level of description (degree within the resolution) processesmay be parallel or sequential, and also the subprocesses within a pro-cess, yet in the combined model the subprocesses might not be directlycomparable with the outside processes to be classified as s/t-correlatedat the level of the outside description 4

4.3. What is missing? A more successful incorporation of GR’sideas will be possible after a more technical development of the Cate-gorification of Mechanics Ch. 8 §3. Indeed, “Since the time of Descartes,we’ve found it very powerful to label points by their coordinates, ...”[G] p.54; it is time to move on ...

4.4. Unruh’s Law. It states that an accelerating observer “feelshimself immersed in a thermal bath of particles” at temperature T =a/2π [W], p.115. It appears paradoxical since an accelerating observer

4Elaborated in terms of DOFs extensions.


asserts “particles are present” when an inertial observer would assertthat “in reality” space is devoid of particles.

This BH Law is particularly intriguing in the context of DWT.Acceleration or “force” is due to interactions, which are (modeled as)relations, so the higher the interaction force the larger the “number”of elementary interactions (“quantized picture”), and correspondinglythe larger the number of vertices: “particles”!

In GR acceleration is equivalent to gravity, which “is” spacetime,i.e. the causal structure. So in the spirit of Einstein’s equivalenceprinciple a = g (ma = mg), “gravity” corresponds to the density ofDOFs (say number of vertices, with a certain weight associated toeach type of vertex); so, roughly speaking, the greater the accelerationthe more particles are observed (modeled) ... (what about uniformmotion”? - see Ch. 10 §4).

The interacting DOFs form only the “topological picture”; “add” aLagrangian to get geometry/dynamics. Then the linking distance willhave a counterpart depending on the momentum flow etc. A possibilityto implement this is, in the graphs (cobordisms) model for QDR (Gn,m

see [FI1]), to use the internal degree n (resolution “depth”) to controlthe spacetime dynamics (acceleration), while the external degree M tocontrol the “actual” number of particles ...

BH are essential in this early implementation stages, since theyare in a sense extreme cases; a BH may be thought of as an “infor-mation sink”, or a causally terminal object which is responsible for atotal “conversion” of dynamics (micro/EDOFs) into thermodynamics(macro/IDOFs).

4.5. The first law of BH mechanics. The first laws, also knownas the “area theorem” [W] p.138, is also striking from the perspectiveof the DWT entropy H: a derivation satisfying the Hologram Theorem3.1. It relates the area of the horizon of a BH with the other “statefunctions”: mass, angular momentum and charge (loc. cit. p.142):

κ∆A = 8π(∆M − Ω∆J).

The LHS is a variation of the Noether charge associated with Killingfield integrated over a bifurcation surface σ:

∆

∫

σ

Q =κ

2π∆S,

with S = 14A. The analogy with the entropy H representing the infor-

mation potential:

I = −dH


with I as a Noether (information) current is striking. Especially sincethe role of the bifurcation surface, similar to the role of the focal pointand surface in Bousso’s result (Ch. 10 §2), is that of the elementarycorolla used in the analysis of the role of entropy (Ch. 5, §6.1):

I(p) =

∫ p

0

ln zdz ↔∫

σ

Q.

The (conceptual) mathematics is the same (symmetries, Lagrangian,Noether etc.).

In fact the “difference” between the discrete picture of a corollaand focal surfaces in semi-Riemannian spaces (or BH as a terminalobject/sink or semi-Riemannian elaborate construct - see the defini-tion of a BH [W], p.134)) is rather “technological” then conceptual.Faraday introduced imaginary lines of force through Newton’s space,and Feynman generalized them; the present author suggested to startfrom the Feynman paths and derive spacetime as a classical limit([Ion00, Ion03]). The resolution approach comes as a more technicalsuggestion regarding how to do this. It seems that the classical (man-ifold) spacetime is “imaginary” (too much “modeling material used”),and Faraday-Feynman lines of force/paths are the “real thing”! (Finetuning: less analysis more algebra; ask Dennis Sullivan!5 And no R; Cor better H! O?).

4.6. The Hwking effect. “A black hole will radiate exactly likea blackbody at temperature a/2π.” [W], p.151. A few additionalcomments will mark the winding road to QDG (see A).

This law is usually interpreted as “surface gravity, a is not merelya mathematical analog of temperature, it literally is the physical tem-perature ogf a black hole.” (loc. cit. p.151). With hindsite, we recog-nize that these “physical identifications” ([W], p.163) are “just” conju-gate variables of the underlying theory: temperature and acceleration(force) T = a∗, entropy 6 and “area” (metric) S = A∗, mass and energyetc. (More in A).

A “coincidence” fueling another speculation is that the “Hodge du-ality” between entropy and area is 2S = A/(4π) where the RHS isthe normalized area in 3D-units (areas of the 3D-unit sphere), whilethe LHS is the double of the normalized entropy; is it because of anunderlying double cover? (SU(2) → SO(3)) 7 ...

5Talk at Henry Poincare Institute - 2002.6Previously denoted as H.7Gauss curvature and angle?


A “difficulty” connected with this law is the so called “loss of quan-tum coherence”. As it was mentioned elsewhere, when properly takinginto account the “whole system”, BH and created pair, it is a naturalconsequence of the “mixture” between its components, BH and pair(see [LB]): each of the two subsystems are in mixed states, but thesystem is still in a pure state. That we have to consider all the subpartsentering the process is natural; in connection with the entropy analysisis called the “generalized 2nd law” ([W], p.163). In fact there is no“generalized 2nd law”, just a correct application of the 2nd law to theIO-process (and forgetting about Eve ...). The “generalized entropy”([W], p.166) is just the entropy of the whole process.

•

BH

γ

e−

~~||||

|||| e+

BBB

BBBB

B

• • •

Regarding the direct calculation of the “entropy from first principles”([W], p.164), it depends on the mathematical model; it was acknowl-edged in [W], p.175, that physically “all the degrees of freedom ofa black hole were concentrated in a Plank length “skin” around thehorizon.”, yet in a not suitable way mathematically (i.e. the semi-Riemannian / differential geometry implementation of GR): “However,these ideas run counter to the notion in classical general relativity ofthe black hole horizon as being a globally defined, mathematical sur-face, possesing no local, physical significance, and thus providing a verypoor candidate for where the em true DOFs of a black hole should lie”(n.a. emphasis). I am glad of this confirmation; as I was saying else-where 8, a semi-Riemanian BH should be thought of, and it is modeledand generalized in DWT, as a terminal object (one EDOF etc.)

Now DWT is not the “toy theory” advocated in [W], p.185 as a“theoretical lab” for investigating these issues, (although it looks likeone :-). It will turn out that “Further investigation of these issues...” ([W], p.175) leads to a insight into the quantum nature of gravity(QDG - Annex A) and the “old semi-R” calculation seems by now ...irrelevant 9.

8It’s too much work to keep track of links!? :-)9Clearly the math model of GR blocks the understanding of the BH physics;

just apply Occam’s razor and abandon GR in continuous spacetime.


Regarding the “enormous entropy” ([W], p.164) of a BH when fun-damental constants are restored:

S =kc3

4G~A, A = 16πM2

it seems that this is consistent with the energy-mass analog situation,together with a huge number IDOFs (yet only one EDOF). As we willsee later (Annex A), the energy-mass relativistic relation hints to asimilar interpretation for the well known relation between free energyand entropy:

Z = e−βE, S = β(E − F ).

In other words, E = S/β+F is a (approximative, i.e. non-relativistic)decomposition of total energy into external/free energy and internal/informational(potential) energy, similar to the classical decomposition of energy intokinetic energy and potential energy. When external and internal dy-namics is modeled, a grain of space-time-matter is the dynamical qbitT ∗qbit (more later on).

The appearance of negative heating capacity ([W], p.165) is naturalin the contex of the full, complexified, theory where energy E is themodulus of quantum information (Annex A):

I = F + iM, E = |I|, MBH ∼ 1/T = kβ.

We should not exclude (such) “radical proposals” ([W], p.183),which introduce “new physics” by keeping the existing physics, yetintroducing another mathematics altogether (basically representationsof PROPs), not just before reaching Plank scale: it should model bothmicro and macro cosmos etc.

The evaporation of a classical BH ([W], p.178) should be thought inDWT of as a convertion between I and E DOFs. A BH, as one EDOF(not one IDOF! see [W], p.183), with a certain mass corresponding tolots of IDOFs, is rather a “reservoir” but with a finite number, yet quitelarge, of IDOFs. The complete evaporation (“empty the correspondingmemory”) is just an insertion of a subgraph (etc.).

CHAPTER 10

Conclusions and further developments

I will be brief here, since I would rather like to know the reader’scomments ([email protected] or Info@VIRequest).

The DWT theory is based on the interaction-communication duplexinterpretation of quantum phenomena, aiming to open the way towardsa unified description of phenomena involving “Mind” and “Matter-Energy”, going beyond Einstein’s unification embodied in E = mc2.

At the conceptual level, the source is a new Unifying “Super-Principle”,only sketched at this stage.

At a more technical level, the unification requires the duality be-tween internal and external degrees of freedom. It comes on top of anew understanding of what “Space-Time” is: “forget the space(-time),all you need is a resolution”. The Quantum Dot Resolution is a resolu-tion of DOFs, representing the causal structure generalizing the “old”approach based on “Space-Time” (manifolds or not).

The focus is now on the Information Flow, generalizing the usualdynamics approach via time-flow in a state space. The crucial conceptis now the Entropy (“information charge/current” etc.), again a deriva-tion, which is supposed to relate the dynamics and thermodynamics ofinternal and external DOFs.

Although the DWT is envisioned as being designed top-down, fromprinciples to implementation, at both levels it draws on the currentlyemerging ideas and techniques of the various existing theories (ST,LQG etc.), viewed from the Computer Science and Homological Alge-bra summits. As always, the “Truth” is somewhere ... “above the mid-dle”; one theory may benefit from the conquests of another (e.g. StringTheory and Loop Quantum Gravity [SciAm/SE]), yet a “clean” newstart seems to be beneficial (DWT, of course :-). Instead of lookingfor a “unique/final theory”, we rather advocate a “flexible framework”capable of adaptation to new scientific discoveries, since informationimplies change, not only as a basis of quantum mechanics 1.

1Information implies “change” and different choice (non-commutativity):AB 6= BA - see Ch. 6§1

121

122 10. CONCLUSIONS AND FURTHER DEVELOPMENTS

1. Dimensions: 1,2,3 ... ∞!

There are various other indications supporting the “resolution withduality approach. The Holographic theories establishing the “samephysics” at different space-time dimensions ([G], p.53), could be “just”Stokes Theorem when embedding the resolution in a specific space-time background. The “resolution approach” resides conceptually atthe level of the cohomology of Feynman graphs [Ion04-1], while em-bedding the graphs in a manifold resides at the level of cohomologywith “coefficients” in a given manifold [FI2]. String Theory is such a“cochain”. An elaboration of the homological and homotopical physics(adapting Stasheff’s terminology) starting with the Cohomology The-ory of Feynman Diagrams (Feynman Categories; see also Hopf diagramsetc.) may lead to a proper classification of (k,d)-QFTs.

There are various indications to support the statement that thenumber of external (space-time) dimensions is ... irrelevant (“So eventhe number of dimensions seems not to be something that you cancount on” [G], p.53). In more technical terms, a certain FormalityTheorem may be at work when classifying QFTs by taking the “co-homology with coefficients” in a given background space-time (Mirrorsymmetry realm?). But the “principal dimensions” (3+1) seam to playa distinctive role, possibly as a “shadow” of spin, the symmetries of thefundamental information unit: the Qbit (d ≥ 3 and t ≥ 1).

Regarding “time” or rather the more fundamental information flow,the PCS-interpretation of quantum phenomena (interaction/communi-cation) suggests the existence of a certain organizational principle atwork (gravity related?) responsible of the increased “stratified com-plexity” we see around us. Although an anthropic point of view, some“anti-entropy law” could be keeping the natural balance, giving a localmeaning to the concept of (arrow of) time as the information gradientin our local bubble of universe 2.

2. The Holographic Universe

Other indications supporting and confirming our ideas (the “collec-tive mind” at work) may be found in [B], p.74.

Indeed “information is just as crucial an ingredient.”. The ideathat world is “made” of more then matter and energy is obvious ifwe think of “hardware” which is “nothing” without “software” (anticeven: body and soul etc.). As we learn from loc. cit., “Indeed, acurrent trend, initiated by John A. Wheeler of Princeton University, is

2Helmholtz/Boltzmann/Eddington?

3. TIME FLOW IS SUBJECTIVE 123

to regard the physical world as made of information, with energy andmatter as incidentals.” 3.

May be not incidental (the sward is double edged - duality etc.),but we definitely should focus on information, to compensate the “lostyears”.

As suggested earlier, black holes, although emerging from GR insingular conditions, seam to be prototypical regarding what matterand energy “really is” 4.

The information bounds (holographic bound etc.) naturally appearin connection with surface terms, expressed as the Holographic Theo-rem 3.1 intends to hint (Ch. 5), as a divergence (entropy is sensitiveto the boundary structure only). Similarly, the 1999 R. Bousso’s result(loc. cit. p. 81), is strongly reminiscent of the behavior of entropy, orrather information flux on the simplest case of a corolla (decision node).The “big step” will consist in abandoning the continuum 5. The devel-opment of the theory (not from scratch: the ideas are portable) in thediscrete realm should be much easier and computationally profitable.

Ultimately, the “game of life” consists in managing the degrees offreedom! (see Fundamental Principles of DWT Ch. 3); and it’s notclear if the resolution (QDR) is infinite or ends up in “degree X” (i.e.is there a “syzygy theorem” here?) 6.

3. Time flow is Subjective

Time does not fly [Davies2] nor flows (in general there is no “time”as we used to think of).

Information does flow and ... we knew this all the time (!) when wesaid “Time passes by”. I picture us (or quantum systems) “exposed” toan information flow (interactions) and depending on our capability ofprocessing (interacting) the (whole) Input (the “present reality”) maypass through us or besides us (opportunities missed, choices we chosenot to make etc., right?). Does this requires a “brain”? yes, of somesort (see Ch.2 §5). But the differences are quantitative rather thenqualitative (if we chose to make a modeling effort we didn’t have toin the past). It is perhaps time “to move” conscience away from thecenter of the universe ...

3Could not find more on John Wheeler’s initiative; my ... thanks for a link,please!

4Loc. cit. p.80/3: “Entropy of the hole - a deeply mysterious concept - equalsthe radiation’s entropy, which is quite mundane”

5Loc. cit. p.81/2-3:“Fields, ..., vary continuously ...”.6“What are the ultimate degrees of freedom? ... level X.” loc. it. p.76/3


What about the distinction between “subjective” (what we 7 think/modeletc.) and “objective” (outside our inside :-)? The ABE-model of mea-surement (and modeling too) can be specialized to B = E and under-stood in terms of [LB], leading to a blurring of this distinction. In aS1 → S2 interaction S2 is subjected to the interaction of/with S1; asa consequence their states change, with respect to Eve’s description.What is special is being Bob and Eve at the same time (... what time?8).

Nevertheless Past, Present and Future are not illusions [Davies2],p.83! Illya Prigogine “have contended that the subtle physics of irre-versible processes make the flow of time 9 an objective aspect of theworld” (loc. cit. p.85). The remarkable book [Prigogine] perhapscame too soon/ahead of its time (QFT, categorical descriptions, quan-tum information flow etc.).

When Alice is disappointed (“She hopes for a white Christmas” –see [Davies2], p.84/2), The Past is “ ... and Bob had no idea!”. ThePresent is the event: the communication of Alice’s (partial) state toBob. And (in) The Future “Bob knows to take Alice to the North Pole(Vail?) for a Merry (white) Christmas”.

The message is that the system is “Alice and Bob” and ... I’m Eve10

:-), therefore an external and global time always exists!Interactions or communications: what’s the difference anyways?

Subjective or objective, are relative (in fact dual) ... As Feynmanwas saying “Physics starts with dividing the universe into two parts”[Fey1]; that’s a “relative” thing to do (subjective), right? yet unavoid-able!

4. Cosmological constant: nowadays Cinderella

The role of the cosmological constant (CC) as the “Einstein ap-proved” 11 change of GR is overloaded ([KT], p.67).

The apparent innocuous change consists in moving CC from onehand to the other: take the CC from where Einstein put it (LHS of hisequation) and interprete it as a matter term; is it related to the Higgsfield (nonzero ground state) or is it dark matter?

7Me, you etc. or correlated “we”.8Interface between conscience and sub conscience, processes and subprocesses

etc. - quite confusing!9I.e. information n.a.10The storyteller - modeler - experimenter or collective mind11Although renegated later!

5. WHAT NEXT? 125

There seems that presently a crisis in theoretical physics is happen-ing 12, despite a certain reassuring attitude that “everything is undercontrol” (“cover-up”?). Old and venerable theories run amok for what-ever corrections might be handy to save the experimental and compu-tational evidence that it’s not OK, and the present theories (no names)are far from being close to the Theory of Everything etc.

Whatever the RHS CC stands for, it corresponds to a “bizzare newform of energy ... ” (loc. cit. p.69/1), which in DWT could meanentropy/information etc. 13. The huge values obtained for the CCin some quantum theories taking into account the nonzero energy ofempty space (Higgs field?) is conceivably consistent with a “missingquantization” of energy/information and spacetime (“grains of space”etc.). Also the impossibility of redefine the zero energy point (now“energy matters”!), is reminiscent of kinetic energy as part of the mass(“small”) and rest mass (“huge”) relationship.

We view DWT as a potential avenue to relate all these pieces of thepuzzle into a nice framework capable of explain both the micro-cosmos(QFT - few DOFs) and the macro-cosmos (cosmology - many DOFs).

GR took shape after a decade-long struggle following his pivotalobservation that gravity and accelerated motion are equivalent (loc.cit. p.68). Or rather “intuition” that such a new fundamental unifyingprinciple will yield a rich “crop”: GR. A collective effort developingthe DWT will need a few years only!

5. What next?

The other Einstein’s “three prejudices” (!?) regarding the model(of spacetime/universe etc.) (a) finite, (b) static and satisfying Mach’sprinciples, most notably that matter should define space (not “just”determine the metric of space) seam to be legitimate in the perspectiveof DWT, with a few amendments (e.g. finite type etc.).

But perhaps one of the “mile stones” of physics: uniform motion,has to go! In some sense it disappeared when introducing covariance,so that any coordinate system is “equally” admissible. Yet our need tohave a “free theory” (even asymptotically) and correspondingly inertialframes, uniform motion etc. is still there.

But a lump of matter traveling through space is contrary to Mach’sprinciple ([KT], p.68/2). In DWT’s DOF picture uniform motion is not“natural”; the resolution of a DOF/system into more DOFs/subsystems

12“Today ... physicists ... explore every avenue possible ...” [KT], p.73/3 etc.13The possibility of realizing mass as an entropic effect was mentioned earlier.


does not involve an ambient space etc. Perhaps the “future” is for the-ories like QCD, which implement confinement as the rule with asymp-totic freedom, as a classical limit, the exception.

DWT approach (“the physics of DOFs and interactions”) is con-sistent with such a view: the modeled universe is confined to relativeproperties between its subparts. Motion of a subpart cannot “continueindefinitely” (makes no sense), since interactions are present (within aconnected component; otherwise “one universe” knows nothing about“another one” etc.). In a way we return at the Aristotle’s point of view,but again at another level: essentially there are only harmonic oscil-lators and “everything is cyclic”; yet processes occur cyclicly within acertain hierarchy (interacting qbits / “stratified complexity”).

If QCD is the “future”, what about Supersymmetry”? At the levelof graphs as models, at least, Poincare duality allows to exchange ver-tices and edges, i.e. “particles and fields”. There should be no newparticles, but rather a unified framework for two dual points of view.14.

6. Epilog

We should also mention Lee Smolin as the advocate of the impor-tance of information exchange among physical processes: “such a finaltheory must be concerned not with fields, not even with spacetime, butrather with information exchange among physical processes” (loc. cit.p.81/3).

We would like to emphasize that “physical” should include “alive”,and there is no clearcut distinction when comes to artificial life (orintelligence; Turing test etc.): If it processes information, then it is(more or less) alive! 15. Indeed, processing (exchanging) matter andenergy with the environment must be “upgraded”, in the perspectiveof the New Unifying Principle of Information/Matter/Energy, to in-clude processing information; in what extent, that’s “details” ... theprinciple counts at this time. How much configurational informationand how much “inert” information is there in a “grain of organizedmatter” is a quantitative issue 16, but qualitatively reminiscent of restand dynamical mass in the context of relativity (E = mc2, m = m0γetc.). Can be use the information “locked” in matter beyond “merely”

14“Tabula rasa” is a clean start :-)15God’s children are playing ... “God”!16“A silicon microchip ... smaller than the chip’s thermodynamical entropy”,

loc. cit. p.76/2-3.

6. EPILOG 127

banding spacetime (worm wholes etc.; not that we can already do this:-)? 17

So (“redirecting” loc. cit. p.81/3), perhaps “the vision of in-formation as the stuff the world is made of will have found a wor-thy embodiment” in the DWT as a “would be theory”, embryonic in[Ion00], announced in 2005 [VIRequest-UP] and presently a GrantProposal/Invitation (n.a.) 18.

Regarding “our position in the universe”, the causal structure re-placing the traditional “Space-Time” (QDR and its representationsetc.) is our computational efficient framework to describe a quantumsystem. Yet the “whole story” is the “observed-observer” interaction-communication process (S−S, S−O or O−O), including the measure-ment process modeled as an S → O (or better ABE-eavesdropping)process. With this in mind and to end on an optimistic tone: “TheUniverse is teaching Us and the Miracle is: we do learn well!”.

17To suggest such a possibility the subtitle THE (Quantum) MATRIX wascoined ...

18Please visit the VIReQuEST! - Annex B.

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2005, p.6.[Stapp] Henry P. Stapp, Mind, Matter and quantum mechanics, 2nd ed., Springer,

2003.[Stat1] Entropy, Helmholtz Free Energy and the Partition Function,

http://theory.ph.man.ac.uk/ judith/stat-therm/node70.html (see also:node71.html, etc.)

[ThD] Internal energy, enthalpy, entropy, free energy, and free enthalpy,http://www.techfak.uni-kiel.de/matwis/amat/def en/kap 2/basics/b2 1 5.html

[V] V. S. Varadarajan, Supersymmetry for mathematicians: an introduction,Courant Lectures in Mathematics, 11.

[VIRequest-UP] L. M. Ionescu, The search for a new unifying principle,www.VIRequest.com/VIReQuest UP.htm, c©2005.

[VIRequest-CS] L. M. Ionescu, Modeling and Quantum Comput-ing: a Comparison Study, Project proposal II (in preparation),http://www.virequest.com/VIReQuest CS.htm.

[Wolfram] Stephen Wolfram, A new kind of science, 2002.[Was Einstein right all along?] Was Einstein right all along, New Scientist, 3 De-

cember, 2005.[WebCopies] Web Copies of Referenced Web Pages,

http://www.ilstu.edu/lmiones/copies[W] Robert M. Wald, Quantum Field Theory in curved spacetime and black hole

thermodynamics, Chicago Lectures in Physics, The University of ChicagoPress, 1994.

[Zee] A. Zee, QFT in a nutshell, A. Zee, 2003.[Woit] Peter Woit interviewed by Susan Kruglinski, Discover, February 2006, p.18-

19.

APPENDIX A

A research diary ...

1. Why is mass energy: E = mc2?

The relation between mass and entropy was conjectured in the con-text of the interpretation of gravity as an organizational principle, moreprecisely as an entropic effect.

We speculate on the mathematical similarity between the black holelaws and thermodynamic laws in the context of the duality betweeninternal and external DOFs asserted by the Fundamental Principle ofDWT (QDR).

Entropy enters the picture as a derivation residing on the surfaceas pointed out by the Hologram Theorem 3.1. The suitable frameworkis that of a Lagrangian with symmetries generating a Noether currentvia Noether theorem (see Ch.5):

H = −dI,with H the information potential (and associated charges: qubits etc.).

The comparison with black hole state functions was started in Ch.9 §4.5. Continuing the comparison and taking as reference for the BHlaws [W], we claim that the flux of the information current :

∆H = Q(∂t) = −I(∂t), I(p) = −∫ p

0

ln zdz.

corresponds to (with the notations from [W], p.146) the integral ofthe Noether charge Q corresponding to the Noether current j = dQassociated with the local symmetries of the vector field/flow N (loc.cit. p.146):

∫

σ

∆Q

which in turn yields the entropy (a is the acceleration denoted in [W]with κ and H is the entropy denoted in [W] with S):

a∆H =

∫

σ

∆Q.

We interprete this as:

133

134 A. A RESEARCH DIARY ...

Theorem 1.1. The flux of the entropic current of a terminal ob-ject (e.g. BH or EDOF) is given by the macro state parameters:

T∆H = ∆M − Ω∆J

where M is the mass, Ω is the surface angular momentum and J is thevolume angular momentum.

The “theorem” relates the “incoming external dynamics” (dynamicsof EDOFs) with the “internal dynamics” (thermodynamics of IDOFs).

The analogy between entropy H and area (BH laws) is clearly jus-tified by the duality (I/E DOFs) since both are “surface terms”. Al-though the area theorem in differential geometry and 2nd of thermo-dynamics seem to belong to different realms (holding true in practiceonly approximately loc. cit. p.148), the latter is also a mathematicalresult at the level of Shannon entropy on corollas for example (focalsemi-Riemannian surfaces ∼= corollas plus “waste empty space”). Atthis point we would like to “normalize entropy” and think of a corollaas a wheel:

Big H bar : H = H/2π.

The “time asymmetry” is just the orientation of the corolla (direction ofthe information flow): no input and n-outputs (rather a “white hole”).

Again we will stress that a BH is a prototypical terminal object(DOF) disguised in semi-Riemannian outfit. The general case is anIO-communication/interaction; the cut and paste operations can beused to relate the general case to the “terminal cases” (duality, traceetc.).

The 1st Thermodynamics Law (1TL) is:

2πT∆H = ∆E + P∆V,

referring to the IDOFs (thermodynamics).The 1st Black Hole Law (1BH) is:

a∆H = ∆M − Ω∆J,

referring to EDOFs (dynamics).These are two coupled equations relating the dynamics and ther-

modynamics; the (future) unified theory is called hyperdynamics (seelater on). There are additional correspondences between temperatureand acceleration (“force” or gradient of an external potential), namelyUnruh’s effect, and between energy and mass, namely Einstein’s equiv-alence principle E = mc2.

Now the Unruh’s Law (effect):

2πT = a,

1. WHY IS MASS ENERGY: E = mc2? 135

may be interpreted as: Temperature is the “hidden” acceleration 1 Itrelates the two frameworks. “... for a given thermodynamic system,one typically obtains both a “physics process” and “equilibrium state”...” [W], p.148, we interprete as disclosing the “conversion” (dual-ity) between internal and external dynamics of DOFs (dynamics andthermodynamics).

Now in order to “explain” E = Mc2, in fact that energy IS massE = M in geometric units c = 1, h = 1, we need to establish a con-nection between the two terms P∆V and −Ω∆J (loc. cit. “The term“−Ω∆J” is closely analogous to the “work term” P∆V ...” p.148).

If we eliminate for the moment 2πT H = a∆H to get:

∆E + P∆V = ∆M − Ω∆J

and use E = M we obtain is a necessary condition:

P∆V + Ω∆J = 0.

In order to justify such a relation, we hint towards general framework:deformation theory.

Definition 1.1. The angular momentum of a DOF is

ω = ω1 + ω2.

2

Then we should have 3:

ω ∧ ω = 2ω1 ∧ ω2,

and ω a solution of the Maurer-Cartan equation 4:

dω + 1/2[ω, ω] = 0, (Dω = 0).

By differentiation one obtains (boundary has no boundary):

d(ω ∧ ω) = 0,

which translates into:

0 = d(ω1 ∧ ω2) = dω1 ∧ ω2 + ω1 ∧ dω2...PdV + ΩdJ (?).

1I.e. T = |∇V| in the context of the duality E/I DOF.2We think: ω1 = J, ω2 = Ω3Research is ... wishful thinking!4The Master Equation is always the key: it provides the duplex interpreta-

tion: deformation theory (perturbative approach/ resolutions etc.) and geometry(holonomy, curvature etc.).


This suggests a conformal setup: complex structure, CR-equations,T ∗M / spin structure, ω a curvature satisfying Bianchi identity: Dω=0etc. 5).

Returning to the “two levels” (internal and external), a general

vertex interaction Inf→ Out (In/Out: external; f : internal), is a role

separation “a la Newton” (F = ma or the 2-dim version of EinsteinT = κG) into a geometric framework (our external dynamics level ofEDOFs) and constituent framework (our internal dynamics - thermo-dynamics - of IDOFs).

At this point it would be better to use the graph invariant (and statefunction) TH rather then H itself, with a tacit assumption T∆H =∆(TH), in order to “allow” temperature to vary:

d(TH) = dE + PdV, d(aH) = dM − ΩdJ.

Dually we should have:

d(aH) = dM − ΩdJ.

That we are in Maxwell’s position 6 (or rather Yang-Mills) is probablytoo much to expect:

dω = 0, d∗ω = 0,

although the duality should be the main ingredient; a Hodge structure(relating H and H now independent complex variables ... twistor pro-gram?) would be nice to have. Indeed we do have information/interactionsources (and sinks) ...

On the other hand the separation between external and internalshould represent a choice/model: In/Out. So the master equationshould rather be thought of in the context of non-abelian cohomology:

Dω = 0 ∂+ω = ∂−ω,

with the two 1st laws rewritten as:

d+ω = δν, d−ω = βδ(aν),

where ν = TH should remind us about the number of configurationspicture (combinatorics) 7.

The dimension these forms leave in should be 4 since all good things(and exotic) happen in dimension 4 8 Again in the spirit of duality(Maxwell/Yang-Mills/Hodge etc.) a Hypersymmetry principle between

5Probably corresponding to the two levels of structure related via Wick rota-tion; see further on!

6... at Ostrogradsky’s examination: not enough time to finish ...7For the purpose of further staring at these pieces ...8Or 1 if working with quaternions.

3. 4D HYPERDYNAMICS 137

space and em time should be expected, besides the Supersymmetry be-tween particles and fields (vertices and edges). Indeed in causal struc-tures s and t-correlations are totally a matter of the way the process isdescribed/represented (information may flows in all “directions”). Inmore technical terms, Wick rotations should be generalized to spacialdirections also, so that Space and Time 9 are finally unified, appearingon an equal footing.

So not only the number of dimensions we like to use is irrelevant(beyond the use of the qubit as the building block), but also what istime-like or space-like is relative 10.

2. Information flow revisited

The classical discussion of Shannon’s entropy (Ch. 5) and inter-preted as a flow of information in the context of Noether’s theorem(information current and charge) are a toy model for the general case:pi = |zi| (Bohr’s rule) (or when a unitary IO-process is decomposedaccording to a spectral partition of unity, analog of

∑

i pi = 1 etc.).The toy model of the probability flow is the colored corolla (prob-

ability distribution π):∑

i pi = 1, H = −∆I, I(pi) = −∫ p

0ln zdz. The

information vector field w = ln z seems to afford the role of “time”(information “Killing” vector field?) and I the role of information po-tential. Then the total surface termH =

∫

∂tI =

∫

tdiv(I) plays the role

of information flux. Indeed, the BH entropy resides on the BH’s surface(BH − entropy = 1

4Area). The information charge is the probability

itself and∑

i pi = p is just the conservation of information equation(ν =

∑

νi is conservation of possible states under the partitioning ofthe micro state space) 11.

3. 4D Hyperdynamics

The internal/external canonical variables are P,Ω (1/2-forms) andtheir canonically conjugate variables (V, J); we will seek guidance fromclassical Hamiltonian/Lagrangian formalism by interpreting the workterm W = PdV as pdq (momentum p as the canonical conjugate toposition q).

Recall that in 4D p = (p,E) and q = (q, t) are conjugate and thePCF is pdq−Edt = pdq. So let’s introduce ω = (P,Ω) and ω∗ = (V, J).

9Once a coordinate system is chosen to describe the information flow is made.10The more symmetries the better!11Rephrase for the amplitude flow and related to Schrodinger’s equation in po-

lar coordinates: Bohmian mechanics interpretation and Hamilton-Jacobi formalism.


Also define ρ = (aH, TH) (or ρ = aH + iTH ?) the information 4-current. Then aH is the “spacial” info current and TH the “temporalinfo current” / info charge (?). We have the “PCF” ωdω = PdV −ΩdJ(recall PdV + ΩdJ = 0 ...).

Then the two 1st laws combine as:

ρ = dE + dM + ωdω.

Here E is not the relativistic energy, i.e. does not include mass. So Eand M are on one hand the two relativistic sides of total energy, whileon the “DWT-side” they reflect external and internal properties.

A complex formalism (or symplectic/Kahler etc.) should bind themas E = p + iM with iM Wick rotated time-like term, so that therelativistic equation would hold (c = 1):

|E|2 = E∗E = p2 +M2.

Then

E +M ≈ p2/(2M) +M ≈√

M2 + p2 = |E|,to get a complex equation unifying the two 1st laws:

ρ = dE + 1/2d(ω2)??.

Alternatively: dF = PdV − TdH since E −W = U − TS etc. and itsconjugate d∗F = ∗d ∗ F = aH + ΩdJ so that Maxwell/YM-equationswould hold:

dF = dE, d∗F = dM.

In this picture, temperature is conjugate to acceleration (“equal” as inUnruh’s Law), and mass is conjugate to energy (“equal” as in Einstein’sLaw).

The conjugation must be due to the duality between internal and ex-ternal DOFs, which in turn requires the duality between space and time(“explaining” the String Theory’s s-t duality [GSW]; more on the “Vi-rasoro picture” later on ...) which is achieved via the generalized Wickrotation (again info flow does not know about a “parallel-sequentialprojection plane”).

The mass is thus an entropy related property (information mo-mentum / info-gradient, i.e. “time-like” ?) and inertia is due to an(imaginary) “internal rotation” (quantum phase?).

A complex version of the above two equations would simplify thetheory:

DF = E, D = d+ id∗ ?

Indeed the complex or symplectic formalism is required by the dynam-ical qubit T ∗H (or T ∗(C ⊕ C not to include the quaternions at this

5. QG AS A DEFORMATION THEORY 139

point); then its symmetries Aut(qbit) = SU2 ⊕ SU2 are the conformaltransformations (Diff(S1) / Virasoro algebra).

Now the question is: “What is the corresponding 12 (Nobel price)Lagrangian (Hamiltonian)?”.

4. Space-Time duality and Hypersymmetry

That info flows in “all directions”, spacial or temporal (i.e. quan-tum computations may be parallel or sequential) suggests a new gaugesymmetry, a sort of generalized s/t-duality (see [GSW]). It allows for abetter analysis of the Bohmian mechanics which separates Schrodingerequation into two equations, when representing the wave function so-lution (i.e. the amplitude of probability) in “polar coordinates”. The

probability p = |ψ| yields I(|ψ|) =∫ |

0ψ| ln rdr the “real/spacial ampli-

tude” of the information flow and the phase lnψ−lnp“imaginary/temporal”component. The complex info potential I(ψ) =

∫

lnψ should be relatedwith von Newmann entropy, playing the role of a generating functionof the info momentum flow (?).

The proper language for a rigorous implementation is that of cat-egories with duality (left 6= right probably allowing for CP-violations,yet preserving CPT).

5. QG as a Deformation Theory

The semi-classical Einstein equation:

Gab = 8π < Tab >

is an “2-dimensional” analog of Newton’s equation maa = Fa, and isonly half of the “full picture”. As Einstein himself said it, the LHS isbeautiful (physics as geometry - Greek ideal), but the RHS is not (so).Indeed, comparing with an IO-process, a Newton type of law F = mais a label for a framework leading to a computational machinery (e.g.RHS ma within differential equations or G within semi-Riemanniangeometry), capable of processing an “input”, the LHS, e.g. a given“force” or energy-momentum etc. (which in turn can be obtained froma Lagrangian formulation etc.).

A “complete theory” should be a “cycle” where O = I∗ (maybethought of as Newton’s 1st law: action=-reaction) label the “cut cycle”:

12See Borcherds p.?


Newton’s theory including gravity (F = kmM/r2), as the only funda-mental force, is indeed a complete theory: the LHS for gravity as a“Coulombian force” is beautiful too!

A complete theory including Einstein’s GR equation should be ofthe form:

tr(I?) = 0 or D(ω?) = 0,

expressing a conservation of information (2nd generalized law?) as aresult of hypersymmetry with its EL-equations (rather some analogDS-equation).

The analog of “average curvature is due to matter” (G = T ) maybe thought of as tr(R) = T ; now what is the law yielding the energy-momentum tensor T ? (We will see that the external dynamics mustbe merged with internal dynamics, later on):

Tr(ω ∧ ω) = dω ... > Tr(TdT ∗) = 0...?

Let’s keep in mind that von Newman entropy is S = tr(U lnU), andexp tr ln = det, so the null average entropy equation is:

< S >= 0 ↔ det(J(O)) = 113.

For example, QFTs (statistical mechanics etc.) are theories steam-ing from such generating equations: partition functions as vacuum-to-vacuum process (amplitude):

< ∅ >= tr(I) (?)

As TQFTs etc. teach us, it is better to have the theory for allHom(I,O)then to sum up everything just “to reduce category theory (transitionsbetween many objects, including symmetries) to ring theory (endo-morphism ring of one huge object, and its symmetries)” (see Mitchell?etc.). That the “partition function” is not convergent (in the L1-norm- see §4), should not be a problem; the partition/generating functionshould be a graded object of finite type (see [K97, Ion03] etc.).

GR is in some sense a “deformation theory” in disguise:

G = T ... > ωdω = j,

with a curvature complex form ω tieing together internal and externalDOFs in duality (although the RHS equation is rather a conservationequation ...).

If allowing a background metric (Minkowski; for a fleeting momentonly), then we are looking at an equation providing a perturbation ofthe background metric etc. In a discrete framework the “background

13Relations with Jacobian conjecture, formal diffeomorphisms, “time” inversionetc.

5. QG AS A DEFORMATION THEORY 141

space” role is played by “vacancies” perhaps ... (see LQG setup /foams).

To include QFT, which is a deformation theory (Dyson-Schwingerequation, as an Euler-Lagrange quantum dynamical equation, is Maurer-Cartan equation in a suitable sense 14, in the DWT QDR frameworkthe analog of Einstein equation is:

D(ω) = 0 ↔ dω +1

2[ω, ω] = 0 G = T.

ωdω = j div(T ) = 0.

The “free theory + interactions” approach leading to the “pertur-bative” interpretation is implemented via Wick theorem as “graphs”(QDR) and Feynman rules (representation: propagators, coupling con-stants etc.).

5.1. String Theory as a Deformation Theory. To interpretST as a deformation theory, we comment on [H].

The creation and annihilation operators A satisfy Maurer-Cartanequation (p.136):

HA+A2 = 0

where H denotes the Hamiltonian (conformal invariance: H2 = 0p.139). It represents the equation of motion (see also [Marcus], p.64).The string field A is the perturbation of a particular string solutionΦ0 (analog of the “vector potential), yielding another string solution(analog of the connection):

Φ = Φ0 +A,

satisfying the Master Equation Φ2 = 0 provided that the Hamiltonianis an inner derivation:

H = adΦ0 , HA = Φ0 ∗A+A ∗ Φ0.

Here ∗ denotes a convolution (e.g. a convolution algebra Hom(C,A)of “Feynman type” - see [Ion03]).

It is remarkable that “half” of the Hamiltonian (left/right “factor-ization”) provides such a solution as Φ0 = HLI, where I is the convo-lution algebra identity, as in the context of torsion algebras [Ion99].

This last condition (H cohomological trivial) “can be viewed as thecondition for the solution Φ0 to correspond to a set of background fields

14Idea “confirmed” by experts.


on spacetime” loc. cit. p.138. Since H corresponds to the energy-matter current (with energy-momentum tensor its flux) and Φ0 the“background spacetime”, then the condition

H = adΦ0

may be thought of an algebraic analog of Newton’s LHS for gravity(constitutive equation): “String theory is a unified theory of gravity...” loc. cit. p.138.

It also plays the role of a (beautiful) LHS of Einstein’s equation.Then the algebraic analog of Einstein’s equation T ∼ G is a MC-equation:

HA = −A2.

In contrast, this is a “complete theory”.A formulation, based on the same formalism (deformation theory)

as string theory, but without a background metric nor an ambient man-ifold topology (see loc. cit. p.141) is the QDR approach of DWT. Therepresentation of a QDR with “coefficients in a certain manifold” willplay the role of a concrete model (e.g. generators and relations versusgroup representations), while the “cohomology class of the theory” willdepend only on a certain class of the manifold. What is essential, isthe “star algebra” (convolution algebra of Feynman type).

At this point we should mention that the symmetric formulation ofdeformation theory studies the perturbation ∂ = D1−D2 (here Φ−Φ0)as the “failure” to have a “double complex”:

D2i = 0 ⇔ ∂2 = [D1,D2] etc.

In conclusion, the Quantum Digital Gravity 6 initiative recorded bellowshould probably be formulated as a deformation theory ...

6. Quantum Digital Gravity

QDG consists in a few new ideas integrated in the common knowl-edge, starting from the motto: “a grain of space-time-matter is thedynamical qubit”. Also, Special Relativity is a “projective theory”,reminiscent of QM, so the possible interpretation I = Eeiθ is kept inmind:

“QFT := SR+QM ′′.

As a first objective we would like to merge the two 1st laws (§4) whichwe will rewrite as (entropy is now denoted by S):

dE = TdS − PdV, dM = adS + ΩdJ.

6. QUANTUM DIGITAL GRAVITY 143

Some additional “coincidences” will play a crucial role. To Einstein’sPrinciples (equivalence of energy and matter, and equivalence of grav-itational mass and inertial mass:

E2 − p2 = M2, M2 = M2g , (8)

we add and interprete Hawking’s Law and Area Theorem:

T 2 − a2 = 0, S2 −A2 = 0. (9)

We interprete them as referring to the two levels of structure, exter-nal and internal, and as in Maxwell’s case, since we think of gravityas an entropic effect, we postulate two charges: informational charge(pure inertial?) Ma and gravitational Mg (“due to entropy”; electriccharges: yes!), Now, the inertial (dynamical) charge/mass equals thegravitational charge

M2 = M2g

since there are no free pure inertial charges (no monopoles! informationcomes in qubits; if there is a “Yes” there is a “No” too (?)). This isrequired by the zeros in the RHSs above.

6.1. Energy-momentum tensors: external and internal. Nowwe assemble an energy-momentum tensor (EP-tensor) with a presenceat the two levels (I/E) (back trace to the action - eventually).

Definition 6.1. The “classical” external energy-momentum tensoris:

TE = (t, q|q, V ), (10)

and the internal energy-momentum tensor is:

TI = (S,A|A, J). (11)

where t, q, V and S,A, J are external and respectively internal variablesof a typical object Q of a Feynman category C (to be explained later).

The conjugate variables are E, p, P and respectively T, a,Ω, of thespace of symmetries of the object: Aut(Q).

The corresponding conjugate EP-tensor is:

T ∗E = (E, p|p, P ), T ∗

I = (T, a|a,Ω). (12)

We will try to “produce” gravity as defined by the gravitationalmomentum a and correlation tensor Ω (to be related with a “met-ric”). Since we model “spacetime” by graphs, we do not just have theEM-tensor of a point or body (including pressure etc.), but we alsohave edges representing interactions (“pressure”; “stress”?), and a cor-responding EM-tensor controlling the internal dynamics (temperatureetc.).


Now the First Law of Hyper-Dynamics, expressing the conservationof energy-matter-information is:

Tr(T ∗dT ) = dω, dω = dF + dM. (13)

Note that “gravity” will not be obtained as in Kaluza-Klein (ST etc.)using additional external dimensions (more room at the classical level)but it is implemented by “moving it” to a completely another level(“internal”), exploiting the I/E duality on top of s/t-duality. To im-plement gravity still as an exchange interaction, use the (postulated)I/E duality (hyper-symmetry) to allow the exchange of bosons not onlybetween “external particles” (and internal DOFs), but also inter-types(to be clarified later on):

int : •I

BBB

BBBB

B

time/phase?// • a

ext : e−γ // //

I

>>||||||||e− p

Note that the pair of 3D-momenta (p, a), representing the currents ofmatter and information, are reminiscent of Maxwell’s EM fields: E,B.Their unification as the curvature obeying Maxwell’s equations dF = 0(Bianchi identity) and d∗F = j will be attempted later on.

We think of T = (TE, TI) as a “bundle morphism” over the classicalEM-tensor:

O ×Aut(O)

T // O ×Aut(O)

O TE // O.

Since quite often “practice” outpaces “interpretation” 15 we willdevelop the conceptual formalism as if it would be an interpretation ofan “Abstract String Theory”.

The first crucial guess consists in interpreting the information chargeQ = −I yielding the entropy (derivation) S, as (related to?) the BRSTcharge and think of the ghosts as playing a similar role to the role playedby neutrinos: carrying the “missing” information (from “entropy in-crease” to a conservation law) and being responsible of generating mass(related to Higgs field?) 16.

Also the “unification between energy-mass and information shouldbe hinted by I = F + iM (E = (F,M)) where E = |I| = (p2 +M2)

12 .

15E.g. Lorentz and Einstein etc.16To be explored later; in many cases “practice” precedes interpretation

(Lorentz and Einstein etc.).


Recall that classical energy E = Z = |Z| (partition function) is relatedto probability p = Ei/Z and that entropy S = I(Out) − I(In) isan information flux, perhaps

∫

div(E) ...? Also, the thermodynamicequation for energy, with the distinction between internal and externalenergy, and work, is:

E − U = W − TS, F = E − U.

It is not clear at this point (nor essential) if the semi-Riemannian for-malism is used (metric/norm etc.) of the quaternionic formalism isused. We suspect that the quaternionic formalism is more powerfuland could provide the pair of above equations as an analog of Cauchy-Riemann equations (holomorphy in the quaternionic algebraic setup -C/H∗ − algebra?; skew field / supersymmetry?).

Returning to the Einstein-Hawking equivalences (“coincidences”),we interprete them as “light-cone gauge constraints” to be dealt withusing BRST or the “more geometric” (but equivalent) approach [H],p.130, which is deformation theory leading to Maurer-Cartan equationetc., as explained in §5.1):

“proper time′′ : |(E, p)| = M, gauge condition : |(T, a)| = 0,

“external′′ metric : |d(t, q)| = ds, gauge condition : |d(S, a)| = 0.

A real EM-tensor T : R4 → R4 (symmetric) has 10 components,quaternionic 1D-bundle morphism restrict the base transformation toa quaternion TE ∼= (E, p).

It is time to list the main ingredients (PCF forms) of the action,and ... entropy definitely is part of the action!

The Legendre transform relates the Hamiltonian H = E and La-grangian L ([]):

L = Ldt = PCF = pdq −Edt.

The other (curvature) 1-forms are:

dF = PDV − TdS, dM = ΩdJ + adA.

Recall that the “light-cone gauge allows to trade TdS and adA. Alsorelativistic energy unifies dF and dM .

The above forms satisfy a relation yielding a (preliminary) “Chern-Simons Lagrangian/Action” is:

L = dE − Tr(T ∗dT ).

The topological WZW-term Tr(T ∗dT ) would make possible the pres-ence of a “information monopole”, or may be interpreting mass as a“kink in spacetime” (instantons etc.).


As mentioned before, a Chern-Simons action yields (the truncated)Maurer-Cartan equation (vanishing curvature, i.e. local integrabilitycondition; e.g. [Marcus], p.64) as Euler-Lagrange equations, as re-quired by a “deformation theory” interpretation of the equations ofdynamics (Master equation Dω = 0 etc.):

dS = L = 0 ↔ dω + ω ∧ ω = 0.

In the geometric picture, the energy-momentum tensor corresponds toa connection ω (geometric quantization etc.); it is tempting to look fora correspondence H = E = F + iM < − > ω ...

The possible alternative incarnation asH2 = 0, related to conformalinvariance should be kept in mind (H(z)dz2 = 0?), i.e. “complex orsymplectic formalism”? (Mirror symmetry: they are equivalent):

Exercise: Does the corresponding (CS) action S built with theabove PCFs (pieces of the CS-Lagrangian 1-form) S =

∫

“Ldt′′ yieldthe above EM-tensors?

T =δSδg.

Recall also that dS ∼ dt ≥ 0, “time and entropy point” in the samedirection (information goes from the system to the observer: In →Out). The proportionality is reminiscent of Lagrange multipliers (Thecorrespondence between probabilities and energy maximizes entropy).

6.2. More clues ... 17 Conformal theories in two dimensions seemto be special. This is consistent with st-duality (generalized Wick rota-tion), and especially with the dynamical qubit as being the fundamentalbuilding block of the theory.

Once spacetime hypersymmetry is broken, a quaternionic approach“E = E + p ” (or rather Clifford algebra approach: E, p, t, q andT, a, S,A span 16 = 24 real dimensions!) seems to be appropriate,(and corresponding with the semi-Riemannian picture).

Indeed, there are too many dualities between the fundamental vari-ables:

External InternalMomentum

Current pE = (E, p) qI = (T, a)Internal

CoordinatesExternal

Coordinates qE = (t, q) pI = (S,A)Info

CurrentThe External-Internal duality should correspond to a bicomplex

structure (δ = d∗ etc.).

17This part is almost a “research journal” :-)


Recall that the (4D) energy-momentum tensor T is the flux of a(4D) current, expressing a certain duality between “spacetime” andinternal DOFs:

flux =

∫

∂M

T =< ∂M,T > .

The external momentum current (E, p) producing the momentum flowTE, are conjugate to the external coordinates (t, q), which seem to bedual to the entropy internal current (S,A) producing the internal flow(energy-momentum tensor TI) relative to the coordinates (T, a) dual to(E, p). Then a “Leibnitz rule” will involve 1-forms (total differentials)like:

d? = dtE + TdS, dqp+ adA.

The 4D-vector valued 1-form is (above notation):

dqE pE + qI dpI = d(qEpE + qIpI) = d(qp),

which is prone for interpretations:

q = (qI,qE), p = (pE,pI),

or better, in view of a Master Equation, should be compared with:

d||q||2, ||q||∗ = q · q∗, q∗ = p.

Recall that:

Tr(T ∗EdTE) = pdq − Edt+ PdV, Tr(T ∗

I dTI) = adA− TdS + ΩdJ.

It exhibits a “breaking of symmetry”, which, when restored, should be:

T ∗EdTE + T ∗

I dTI = dSE + dF + dM

= (−Edt+ pdq) + (−TdS + PdV ) + (adA+ ΩdJ)

= pdq + adA− (Edt+ TdS) + (dV P + Ω dJ).Here SE denotes the external action (free energy term?), dF or rather“dU” is the analog internal energy term (work term?) and dM appearsassociated with the internal DOFs, including an angular momentumterm.

The “correct order” when merging internal and external DOFs,appears to be:

dS = “LEdt+ LIdS′′ =

(−Edt+ pdq + PdV ) + (−TdS + adA+ ΩdJ) =

−Edt− TdS + pdq + adA+ PdV + ΩdJ18.

18Corresponds to the internal and external momentum and information cur-rents together with angular momentum tensors associated to the internal and ex-ternal symmetries: translational and rotational.


Now the graded 1-form (vector and tensor valued):

ω =1

2d||T ||2 = ReTr(T †T ) = T ∗

EdTE + T ∗I dTI = ω1 + ω2

19

is a nice candidate as a WZW-term for a Chern-Simons action, whichmay generate interesting physics under “dimensional reduction” ([GIJP];see later on).

Note the satisfied “light-cone gauge” constraints:

T = ||a||, S = ||A||, ||(E, p)|| = m = M0

⇒ TdS = adA, etc.

as well as Lagrangian submanifolds constraints:

p = Eq, ⇔ pdt− Edq = 0.

The “internal analog” is:

a = TdA

dS, ⇔ adS − TdA = 0,

which is satisfied under the “light-cone gauge” conditions from above.6.2.1. An other possibility ... On the other hand the two original

1-forms dF and dM suggest a complex setup (towards a CR-equations,Calabi-Yau etc.):

Re : dF = PdV − TdS, Im : dM = adA+ ΩdJ...?

In view of the duality between I and E DOFs, there should be a “La-grangian submanifold” constraint providing a basis for the new princi-ple of equivalence between energy and information:

Edt+ kTdS = 0, (14)

where k denoted Boltzmann’s constant, having the meaning that the“energy per interaction of a DOF” Et/kT (quantum jump”) is equiv-alent to information transfer (flow) 20.

6.3. Quantizing I/E DOFs. Speaking of quantization, the “in-ternal DOFs” analog of the Dirac quantization based on a Poissonbracket a,A = 1 (analog of q, p = 1) should be a = 1

i∂∂A

. Thederivation of the Schrodinger equation in the “combined picture” (I/E)should be compared with the Bohmian interpretation in terms of a po-lar representation of the wave function (later on).

19Tr(T †T ) should be real somehow ...20“Quantization of energy/information”.


6.3.1. The string field and entropy. A parallel between String The-ory and Entropy field (entropy as the flux of the information currentetc.) might be rewarding. To start with, consider a RS as the ana-log of the corolla: the system’s I/E DOFs “branch” (one string intomany, or splitting internal states into a partition etc.); then follow the“quantization of a string” [H], p.130.

6.4. Currents in String Theory. The basic ideas of QFT (andalike theories) can be expresses in different languages, as in manyother cases, for example in geometric language (geometric quantization,vector potentials as connections etc.) or algebraic language (VOAs,TQFTs, A/L-infinity algebras/categories etc.). A bridge is alwaysthere: Yoneda Lemma which allows to interpret (visualize) a repre-sentable functor as an irreducible connection 21.

The geometric picture (physicists’ gauge theory) is wide spread; toprovide a few links with deformation theory, we comment on [AW].

The article starts from the (natural) interpretation of the partitionfunction as a norm (as noticed earlier §4), except in the context of gaugetheory the finite index set must be replaced by a line bundle (etc.).Then the energy-momentum tensor (EP-tensor) is the flat connectionproviding the parallel transport implemented by the partition functionas a section. As expected, the flatness condition (local integrability) isexpressed by Maurer-Cartan equation.

The flatness condition is proposed as a fundamental dynamicalequation (p.77); this is natural, since MC-equation appears as EL-equation of a CS-action (see §5.1):

CS −Action

FPIuujjjjjjjjjjjjjjjV ariation

))RRRRRRRRRRRRRR

Partition function ↔ EP − Tensor.

The result is established by deriving the partition function from (acollection of) EP-tensors. The flux of the generalized current j(A)associated with the background fields (p.79):

ω =

∫

X

j[A]dA

satisfies MC-equation (deformation equation):

Ω = dω + ω ∧ ω = 0 ↔ D2 = 0,

21Grad school observation.


where D = d+ ω, which is the dynamical equation of a CS-action (see§5.1).

Indeed, if Z[A] = Z0e−ωA is the solution of dZ = ωZ (p.78), then

differentiating one obtains the MC-equation for ω.The relation between the “standard definition of the energy mo-

mentum tensor density” (p.81):

d log Z[g] =1

2

∫

T abdgab

and the entropy flux will be considered elsewhere. In the context of[AW] (p.81), the (complex) connection is related to the partition func-tion by:

ω = −∂Z, ω = −∂ logZ,

where d = ∂ + ∂ splits.As mentioned before, if d is rather thought of as the perturbation

(i.e. satisfying MC-equation) in the symmetric deformation theory pic-ture, then

d2 = ∂∂ + ∂∂ = [d, ∂] = add(∂),

i.e. we have a double complex modulo homotopy.The condition of being a complex corresponds to d2 = 0, and in the

context of [AW], is equivalent to:

∂ω + ∂ω = 0 (dω = 0)

i.e. with the assumption that ω∧ω = 0 (ω is related with Z as above).As an example consider Quillen’s choice (p.87) given by:

ω0 = Tr(D−1∂D).

The correspondence with the algebraic picture is via the interpretationof the Chern class of the line bundle as the central charge of the Virasoroalgebra (central extension etc.). ¿From the physical point of view, thetopological WZW-term implements “topological particles” (monopolesetc.).

6.5. Gravity and electromagnetism: the revival of an oldstory! It seems that gravity and electromagnetism can be unified af-ter all (Einstein, Kaluza-Klein etc. were almost right), except theexternal space must be “enlarged” to include internal space too! Inother words, DWT with its duality between external and internal space(DOFs; QDR etc.) may be thought of as a “super” Kaluza-Klein theory(or rather “hyper” ST, since “spacetime” is replaced by the FeynmanCausal structure - paths/transitions etc.). External dynamics and In-ternal dynamics (an upgrade of the Standard Model to include entropic


effect, duality etc.) seem to lead to a pure gauge theory 22 with a Chern-Simons action on the “double” of traditional space (E, p, S,A etc.). Areduction of the CS-action to external space, in the traditional setup(e.g. 3+1 to 2+1 dimensions [GIJP]) is known to afford solutionswhich break the symmetry and produce topological kinks [GIJP].

It would be interesting to try to reduce gravitational Chern-Simonsterms in the DWT framework (with energy-entropy duality (E,S), orrather momentum-entropy 4-current-4-current interactions) to see ifgravity is produced, incorporating EM and GR. Then accounting forthe other interactions (SM and GR) should be ... just hard work :-)

Indeed, as mentioned before, there is an alternative regarding theinterpretation of the entropy flux: 1) either a 4-current (S,A) withconjugate 4-current (T, a) or 2) S = |A| and T = |a| with a entropy-temperature field F = Alt(A, a).

6.5.1. The Quantum Temperature-Entropy Field Theory. In thisMaxwell like interpretation (pure gauge theory with CS-action), thetemperature T and the gravitational acceleration a could be the ana-log of the electric field (and energy), while the entropy S and “area” A,or rather the entropy current (divergence of probability amplitude ina corolla etc.) is thought of as the analog of the magnetic field (somekind of curvature, therefore suited to model gravity 23).

The entropy tensor as the flux of the information 4-current, prob-ably corresponds to Wick hypersymmetry as an enlarged local gaugesymmetry, dual to Weyl symmetry:

Hom(Ext× Int,C).

“Ext” represents EDOFs (say a manifoldM in ST or a Feynman causalstructure as a QDR) while “Int” refers to IDOFs (gauge theory prin-ciple bundle or Feynman-Kontsevich rules etc.).

The external local gauge symmetries (translation and rotation in-variance) of a Lagrangian determine the EP-tensor and angular mo-mentum tensor. A U(1) local symmetry on the target space C yieldsEM (current, tensor (E,B) etc.). By duality the complex phase maybe traded between internal and external spaces, allowing to couple EMand entropy; is the resulting theory gravity? But space is 3D!?

Due to the I/E-duality, maybe SU(2) “is” the 3D-space (i.e. thehomogeneous space of SO(3)) when viewed as external, and the localgauge symmetry of external space leads to particle dynamics (momenta:

22Are massless two-spinors [Horvathy] ...“infons” (quanta of information)?23Speculation level “argument” ...


(E, p) and angular)

“Hom(graphs ⊗ SU(2),C) ∼= Hom(graphs, SU(2)).

As internal, it enables the U(1)-gauge symmetry (gravity/entropy as apure U(1)-gauge topological theory).

Looking around, one finds such attempts to modify Einstein equa-tions by including CS-actions [JP, J](WZW-topological terms), butagain “a la” Kaluza-Klein. It would be instructive to try to “trans-mute” these attempts in the context of I/E-duality (hyper-spacetime-energy symmetry, including s/t-duality).

The formula yielding the “total entropy” change (per “unit of time”):

“d/dtH ′′ : ∆S = −I(t+) + I(t−), I(p) =

∫ p

0

ln zdz.

should probably be thought of as “Gauss Theorem”, including a unitinformation charge I(t−) (source term) besides the information fluxterm I(t+).

6.5.2. Is the merger of EM and GM possible? So, u = (E, p) is the4-momentum current and (a,A) may be interpreted as an analog ofthe electromagnetic field (E,B), with entropy S = |A| associated tothe information potential and T = |a| as a “curvature” (magnetic field:Hodge dual of EM-tensor/curvature F [R], p.18). The energy densityof EM is

1/2(E2 +B2) =1

4F2 −F2

0 (∼ g +Ric)

tr(F ∧ F =1

2F2 = B2 −E2.

The analog expression of T and S is not “OK”; may be T and Srepresent rotated coordinates:

TS =1√2(e+ ib)

1√2(e− ib) ?

The electric field is the momentum conjugate the vector potential A,and the scalar potential A0 appears as a Lagrange multiplier, like β (orkT ; Hmmm ...).

As a spin 1 bosonic U(1) gauge theory with fixed spacetime M , onegets electromagnetism (coupled with a particle with charge). But quan-tum information (amplitude of probability) couples with everything ...Gravity, as an entropic effect, could mean information as a conformalstructure with action determined by von Newman entropy (see §6.7.1)with a metric is a gauge field. The electrons and ions plasma confine-ment CS-model of loc. cit. p.27 might be start up point (a wild guess:


model for the particles and vacancies of the QDR; it supports “stableknot like solitons” loc. cit. p.33).

Now allow the topology of M to vary (graphs, RS/Galois coversetc.) together with the complex structure. If the “target C is fixed”,the entropy should be invariant under local “conformal gauge”, as wellas “topological gauge” transformations (H(π) or quantum informationpotential S(RS), including a WZW-term), possibly introducing massas a topological kink (central charge?).

Now the metric defines a conformal class (complex structure / RS),but if SU(2) “is” space (pointwise, or rather vertex-wise), then no met-ric is necessary. It is rather misleading (mechanicist point of view) andthe key structures are the (dynamical) qubit and its symmetries SU(2)together with the causal network (transmute Calabi-Yau formalism).

6.5.3. Variation of the action: manifold too! The duality in space-time (Stokes Th.) plays a crucial role, especially in connection with di-mensional reduction of WZW-terms [GIJP] (j∗dG may count in lowerdimensions; dG exact term in the Lagrangian). What new phenomenadoes I/E-duality bring? If action is a pairing (integral) but spacetimeis not fixed, not even topologically (NOP/subsystems change due to achange in the scale in the QDR), then a variation of the action (underfixed boundary conditions) involves internal and external terms:

δS = δ < M,L >=< dintM,L > + < M, δL > .

In the Feynman process framework the role of “spacetime” M is takenby a colored graph Γ. It is worth investigating the non-classical termcorresponding to the variation of the path itself as a quantum correction(e.g. Jacobi fields in semi-Riemannian geometry [O’Neill] etc.). 24.

An exact term, under duality, is not just a boundary term:

< M, dG >=< dintM + ∂M,G >=

∫

dintM

+

∫

∂M

G.

Exploring the current literature [JP], maybe EM and GR can be uni-fied, not as Einstein tried as a “pure” Kaluza-Klein theory, but in thecontext of DWT with its new principle of equivalence between internaland external DOFs (hyper-space-time symmetry).

6.6. Bohmian mechanics: a “hidden” message. Broadly speak-ing, information (∼ energy!) can be created and destroyed, and theprobability (↔ energy) is just the “modulus” of quantum information

24A categorical interface to Field Theory as Feynman processes is overdue bynow :-)


(the wave function!):

ψ = Ee−iS.

The 4-momentum flow of the wave function (classical energy and mo-mentum) represents half of the picture; the other half should probablybe the 4-entropy flow, after a closer inspection of the Schrodinger’sequation written in polar coordinates (Bohmian mechanics [Gosson],p.29, [HCM], p.15). Then the quantum potential Q is conceivably anentropic term: TS.

Recall that the real part of the Schrodinger equation is:

∂S

dt+

1

2m(∇S)2 + V +Q = 0

with an associated conservation energy:

E =1

2mp2 + V +Q,

under the correspondence:

E = −∂S∂t, p = ∇S.

The quantum potential is given by ([HCM], p.15):

Q = − 1

2m

∆R

R.

The imaginary part is

∂P

∂t+ ∇ · (Pv) = 0,

where R2 = P is identified as the probability.This last equation is a conservation equation, with P the probability

density and Pv the probability current:

div(P,Pv) = 0,

i.e. there are no “probability sources” 25. The relation between thequantum potential and our probability current ln p is under investiga-tion:

∆(lnp) =∆p

p− (∇p)2

|p|2 .

We should recall that the ‘’correct” state function is TS, not S: Ψ =Ee−iTS.

That energy is a “modulus” is suggested by the Boltzmann cor-respondence with the probability picture, as well as the relativistic

25No “floating daemons” to measure/ask questions :-)


version (E2 = p2+m2). In order to implement the generalized Wick ro-tation (imaginary time, s/t-duality, euclidian-statistics mechanics ver-sions etc.) perhaps one has to complexify the action (energy and en-tropy as modulus and phase) as a meromorphic function (on “Calabi-Yau manifolds”?) such that the Cauchy-Riemann equations would playan essential role.

6.7. Heat transfer and entropy flux. Conservation of internalenergy gives the classical heat equation:

d

dt[

∫

edV ] = heat flux+ heat generated(sources).

Heat/temperature are the 4th component of the information 4-current(the “density of information”). It is not clear if the I/O-flux through acorolla should be understood as a In/Out flux or as consisting of twoterms: an information charge and an outward information flux:

∆H = −Q(t+) +Q(t−) = flux + produced entropy ?

On the other hand conservation of probability:

p =

n∑

i=1

pi

may be viewed as a conservation of informational charge, towards a“string theory” picture of the above equation: the corolla represents“Faraday’s lines of forces” on a punctured Riemann sphere with onepositive charge and n negative charges (in equilibrium), and Shannonentropy is the information potential of the configuration.

As another (possibly helpful) analogy, consider a particle (system)breaking up into n particles (subsystems) (“mass” m is really our pre-vious ν):

m =n

∑

1

mi.

The non-relativistic Lagrangians (internal and external) are (could be):

LE = mq2/2 − 0, LI = 0 − T, dS = mv2/2dt − TdS,

i.e. no external potential energy and heat has no kinetic term.Then the action is:

S =

∫

mv2/2 −∫

TdS = kinetic energy + entropicpotential26.

26Or rather lnΨ in view of Bohmian mechanics ...?


To relate with Shannon’s entropy, further represent entropy as asurface term:

Q(p) =

∫ p

0

ln rdr =

∫ p

0

∫ r

1

dA

|A|and think that normalized probability p runs from [0, 2π] and r is a “po-lar radius”, while W (A) = 1

Ais an information potential. Then Shan-

non’s entropy should be expressed in terms of this potential (analogywith gravitational potential):

Q(pi) =

∫ ∫

Σi

dA

|A|

∆H =

∫

S2

W (|A|)dA, W (|A|) =1

|A|.

The above interpretation may be of use in connection with the ABE-model for the measurement problem. The conservation of energy andinformation:

d/dt(total) = I/O − flux+ I/O − sources

involves a deterministic and unitary Schrodinger flow of amplitude ofprobability, as carrier of energy and information/entropy, as well assource terms (sinks: info “leaking” out of the A → B quantum inter-action/communication) due to the observer’s change in entropy (Eve),and related to the “collapse of the wave function” 27.

The Hamilton-Jacobi formalism of the above internal dynamics willbe considered later on. In a sense Fourier’s law φ = −∇U for the heattransfer implies that “heat/entropy has no inertia” (U ∼ E ∼ kT ),which should be related with the equivalence between the gravitationaland inertial mass: mGR = Mext + Mint = Mext. The internal analogof Fourier law should relate “acceleration” (info current) and entropypotential a = −nablaW (|A|).

6.7.1. Is it you, String Theory? The equipartition probability dis-tribution, as a colored corolla, is reminiscent of the Galois cover f(z) =zn. It is conceivable that a potential theory analysis (ln z =

∫

1r,

H =∫

ln z etc.) might reveal entropy as the flux of the informationcurrent produced by a “Coulombian” quantum information potentialwith standard Green function

G(A→ B) = 1/r, ∆G = δ

and dynamics (information flow) governed by Schrodinger diffusionequation (or rather relativistic version). The metric is a convenient

27The other n − 1 possible outcomes branch into ... other parallel universes!?


“devise”, and the above equation could be read the other way around:

“|A−B|′′ = G(A→ B)−1.

The primary concept (besides Feynman paths, the category) wouldbe transition amplitude “table” (in an axiomatic approach: the “S-matrix”; see also §6.5.2).

Information theory based on the combinatorics of labeled configu-rations is in fact Galois theory (extensions/bundles and their symme-tries). This sheds light on the role of (pointed) Riemann surfaces forString Theory. After all the WZW-term of the CS-action is a windingnumber ([JNP], p.1):

CS(A) ∼∫

tr(A)3 = W (g), A = g−1g (FP − ghost)

detecting the “jump distance” between the topological branches 28. Thefundamental structure is the bifield with its antipodal map. On one side∫

r = r2/2 “yields” the kinetic term (and its inverse, the propagatoretc.), while on the other side

∫

1/r = ln r yields a current etc.As an “inverse scattering problem”, the process modeled by f(z) =

zn has a partition function as a section (see §6.4). Measuring the wavefunction Ψ (also a section), “localizes” the branch, collapsing the wavefunction.

The more general problem leads to meromorphic functions, Ga-lois covers, local systems and flat connections, Fuchsian systems, KZ-equations etc.

An analysis of the potential theory of the amplitudes of probabilitywill be started in the context of the Bohmian interpretation of QM.

6.8. The Bohmian mechanics interpretation. The Bohr’s rule:

probability = |amplitude|2

becomes clear as a requirement that the L1-norm of a probability distri-bution (partition function in statistics formalism) equals the L2-norm ofthe complex formulation in QM. Probabilities can only decrease “withtime” (succession of cause-effect), yet the quantum reality shows thatinformation can be not only destroyed, but also created (interference:destructive but also constructive!).

So, the wave function Ψ = ReiS (density of amplitude of proba-bility), thought of as the quantum information potential, satisfies theSchrodinger equation and the probability density (distribution) associ-ated to it is P = R2. Assume Ψ is a superposition of basic measurementstates corresponding to some Galois cover of a meromorphic function

28The role of the Faddeev-Popov ghost will be investigated later on.


f . Using a metric is a classical way to specify the potential of to eventsto be correlated (interact) in some way, but ultimately the conformalstructure is the primary structure (more than topology yet less thangeometry: information theory :-).

6.8.1. The quantum dice. ‘Up or down” is the basic question in thequantum world. God plays quantum dice after all, whether we like itor not. The quantum dice is the “unit quantum configuration space”,which is the projective space of the qubit C⊕C: the Riemann sphere(it has no corners, faces etc.; it is the perfect shape, as the Greekswould agree).

A coordinate system amounts to an axis; then “Up or Down” is thebasic quantum question (Shannon/von-Newman sense).

Conservation of the number of configurations (or energy etc.) un-der partitioning ν =

∑

i νi (actual space) corresponds to the normalitycondition for classical probabilities

∑

i pi = 1. This is analysis (L1-norm picture etc.). In the corresponding geometric picture of ampli-tudes, the conservation equation is Pythagoras’ theorem. More pre-cisely, The possible transitions (paths/histories) between two “effec-tive” state spaces Hom(A,B) are orthogonal (measurement basis), andthe lengths/weights are the amplitudes. The description is a diagonalrepresentation relative to an observable/observer.

The “actual size model” is probably the pointed RS picture deco-rated with vertex operators. Then a String Theory with a backgroundindependent (Shannon-von Newman) entropy CS-action (WZW-term)could be the answer of a massless bosonic pure gauge field: gravity.The Kontsevich graphs (and homology) [K92] in the light of the Feyn-man graphs cohomology interpretation of Ionescu [Ion03, Ion04-1,Ion04-2] will be called the Kontsevich-Ionescu QDR. In the context ofthe appropriate pairing (Hochschild DGLA or A∞-algebra etc.) can berealized as a double complex, and quasi isomorphic as total complexwith the RS-complex (relative some differential; moduli space as thecohomology).

So RS are essentially configurations of quantum information (“quan-tum questions”:interactions/communications). Knowing the “real” prob-abilities amounts to knowing the factorization Pi = ZiZi, where Zi be-longs to some quantum dice (Hopf filed is better then “bifield”). Still,


why can’t we “see” the phase? If it’s just a mathematical model to im-plement creation of quantum information (and destruction; everythingis periodic at some scale ... 29), what is its meaning? 30

RS exhibit a rigidity which comes together with the maximum prin-ciple for analytic functions (observables), which are determined by theirvalues on the boundary. This maybe thought of as the perfect encoding-decoding scheme for quantum communications (interactions). In somesense a complex structure is such a scheme, and the entropy of a RSmeasures the quantum information capacity of the quantum interactionchannel (see [NC]). As mentioned before, a background independentaction is therefore the entropy of a RS which, if there is any justice atall, should be Polyakov’s action: the area of the world sheet under anembedding. And this is the “hidden massage” of the black holes AreaTheorem, that entropy “is” area. Our approach shows that I-entropy31 is the flux of the information current. It should be related with thevon-Newman entropy (later on: f ′(z), Jacobian etc.)

6.8.2. Entropy and String action. Probability/entropy reflects a par-titioning of the internal space and time reflects a partitioning of theexternal space 32. We used to think that these two “realms” are totallydistinct; but eventually it is a matter of scale (resolution: zoom in andout).

I/E-duality goes beyond the generalized Wick rotation and tradesI and E DOFs; at the level of QI this is “just” a complex structure.A quantum interaction/communication is modeled by a RS and thecomplex structure is a shadow of the I/E-duality.

Since we do not believe in coincidences, we state the following con-jecture.

Theorem 6.1. The background independent String Action is:

S = Entropy(Riemann Surface) (Quantum Information Capacity).

29The “life cycle”; “All universe is in an atom” and life, i.e. quantum informa-tion, starts with a roll of dice :-)

30In some sense (periodicity: “birth and death”), the quantum phase is a properquantum time (proper time is a trademark of relativity :-) of the local quantumprocess. Each qubit/q-dice is an “elementary” Q-processor with its own “clock”;network enough of them, and “ta-ta”: The Universe!

31It is not Shannon and not automatically von Newman; of course, it’s Ionescu’s:-)

32We have a “superspace situation” here: external, “visible” space, is associ-ated with our classical understanding of reality, and internal, “hidden” space, isassociated with our quantum description of reality.


The Euler-Lagrange (Maurer-Cartan) equations maximize entropy, i.e.the Quantum Information Capacity of the Quantum Channel (interac-tion/communication).

In some sense it is the “perfect action” Nature could thing off: maxi-mizing the interaction / communication efficiency. The duality betweenI/E includes both traditional maximization principles: minimum actionprinciple in mechanics and maximum entropy in thermodynamics.

Glancing at [Gosson] p.30:

HΨ = H +QΨ

confirms that the action (as wished for before) should be of the form

S = Lextdt+ Lintds.

The corresponding Hamiltonian should allow a “merger” of the heatequation with Schrodinger’s equation (relativistic form), i. e. mergingdynamics and thermodynamics first, and only then quantize the theory.

A corollary regarding the measurement paradox is that althoughinformation “diffuses” internally and externally, the information flowis conserved:

∆Total QI = flux through RS + charges at ∂RS

and deterministic “in between measurements” (rigidity: I/O-boundarydetermined). The measurement produces a “rotation” from external tointernal space.

The simplest idea for implementing the I/E-duality is to mergeenergies (Hamiltonians), i.e. the external and internal:

H = E + iU

Looking at [HCM] p.15 (the use of polar coordinates) suggests that Sas a flux, is a current on the RS, say the complex plane; the “nodes”[Gosson], p.30, are the poles of information charges (punctured RS),with an associated Green function (the Hamiltonian?). If there is justa pole at 0, locally:

G(0, z) = lnΨ = ln |z − 0| + iSloc

while globally dG = d|z|/|z| + iα defines a closed “angle form” α.So the Hamiltonian dynamics equation of our complex Hamiltonian

is analyticity, the wave function is the quantum potential (the internaldynamics has no kinetic term: zero mass field), and the real part (clas-sical Hamiltonian equations) are Cauchy-Riemann equations (harmonicfunction with sources).


Externally, time corresponds to the “real” Hamiltonian flow: exter-nal energy. Internally, we have the entropy / quantum phase and inter-nal energy/temperature. The entropy is an “imaginary time-direction”corresponding to the imaginary Hamiltonian: internal energy, whichcomes from a quantum potential only (no kinetic part). An importantdifference between gauge theory and DWT, is that in gauge theorythe distinction external space - internal space is definitive (althoughallowing twisting) as a principle bundle. The internal symmetries andexternal symmetries (via the homogeneous space) are separated (Nogo theorem: cannot “merge” Poincare and gauge groups). In DWT,with its QDR with duality approach, the hyper-symmetry between I/E-spaces allow for a “rotation” between external motion (time) and inter-nal motion (entropy/phase). In this way the Wick rotation, imaginarytemperature, s/t-duality etc. “tricks” become mandatory for under-standing reality.

Reality is not, and will never be what it seems, since our under-standing has its own ... dynamics!

6.8.3. Information current revisited II. Whether we work in a “Hamilton-Jacobi gauge” (p and E conjugates: p = ∇E) or “Lagrangian gauge”(p and q conjugates: p = mq) should be irrelevant, due to the algebraic-geometric correspondence: Space = Spec(Observables) (states-observablesduality; Pontrjagin duality etc.):

E pt q

Then we “amend” the internal/external correspondence stated ear-lier having in mind an “analytic Hamiltonian”:

H : H → C33.

Internal energy U ∼ T (temperature) is I/E-dual to E and a = Mvan internal momentum is I/E-dual to p. Together j = (U, V ) is the2D-entropy current (I/E-dual to 4D-momentum; is dH a pairing!?).The corresponding coordinates are (τ, r), i.e. the proper local time (thequantum phase) and (essentially) the probability density. U, v should beanother pair of conformal coordinates, so why not identifying (anothergauge ...) U ∼ T ∼ τ (related with time via Wick rotation τ = it?):z = reiτ ∼ (U, V ). Recall that “the ST practice” is to use independentcomplex variables z and z, which reduce on shell: z = x + iy thenz = x − iy (PCF form reduces to Lagrangian). The Shannon entropyH(P ) as the flux of probability

33WZW-like: Tr(H−1dH = Trd lnH ... is von Newman entropy?


In the HJ-gauge a = ∇T (heat / internal energy current) and Uand V are harmonic conjugates.

Also, if energy is complex H = E + iT (imaginary temperature ∼internal energy U), we should have corresponding complex coordinates:the complex time is z = t + is, where s is entropy (related); a polar-ization should restore our conviction that “time” and “entropy” havesimilar arrows (TdS corresponds to Edt).

6.8.4. Again, why is energy mass? Energy is mass and TS is aninternal energy. Is there an obvious relation?

E2 −M20 = p2 ↔ (E −M0)(E +M0) = p2

Are mass and entropy directly related M0 = iT (or M0 = iU)?

Edt+ Tds = pdq ?

What about mass, as a coupling constant? Does mass couple externaland internal energy?

H = He + iMHi?

What “natural” maps from quaternions z, z to the complex field arethere? (Use Edt− pdq and Tds ...):

H(z, z)dz = ...?

In other words, what is the “obvious” complex Hamiltonian flow?We should look at what a Calabi-Yau manifold is, since if a polariza-

tion (Lagrange multiplier/manifold etc.) relates time t and entropy sand harmonicity relates the real and complex components of the Hamil-tonian, then, out of 8 dimensions only 6 are left; is it a coincidence?(Ditto!)

Regarding the number of dimensions of external space (which isessentially irrelevant, as it is by now acknowledged), once the mani-fold approach is abandoned (framework of the QDR) mirror symmetryshould appear as an I/E-duality (the algebraic Kontsevich’s version).

I quick glance at [M] revels two things: the Hodge diamond, rem-iniscent of the dualities from §6.2, and that they are too complicated(as a theory of manifolds)!

6.8.5. Bohmian mechanics’ Schrodinger equation. Since (x, y) and(r, φ) are coordinates conformal related, we will interprete first Schrodinger’sequation (~ = 1):

i∂Ψ

∂t= −1/2m∆Ψ + (u+ iv)Ψ

with a complex potential U = u+ iv as a CR-equation, assuming thatthe real and imaginary parts of Ψ = X+iY satisfy a system of coupledheat equations.


Recall that the external coordinates are (t, q) with q = (x, y, z) andthe internal coordinates are (s, a) 34 (previously denoted as S and A,with conjugate variables (T, a), since the PCF’s are Edt − pdq andTdS − adA; or should it be (S, T ) since S = A and T = a ...?).

Then, taking the real and imaginary parts and assuming heat equa-tion with coupled sources are satisfied:

∂X

∂s= (1/2m∇− u)X + vY ⇒ ∂Y

∂t=∂X

∂s,

∂Y

∂s= (1/2m∇− u)Y − vX ⇒ ∂Y

∂s= −∂X

∂tso CR-equations are satisfied. The need for more independent coordi-nates (reflecting the extention of observables from external to externaland internal: “Space = Spec(Algebra of Observables)”) is clear: (t, q)(H) and (s, a) (internal Hom dual to C; Hodge diamond and PCFto Lagrange reduction etc.) are consistent with a generalized Wickrotation (a more symmetric notation is used):

Ψ(t, t, q, q) ∈ C...?

Yet to understand the significance of the Schrodinger equation in thespirit of the Bohmian mechanics, we must revert to polar coordinates(R,φ) (S ↔ φ and A ↔ R), i.e. Ψ = Reiφ, where the probability isP = R2 and the local proper time (entropy/ “information arrow”) isφ.

We will try to rewrite the imaginary and real parts involving theGreen function (potential; see [Conway], p.275):

lnΨ = lnR + iφ.

A Green function is associated to a RS and a one distinguished point(see Dirichlet problem etc.). To suggest the connection with the en-tropy flow and entropy determined by the boundary values, we will callsuch a (genus 0) RS (Dirichlet region) an RS-corolla.

The real part of Schrodinger’s equation for Ψ(R,Φ) (polar coordi-nates) is (equivalent to [Gosson], p.29; [HCM], p.15):

∂φ

∂t+ 1/2m(∇φ)2 + U − 1/2m (∆R)/R = 0. (15)

Substituting lnΨ and using:

∆ lnR = ∆R/R − (∇R)2/R2, (∇ lnR)2 = (∇R)2/R2

and therefore:∆R/R = ∆lnR + (∇ lnR)2,

346 dimensions!


we obtain

∂φ

∂t− 1/2m ∆lnR + 1/2m ((∇Φ)2 − (∇ lnR)2) = −U.

(16)

Since

(∇φ)2 − (∇R)2 = Re(lnΨ)2

we obtain:

∂φ

∂t− 1/2m(∆ lnR) + 1/2m Re(∇ lnΨ)2 + U = 0 (17)

Introducing the complex operator (parabolic case/ non-relativistic):

HC = 1/2m ∆ + i∂t,

then the equation is:

Re[HC lnΨ + 1/2m (∇ lnΨ)2] = U.

(probably) yields a Klein-Gordon equation:

lnΨ +m(D lnΨ)2 = U (18)

for the (infinitesimal) current A = lnΦ corresponding to the infor-mation flux (in view of our flux interpretation of entropy Q(p) =∫ p

0ln zdz).Since we must have = D2 (at least at the level of a corresponding

Dirac equation), then the above “Klein-Gordon equation” 18 is “just”a Master Equation with Matter-Information Sources 35 (U = u+ iv toinclude internal and external sources: information and energy-matter).

Since the above “issue” is important, namely trying to interpreteSchrodinger’s equation from the physical point of view we need to revertto the alternative conformal coordinates having a physical meaning(internal entropy and proper time), a “fresh start” might help.

6.8.6. Interpreting the new “Klein-Gordon equation”. Although wewill start with Schrodinger’s equation, we aim at a KG-equation (andultimately a Dirac/ spinorial version, perhaps with a “stop” at a quater-nionic form).

In contrast with the Bohmian approach which focuses on probability(half of the “visible” picture), we focus on the quantum informationcarries by the wave function, and are looking for a Green function(2-point function and the corresponding equation):

G(z, z0) = ln |z − z0| + g(z0, z)

35Non-homogeneous Maurer-Cartan equation is an algebraic substitute for theEinstein’s equation “Average Curvature=Energy-Matter Tensor”.


When z0 = 0, then our candidate (the infinitesimal generator of thewave function) is (the 2-point function):

G(z) = lnΨ = ln |z| + iΦ = lnR + iΦ.

Therefore the real part of the Schrodinger equation 15 in terms of theinfinitesimal generator G(z) is Equation 17.

In view of a “Maxwellian interpretation” of entropy/gravity, in-stead of the Bohmian Hamiltonian H +QΨ [Gosson], .p 30, let us trythe Hamiltonian which besides a scalar potential U includes a vectorpotential A (loc. cit. p.17):

H = 1/2m(p −A)2 + U.

We have in mind A = ∇S (conjugate to entropy), towards a gauge the-ory framework where information/entropy is a connection (topological/ WZW-term etc.) and the curvature is related to mass in the spirit ofGR.

Then the real part of the deformed Schrodinger equation (see Hloc. cit. p.18) by including a quantum vector potential, is our aboveequation 16:

−∂φ∂t

+ 1/2m (A2 − (∇Φ)2) = U + ρ. (19)

where the Laplacian term for lnR corresponds to the distribution ofinformation charges:

1

2m∆lnR = ρ.

It is tempting to rewrite it in terms of a probability potential:

∆ lnP = mρ.

The above “KG-equation” 18 can now be interpreted 36 as including avector potential corresponding to a connection D = D + imA 37

D2G = U . (20)

A Dirac-like equation should be “around”; a quaternionic form (beforetackling a full Clifford algebra version) should involve the “obvious”operator:

H = ∂t + h · ∇, h = (I, J,K)

36... with “confidence” :-)37To “pack” (∇ · ∇)G + (∇G) · (∇G) one may use a coproduct trick:

< ∇ · ∇, G > + < ∇ ∗∇, G >= (∇ ? ∇)(G)

with ∗ the “convolution” dual to ⊗ and ? = · + ∗; ... too “convoluted” :-)


where the pure quaternionic basis elements I, J,K act as a complexstructure in the plane C:

· : H → End(C).

It is still an incomplete form without a generalized Wick rotation, so acomplexified version should be considered:

∂t 7→ ∂τ = ∂t + i∂T , ∇q 7→ ∇f = ∇q + i∇a.

Indeed G“=”E+iS under the probability-energy correspondence (nor-mal distributions etc.). Note that the “natural” symmetries betweenthe internal and external variables and their conjugates used abovecorresponds to the Hodge diamond of a Calabi-Yau manifold (...?).

To conclude, we state the “result” of the above discussion, as a goalfor the future implementation (see Annex B).

Theorem 6.2. The Quantum Digital Gravity dynamics equation(20) is the Master Equation (non-homogeneous Maurer-Cartan) satis-fied by the quantum information potential (2-point function):

D2G = Uwhere U represents the sources: energy-matter and information.

Proof. If the background free String Theory action is not enough,use some Chern-Simons modification of the Hilbert-Einstein action[JP, J], but on a foamy causal structure [Rovelli-1] as a QDR :-)

6.9. Entropy entanglement and Riemann surfaces. Asking“The Oracle” (i.e. the Web, or course) “What is the entropy of a Rie-mann surface”, we found that the link between entropy and RS is notnew [Gaite]. That a RS is a merger of the input systems not just intu-itively (level of pictures), and it entails entanglement, is confirmed bythe point of view of [LB] that an interaction of pure states is a merger,the system remaining “pure”, even if the parts become mixtures (inthe “effective language” of Hilbert spaces).

This confirms that a Riemann surface carries entropy, as noticedearlier as the Riemann-corolla analogy, and leading to an interpretationof the string action.

The crucial issue remains: what is the relation between quantuminformation (and therefore entropy) and time. The various dualities(Wick, s/t, I/E) require a complexified description even at the level ofspacetime (twistor program?). More precisely, the I/E-duality should


be implemented as a Hermitean pairing, and “hiding” EDOFs by du-ality into IDOFs should correspond to a complex rotation; e.g. iq, ipbecome internal coordinates etc.

Time is special; it is an external causal correlation. The I/E-DOFsmodel, we picture internal DOFs as the IO-quantum computation withthe IO (input/preparation and output/the result or observation) asthe external DOFs as modeled/associated with the observer. A crudediagram of the “universe” (the two parts - Feynman or just the Q-computation loop 38 etc.) is suited for a “convolution framework”:

C∆→ I∗n ⊗ In

Ext⊗Int→ O∗n+1 ⊗On+1

<,>→ C

graded by “external time” (a “Markov-Feynman” chain of pairings:I/Os, of the system and observer).

Here Ext and Int denote the quantum computing processes (“hard-ware and software”) of the corresponding system or observer. Themodel tends to favor (is appropriate for) processes exhibiting a heavyinformation processing (“leaving systems”) rather then “mechanicalprocesses” (motion/change of interaction capability with a “fixed pro-cessing frequency” internal clock - de Broglie pilot wave!). That theinternal time of leaving organisms is naturally modeled in a flexibleway, is well known.

It appears that time especially, is indeed only a coordinate of themodel, used for labeling purposes in a quantum communication (inter-action). It reflects causal correlations (relative to the observer), whilethe other correlations appear to the observer as non-local/spacial; onemay chose to “collapse” the corresponding non-locality and representit by adjunction as internal (imaginary indeed!). The relativistic uni-fication of space and time is by now misleading (era of quantum inter-action/communication modeling).

The need for a complete complexification of QM is now apparent(beyond Wick rotation etc.). If we inspect the classical picture versus

38The QC-loop as a black-box processor plays a crucial role in many theories,including Loop Quantum Gravity.


the quantum picture:

CM QM

ProbabilitiesR Amplitudes C

Observables Real numbers Real eigenvalues

Energy/mass ≥ 0 +Phase

we see that the observables are kept real in QM although the com-plex wave function contains the “real description” of the system. Thesymmetry when considering Bohr’s rule P = Z∗Z (and probabilitydistributions, random variables etc.) as opposed to “mechanical ob-servables” 39 is restored if one allows the Quantum Observables to havecomplex eigenvalues (Bohm-Aharonov effect-like question: is the com-plex quantum potential (vector potential) “real”? Yes, it is part of the“real model”).

CM QM

External Internal

ProbabilitiesR Amplitudes C

Observables Real eigenvalues Complex eigenvalues

Energy/mass : O∗O Quantum Information : I/O

Then the “classical description” and “quantum description” are uni-formly separated.

39Random variable and wave functions.


Classically P = Z∗Z and external space/paths corresponds to mix-tures and classical logic (path can be concatenated and stacked in par-allel: sequential and parallel computing). The internal space is “quan-tum” pure states, superposition with interference etc. The classical(external) observables are the real part of Q-observables C = Q∗Q etc.

But “How do we factorize classical observables (“quantize”)?”, thisis the question.

With a totally complex framework, there are lots of possibilities:energy and temperature can be mixed into E + iT , time and entropyrelated by a “Lagrange multiplier” t = is etc. Or perhaps a Wickrotation relates energy time-flow and entropy-information flow (holo-morphic flow) in a “Dirac quantization” paradigm:

(t, i∂t) (T, i∂S).

to implement the Heisenberg CCR as required by the categorical viewof transitions (see quantum jumps: q and p = ∂q do not commute).Whether the Dirac spinors or quaternions are used for the implemen-tation, remains to be seen.

The external metric (no time variable) describes external correla-tions (angle form, PCF etc.). Time is an information processing coor-dinate; it’s “user dependent”. The quantum information flow is rathera holomorphic flow (2 real dimensions).

But these require rethinking the basics (relativity, Dirac quantiza-tion etc.). And if the external causal structure is 3D after all (SU(2)really), but as part of a double ... we should still try to understand themessage encoded in the Calabi-Yau 3-folds. 40

A look at [Davies2] (what a coincidence; there has to be fate ...)confirms the above “impression”: for a Calabi-Yau three (complex di-mensions) fold, the quantum pre-potentials (or rather the whole sum)seems to play the role of the infinitesimal current of quantum informa-tion (our G = lnΨ; loc. cit. p.164):

lnZ ∼∑

g≥0

g2g−2s Fg(q).

The conifold (diffeomorphic with T ∗S3) could be related with the quan-tum qubit with its three external internal and three internal coordinates,since the associated Riemann surface (loc. cit. p. 159): q = x2 + y2

is reminiscent of the Bohr’s rule. The connection between matrix in-tegrals and ribbon graphs exhibits q as a “partition function” withqi/q = Ni/N = pi as asymptotic probabilities of a corresponding dis-tribution of probabilities.

40“Reality” is not quite what it seems ...


Now the idea of complex eigenvalues appears too (p.165): “To reallymake sense of these ... consider a generalized matrix integral, in whichone does not integrate over Hermitean matrices, but over a particularcontour in the space of complex matrices. 41”.

The Riemann surface of p.166 is in our opinion a QC-routine, com-posed of two RS-corollas “B∗A” (see A→ B processes in our Part II).It also suggests a flow of quantum information entangling the In states,and also reminiscent of a “three-slit experiment”. The first quantumpre-potential is probably just a Green function:

F1(q) = −1

2ln detA− 1

12D.

So, what is the entropy (as a flux) of such a RS-corolla? (and manyother questions: PCF is ydx, what it means to “select” a space-timecoordinate? Is it equivalent to split the I/E DOFs? etc.).

41I.e. “full” quantum computing!

APPENDIX B

VIReQuEST: a Virtual Institute

The Virtual Institute for Quantum Entropy and Space-Time hadinitially the role of a non-standard, permanent, research proposal inQuantum Mathematical-Physics and Computer Science with emphasison the role of information (entropy) in the description of reality.

The Virtual Institute plays the role of an interface between spon-sors of all categories and the Focused Research Group coordinating theimplementation and development of the Digital World Theory underan “open source” research strategy (Linux project worked, why notDWT?).

Depending on the funding available (sabbatical, NSF grants, WWWsupport etc.) a technical version of DWT (ver.2) is scheduled for thesummer 2007 (on a web site near you; relative the clicking distance :-)1).

For more details, please visit our web site http://www.virequest.comand the DWT dedicated site http://www.thedigitalworldtheory.com/.

Comments regarding this “uncensored” version and other kind ofsupport 2 are most welcome!

1. Mathematical-physics and top-down design

To justify the possibility for a larger team to actually designingthe theory, not just “discovering” it, we will sketch a parallel with thedesign process of expert systems.

Physics theories are implemented in mathematical language, to al-low precise computations / processing both for computers and humanmind.

The design of physical theories (“the product”) by the theoreticalphysicist (designer) with help from the mathematician (the implemen-tation specialist) is supposed to be used by the experimentalist (“thebeneficiary”).

1A bookmark away: http:www.theDigitalWorldTheory.com.

2http://www.virequest.com/VIReQuest Sponsors.htm

171

172 B. VIREQUEST: A VIRTUAL INSTITUTE

The application analyst acts as a mediator between the beneficiaryand executant (designer and implementation specialist): collects infofrom the actual site (phenomenon), benefitting from how the benefi-ciary perceives the phenomenon (The beneficiary does not always knowwhat they want in terms of product capabilities ...). The applicationanalyst after the collection of data phase is over elaborates a projet de-scription. The executant (team) develops the project description intoan implementation specification (“blue prints”); implementation spe-cialists (“programmers”) build the system (theory) according to theimplementation specifications. Other phases follow (testing, back tothe drawing board etc.).

A systemic approach in developing physical theories could be a stepfurther in achieving several improvements: portability, longevity, betterperformance (prediction).

The interface between user and product (physics interface) shouldbe “simple”, not reflecting the internal complexity of the implementa-tion (“buttons” and “switches”, not all the wires visible!). The user’sinterface should be device independent.

The implementation of the user’s interface is normally not done bythe designer of the user’s (physics) interface (e.g. let mathematicianschose the right language/technology etc.). The implementation shouldjust make the interface work as specified.

The two levels alluded above (interface/implementation) constituteonly one stratum in a complex application (e.g. ST, QFT, QG etc.).Care for “less convection” within this “stratified complexity” shouldbe exercised.

All the above are more or less common (instinctive) practice; yeta deliberate effort towards enforcing a better methodology should payoff quickly.

The main goal of VIReQuEST (the mediator), is to “prove” thatsuch a methodology can be applied to mathematics and physics, whenit comes to building a better theory, and that such a methodology ismore efficient.

The “Platonic way” is not the only way; mathematics can be dis-covered/uncovered etc., but it can also be designed according to the“application’s needs”. Physics (the application) is driven by obser-vation (conforming to reality as the ultimate “beneficiary”), but thedesign of a theory should conform to the designer’s expectations andstandards.

2. DWT VER.2 IMPLEMENTATION GOALS 173

2. DWT ver.2 implementation goals

A few more specific goals will be sketched bellow.

To convert the connection between energy and entropy, an old storyindeed, into a fundamental new equivalence principle, we will tenta-tively claim the relation between Heisenberg CCR and entropy at thelevel of the QDR.

The QDR, say in its graph (co)homology incarnation, enables thecreation and annihilation operators a = p + iq and a∗ = p − iq to beextended from the internal (Fock) space to the external space (DOFs),as the graph homology and cohomology differentials. A superspacestructure on the QDR should allow to implement Heisenberg CCR andthe Hamiltonian (Laplacian) ([LP], p.3 3):

Resolution : a, a† = 1 (HCCR↔ Entropyquanta)(21)

Laplacian : [a, a†] = H (Hamiltonian↔ Energyquanta)(22)

The entropy is related to the number of micro-states, S = k lnW , whichin turn appears as the size of the symmetry group of the object. In thecombinatorial picture, for example: W = n!/

∏

ni!, is the size of thebundle |Aut(E)| corresponding to a given partition, etc. We will thinkof the “total information” (rather then the average represented by H),as representing a measure of the symmetry at hand: NH = ln |Aut(E)|.4

Then, creating a new “edge” (a particle-antiparticle pair) in a “space-time graph” Γ of degree N (NOP: N → N + 1), and then annihilat-ing an edge (N + 1 → N) will produce a variation of the symmetrygroups which is different from the other scenario, when first the annihi-lation operator is applied (N → N − 1) and then the creation operator(N − 1 → N). The physicists say that the entropy is carried by thefield (... ?).

This should be consistent with the role of the entropy as part ofthe action principle :

K(A,B) =

∫

Γ∈Hom(A,B)

eiS(Γ)/|Aut(Γ)| =

∫

Γ∈Hom(A,B)

e(NH+iS)(Γ).

3Forget about some constants/parameters for now ...4Not surprising, if recalling that a standard mechanics for generating mass

involves breaking the symmetry ...

174 B. VIREQUEST: A VIRTUAL INSTITUTE

The superspace approach for the QDR is also consistent with thedemand 5 of doubling the parameters (internal/external: hyperdynam-ics). But the “holly grail” remains the theorie’s ability to explain (gen-erate) mass (and gravity etc.) as an entropic effect.

In order to implement the information flow, with its quanta thequbit, as a generalization of time flow and causality, “time” probablyneeds three dimensions (associated to the internal symmetries of theinformation unit: SU(2)) 6. In other words, to model sequential versusparallel quantum computing as the internal description of causality, itis not space in need of “extra-dimensions”, but rather time.

This would allow to implement a full set of Feynman-Wick rota-tions: one particles’ time rotated into space, and further into its anti-particles time via conjugation of the qubit’s symmetries (the quantumdot), or better using a generalized complex structure allowing to lookfor a precise meaning for the fundamental particle-wave duality.

¿From a local space-time structure x, y, z, x, y, z (like Darbeaux co-ordinates in the symplectic case), special relativity should emerge asresult of collapsing the three space-like orthogonal dimensions, wheretime t2 = x2 + y2 + z2 represents the classical external parameter la-beling quantum events from the point of view of the observer.

The resulting super-resolution together with the I/E-duality andthe generalized complex structure, would provide enough structure toaccount for the classical symmetries: conformal symmetry, st-duality(Feynman-Wick rotations) and super-symmetry.

On the other hand, at the conceptual level, the classical fundamen-tal principles would be represented in a unified framework: Heisen-berg’s uncertainty relations and particle-wave duality, space and timewithout the 3-1 assymetry. The new matter (energy) and information(entropy) unification should be able to tame gravity, one way (as aforth interaction) or the other (as an “entropic effect”) 7.

The (implementation) goals sketched above will be investigated aspart of the DWT ver.2 8.

5!? “research on demand” ... hm ... why not?6Blow up the time!7... hope is a catalytic ingredient:-)8Coming up the summer of 2007, on a web site near you :-)

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