Change Without Time Relationalism and Field Quantization Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) der naturwissenschaftlichen Fakult¨at II – Physik der Universit¨at Regensburg vorgelegt von Johannes Simon aus Schwandorf 2004 Promotionsgesuch eingereicht am: 19.04.2004 Die Arbeit wurde angeleitet von Prof. Dr. Gustav M. Obermair. Prufungsausschuss:¨ Vorsitzender Prof. Dr. W. Prettl 1. Gutachter Prof. Dr. G. M. Obermair 2. Gutachter Prof. Dr. E. Werner weiterer Prufer¨ Prof. Dr. A. Sch¨afer Tag der mundlichen¨ Prufung:¨ 23.07.2004 Tempus item per se non est [...] Titus Lucretius Carus [Car, 459 ff] Preface This thesis studies the foundations of time in physics. Its origin lies in the urge to understand the quantum measurement problem. While the emergence of classical- ity can be well described within algebraic quantum mechanics of infinite systems, this can be achieved only in infinite time. This led me to a study of quantum dy- namics of infinite systems, which turned out to be far less unique than in the case of finitely many degrees of freedom. In deciding on the correct time evolution the question appears how time – or rather duration – is being measured. Traditional quantum mechanics lacks a time observable, and for closed systems in an energy eigenstate duration is indeed meaningless. A similar phenomenon shows up in general relativity, where absolute duration (as well as spatial distance) becomes meaningless due to diffeomorphism invariance. However, by relating different parts of a closed system through simultaneity (in quantum mechanics as well as in general relativity), an internal notion of time becomes meaningful. This similarity between quantum mechanics and general relativity was recognized in the context of quantum gravity by Carlo Rovelli, who proposed a relational concept of quantum time in 1990. He showed in a two-oscillator model that, by using an energy constraint instead of time evolution, the algebra of constants of motion can be quantized and used to relate so-called1 partial observables. The main problem with the relational concept of time turns out to be the lack of a fixed evolution in the quantum domain, where arbitrary superpositions are allowed for the total system, leading to a possible superposition of different instants of the internal time. At this point a question naturally arises, which to the best of our knowledge has not been asked so far: If the system becomes infinite, can we reconstruct a classical notion of time from algebraic quantum mechanics; and what do inequivalent representations of the algebra of observables mean for time? – When trying to find rules, which guarantee agreement of a general notion of time with the empirical time of classical observers, one is lead to a number of further questions, whose very meaning is not easily clarified: What is time? Why is time a totally ordered set, even a one-dimensional differentiable manifold? Why does time pass by? What determines the direction of time? What is a clock? – During the last decade foundational physics has seen much progress on the subject of time; the central issues are however still unsolved – there is still pretty much to do. This thesis analyzes the foundation of the relational concept of time in view of various meanings and problems of ’time’, and it asks whether a ’relational quantization’ of the free quantum field is possible; see the overview on page vi. 1The notion of partial observable was introduced only recently [Rov01b], thereby putting the conceptual foundations of relationalism on a solid basis. i It could not accomplish its initial goal, a solution of the age-old quantum mea- surement problem. Nevertheless we feel confident that a solution of this problem will be possible with a new, physical understanding of time. This will require to free oneself from preconceptions, to doubt the foundations instead of believing in them, an act of pleasant emancipation, which allows to discover the real world behind the shadows we see and too often take for real. Wurzburg,¨ spring 2004 ii Contents Chapter 1. Concepts of time 1 1.1 Modelling time........................................ 4 1.2 The role of time in physical theories ................... 7 1.2.1 Newtonian mechanics . 7 1.2.2 Homogeneous and presymplectic formalism . 9 1.2.3 Special relativity . 16 1.2.4 General relativity. 20 1.2.5 Nonrelativistic quantum mechanics. 25 1.2.6 Relativistic quantum mechanics . 26 1.2.7 Summary .............................................. 26 1.3 Relational time........................................ 27 1.3.1 Relationalism . 27 1.3.2 Relational theories. 29 1.3.2.1 Classical mechanics . 29 1.3.2.2 General relativity. 31 1.3.2.3 Quantum mechanics . 32 1.3.3 Relational time from simultaneity . 34 Chapter 2. Problems with time 41 2.1 The arrow of time ..................................... 42 2.1.1 Time-reversal invariance . 42 2.1.2 Irreversibility . 46 2.1.3 The arrow of time in classical physics. 48 2.1.4 The arrow of time in quantum physics . 52 2.2 Time measurement .................................... 58 2.2.1 Quantum time observables . 58 2.2.2 Classical oscillator clocks . 61 2.2.3 Quantum phase observables . 63 2.2.4 Quantum oscillator clocks. 65 2.3 Quantum gravity ...................................... 67 Chapter 3. Relational field quantization 69 3.1 Relational classical mechanics ......................... 69 3.1.1 Two oscillators . 70 3.1.1.1 Hamiltonian formulation. 70 3.1.1.2 Constants of motion . 71 3.1.1.3 Presymplectic formulation . 72 3.1.1.4 Complete observables . 73 3.1.1.5 Partial observables. 75 3.1.1.6 Limitations of the model. 76 3.1.1.7 Evolving constants of motion. 76 3.1.2 Freefield............................................... 77 3.1.2.1 Hamiltonian formulation. 78 3.1.2.2 Constants of motion . 79 3.1.2.3 Presymplectic formulation . 82 3.1.2.4 Complete observables . 83 3.1.2.5 Partial observables. 84 3.2 Relational quantization ................................ 86 3.2.1 On quantization . 86 3.2.2 Two oscillators . 88 3.2.2.1 Quantization based on the angular momentum algebra . 88 3.2.2.2 Quantization based on oscillator algebras . 90 3.2.2.3 Discussion . 90 3.2.2.4 Directions of research . 91 3.2.3 Freefield............................................... 92 3.2.3.1 Quantization map . 92 3.2.3.2 Groenewold-van Hove theorem . 95 3.2.3.3 Constraint............................................ 99 3.2.3.4 Complete and partial observables . 99 iv Chapter 4. Change without time 101 4.1 Classicality of time ....................................101 4.1.1 Physical time . 102 4.1.2 Instantaneity . 102 4.2 General dynamics and quantum measurement . 104 Summary 107 Bibliography 109 v Overview Chapter 1 After a brief historical introduction to the phenomenon of time we present math- ematical structures often used in talking about time and we discuss the concepts of time as used in Newtonian mechanics, special and general relativity and quan- tum mechanics. Thereafter we focus on the relational concept of time and reduce time to its essence: a simultaneity relation. Chapter 2 An introduction to three of the main problems of time is given: 1. The arrow of time in classical and quantum physics from the absolute and relational point of view. 2. The measurement of time with quantum clocks. 3. The meaning of time in quantum gravity (without a fixed background metric). Chapter 3 We discuss Rovelli’s model of two oscillators, which shows the very meaning of the relational concept of time at the quantum level: Via coherence one oscillator acts as a clock for the other one. In the first section we generalize this model on the classical level to the free massless scalar field in one dimension. The second section is concerned with quantization. Chapter 4 The last chapter briefly discusses the classicality of time and gives an outlook on a general dynamics compatible with quantum measurement collapse. vi On arrangement and notation Each chapter is divided into sections, sections are divided into subsections and, possibly, subsubsections; numbers include the chapter, e.g. /1.2.3.4/ refers to subsubsection 4 of subsection 3 of section 2 of chapter 1, and /1/ refers to the first chapter. Equations are consecutively numbered within each section and contain the number of the chapter and section, e.g. (1.2.3) denotes equation 3 in section 2 of chapter 1. References are cited in square brackets, e.g. [Rov90]. In cases where the literature is too much to be cited completely, we have tried to include in the references at least recent reviews or original articles. By “matter” we mean all forms of energy, not only fermionic matter. If not otherwise stated, by finite (infinite) quantum systems we mean quantum systems with a finite (infinite) number of degrees of freedom. We use“quantum mechanics” and “quantum theory” synonymously. With “state” of a quantum system in the traditional Hilbert space formalism we usually mean a state vector (and not a ray), or a density matrix. In /2.1/ “state” does mean a configuration, not a point in phase space. We use the terms “two-oscillator system” and “two oscillators” instead of and synonymously with “double pendulum”. In the context of general relativity, as is customary, we use Einstein’s summation convention and the range of greek indices is 0, 1, 2, 3, while roman indices take the values 1, 2, 3 corresponding to 3-space. Throughout the text we use the following symbols and abbreviations: T instant T set of instants T time operator t time parameter, or time coordinate x0 time coordinate in relativity 1S identity mapping on S V Poisson algebra C Poisson subalgebra of constants of motion span(A) vector space spanned by A gen(A) algebra generated by A Ran(A) range of A S(R, R) real-valued Schwartz functions with domain R SYM(H) symmetric operators on the Hilbert space H V,V2,V∞ Poisson algebras C,C2,C∞ algebras of constants of motion C∞,g set of generators of C∞ + an , an creation and annihilation operators ± ± aij, bij constants of motion of the free field involving triples of field modes θ(x) Heaviside step function (θ(x) = 1, if x ≥ 0 and θ(x) = 0 otherwise) iX Y inner product of the tensors X and Y genLie(A) Lie algebra generated by the set A w.r.t.