An Analysis of the `Thermal-Time Concept' of Connes and Rovelli Eine Analyse des `Thermal-Time Konzepts' von Connes und Rovelli Diplomarbeit vorgelegt von Tim-Torben Paetz aus Braunschweig Angefertigt im Institut f¨urTheoretische Physik der Georg-August-Universit¨atG¨ottingen Juli 2010 Gutachter der Diplomarbeit: Prof. Dr. Karl-Henning Rehren PD Dr. Manfred Requardt Contents Notation v 0. Introduction 1 1. The Notion of Time3 1.1. Introduction........................................3 1.2. Time and Philosophy...................................4 1.3. Properties Ascribed to Time...............................6 1.4. Non-Generally Covariant Notion of Time.......................7 1.5. General Covariance and the Notion of Time...................... 10 1.6. Some Special Notions of Time.............................. 12 1.7. The Problem of Time in Quantum Gravity...................... 14 1.8. Fundamental Timelessness................................ 15 2. Algebraic Quantum Field Theory 17 2.1. A Historical Look Back at Quantum Field Theory.................. 17 2.2. The Wightman Framework............................... 18 2.3. Issues of Conventional Approaches and the Local Viewpoint............. 18 2.4. The Algebraic Approach................................. 20 2.5. AQFT in Terms of von Neumann Algebras and Reeh-Schlieder Theorem...... 23 2.6. Nuclearity Condition and Split Property........................ 24 2.7. Types of von Neumann Algebras Appearing in AQFT................ 26 2.8. Generalisations to Curved Spacetimes and Generally Covariant AQFT....... 27 3. Thermal Equilibrium and the KMS Condition 29 3.1. The Traditional Approach and its Limitations..................... 29 3.2. Time Evolution in AQFT and the Complementary Role of C∗- and von Neumann Algebras.......................................... 32 3.3. Kubo-Martin-Schwinger Boundary Condition..................... 34 3.4. Some Basic Properties of KMS States......................... 37 3.5. Stability Properties and the KMS Condition...................... 38 3.6. Passivity.......................................... 45 3.7. Concluding Remarks................................... 46 4. Tomita-Takesaki Modular Theory 47 4.1. Tomita-Takesaki Theorem................................ 47 4.2. Modular Theory by Example.............................. 50 4.3. Link between Modular Theory and KMS Condition.................. 53 4.4. Cocycle Radon-Nikod´ymTheorem........................... 54 4.5. Geometrical Meaning of the Modular Flow and Bisognano-Wichmann Theorem.. 56 4.6. Non-Geometrical and Non-Local Modular Flows................... 63 4.7. Physical Applications of Modular Theory....................... 64 5. Thermal Time Hypothesis 65 5.1. The Starting Point.................................... 65 5.2. Formulation of the Thermal Time Hypothesis..................... 65 5.3. Alternative Versions of the Thermal Time Hypothesis................ 70 iii CONTENTS 6. Tests of the Thermal Time Hypothesis 71 6.1. Equilibrium States and Tolman-Ehrenfest Effect................... 71 6.2. Special Relativistic System............................... 72 6.3. Robertson-Walker Space Filled with a Dynamical Maxwell Field.......... 73 6.4. Rindler Wedge and Unruh Effect............................ 76 6.5. Minkowski Diamond................................... 78 6.6. Minkowski Forward-Lightcone.............................. 81 6.7. Static De Sitter Space and Gibbons-Hawking Effect................. 82 6.8. De Sitter Diamond.................................... 84 6.9. Schwarzschild Space and Hawking Effect........................ 86 7. Discussion of the Thermal Time Hypothesis 89 7.1. Introduction........................................ 89 7.2. Compatibility between Thermal Time and Relativity................. 90 7.3. Significance of the Examples of Chapter6....................... 92 7.4. Thermal Time as the Origin of Time.......................... 94 7.5. Thermal Time as an Entirely New Concept of Time................. 96 7.6. Intrinsic Equilibrium................................... 101 7.7. Relation between Thermal Time and Physics..................... 102 7.8. Conceptual Issues..................................... 104 7.9. Thermal Time Hypothesis in the Covariant Formalism................ 106 7.10. Symmetry Properties Related to Thermal Time.................... 108 7.11. Thermal Time and Irreversibility............................ 111 7.12. Dispersive Properties of the Thermal Time Flow................... 112 7.13. Thermal Time Flow on Type I Factors......................... 113 7.14. Thermal Time Flow on Local Algebras......................... 116 7.15. Fixed Points and the Thermal Time Flow....................... 120 7.16. Relevance of the Asymptotic Behaviour and the Role of Type I Factors...... 124 7.17. State-Dependence of Thermal Time.......................... 125 7.18. Concluding Remarks................................... 128 8. Conclusions and Outlook 129 Appendices 130 A. Mathematical Preliminaries 131 A.1. Functional Analysis.................................... 131 A.2. Almost Periodic Functions................................ 135 A.3. Algebras, Representations and States.......................... 136 A.4. Von Neumann Algebras................................. 140 A.5. Classification Theory of von Neumann Algebras.................... 144 B. Covariant Formalism in Classical and Quantum Mechanics 147 B.1. Covariant Formulation of Classical Mechanics..................... 147 B.2. Covariant Formulation of Quantum Mechanics.................... 151 B.3. Concluding Remarks................................... 155 Bibliograhpy 157 Acknowledgements 163 iv Notation We work in the geometrized unit system with the speed of light c, the gravitational constant G, the Boltzmann constant kB, and the reduced Planck constant ~ equal to unity. The signature of the metric gµν used throughout this thesis is − + + +. The index notation uses Greek indices µ, ν; ··· = 0; 1; 2; 3 to indicate 4-dim. spacetime indices, Latin indices i; j; : : : describe the spatial part and run from 1 to 3. We exploit Einstein's summation convention. If not explicitly mentioned, we assume to be in 3+1-dim. Minkowski space. The (complex) Hilbert spaces we are dealing with are supposed to be separable (only in appendixA this assumption is explicitly stated whenever it is necessary). For the benefit of the reader, we list below a number of symbols appearing more or less frequently in the course of this thesis. H; K;::: (separable) Hilbert spaces A; B; : : : operators (observables) D(A) domain of the operator A A ⊂ B D(A) ⊂ D(B) and AjD(A) = BjD(A) A ≤ BA − B is a positive operator ker(A) kernel of the operator A ran(A) rank of the operator A E(λ) spectral decomposition ρ(A) resolvent of A −1 Rλ(A) resolvent of A at λ, Rλ(A) = (λ − A) σ(A) spectrum of A P; Q projection operators PΩ projection operator on the subspace generated by the vector Ω 2 H O spacetime region O0 causal complement of O O1 ⊂⊂ O2 closure of O1 is contained in the interior of O2 M Minkowski space D double-cone W wedge region V± forward / backward lightcone A algebra A+ positive cone of A A (quasi-local) C∗-algebra R von Neumann algebra M, N von Neumann algebras, mostly type I factors R0 commutant of R R∗ dual space of R R∗ predual space of R Γ(R) Connes spectrum S(R) spectral invariant 00 _iRi ([iRi) B(H) von Neumann algebra consisting of all bounded operators acting on H O 7! A(O) abstract local net O 7! R(O) weak closure of the representation of an abstract net ! state ρ density matrix associated with a normal state ! R(O) partial state obtained from the restriction of ! on R(O) C! centralizer of ! !P pertubed state, constructed from a perturbation P = P ∗ 2 R of ! π representation (H!; π!; Ω!) GNS representation induced by ! (H0; π0; Ω0) vacuum representation Ω0 2 H0 vacuum vector !0 = hΩ0; · Ω0i vacuum state v ∗ τt one-parameter group of -automorphisms Rτ fixed point algebra of τ Rτ set of entire analytic elements for τ spec τ Arveson spectrum d δ(A) infinitesimal generator of τt, δ(A) = dt τt(A) t=0 P ∗ ∗ τt perturbed -automorphism group with perturbation P = P 2 R αt physical time flow H (physical) Hamiltonian Ad(eiHt) Ad(eiHt)(A) = eiHtAe−iHt Hc covariant Hamiltonian S = J∆1=2 Tomita operator ∆ modular operator ∆~ modular function K modular / thermal Hamiltonian ! σs modular flow / thermal time flow induced by ! O Λs automorphism group describing a geometrical action of the modular group in O σs(O) modular flow acting on R(O); we reserve this expression for the vacuum modular flow σ~s canonical flow associated with a von Neumann algebra J modular conjugation j! ∗-anti-isomorphism from R onto R0 induced by ! s modular parameter / thermal time " P+ proper orthochronous Poincar´egroup Λ(λ) Lorentz boost with boost parameter λ L generator of the unitary representation of the Lorentz boosts a acceleration τ proper time T clock time Γ phase space Γex extended phase space Γc covariant phase space β inverse temperature vi Chapter 0 Introduction \I strongly suspect that the key is time." Lee Smolin, American physicist (*1955) It is a long-standing dream of mankind to disclose the most fundamental secrets of Nature and to understand how it works. The ultimate goal is a \theory of everything" which gives a compre- hensive picture how Nature is build up. In a first step one would like to unify general relativity with the principles of quantum theory into a theory of quantum gravity. One of the main issues appearing in attempts to do that is an incompatible treatment of the notion of time underlying