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Imperial/TP/91-92/25 Canonical Quantum Gravity and the Problem of Time 1 2 C.J. Isham Blackett Laboratory Imperial College South Kensington London SW7 2BZ United Kingdom August 1992 Abstract The aim of this paper is to provide a general introduction to the problem of time in quantum gravity. This problem originates in the fundamental conflict between arXiv:gr-qc/9210011v1 21 Oct 1992 the way the concept of ‘time’ is used in quantum theory, and the role it plays in a diffeomorphism-invariant theory like general relativity. Schemes for resolving this problem can be sub-divided into three main categories: (I) approaches in which time is identified before quantising; (II) approaches in which time is identified after quantising; and (III) approaches in which time plays no fundamental role at all. Ten different specific schemes are discussed in this paper which also contain an introduction to the relevant parts of the canonical decomposition of general relativity. 1Lectures presented at the NATO Advanced Study Institute “Recent Problems in Mathematical Physics”, Salamanca, June 15–27, 1992. 2Research supported in part by SERC grant GR/G60918. Contents 1 INTRODUCTION 4 1.1 Preamble ................................... 4 1.2 PreliminaryRemarks ............................. 4 1.3 Current Research Programmes in Quantum Gravity . ..... 6 1.4 OutlineofthePaper ............................. 8 1.5 Conventions.................................. 9 2 QUANTUMGRAVITYANDTHEPROBLEMOFTIME 9 2.1 Time in Conventional Quantum Theory . .. 10 2.2 Time in a Diff( )-invariantTheory . 12 M 2.3 ApproachestotheProblemofTime. 16 2.4 Technical Problems With Time . 19 3 CANONICALGENERALRELATIVITY 20 3.1 IntroductoryRemarks ............................ 21 3.2 Quantum Field Theory in a Curved Background . ... 22 3.2.1 TheCanonicalFormalism . 22 3.2.2 QuantisationoftheSystem . 23 3.3 The Arnowitt-Deser-Misner Formalism . .... 25 3.3.1 IntroductionoftheFoliation . 25 3.3.2 The Lapse Function and Shift Vector . 27 3.3.3 The Canonical Form of General Relativity . .. 28 3.3.4 TheConstraintAlgebra . 32 3.3.5 TheRoleoftheConstraints . 34 3.3.6 Eliminating the Non-Dynamical Variables . ... 36 3.4 InternalTime ................................. 37 3.4.1 TheMainIdeas ............................ 37 3.4.2 ReductiontoTrueCanonicalForm . 40 3.4.3 TheMulti-timeFormalism . 43 1 4 IDENTIFY TIME BEFORE QUANTISATION 44 4.1 Canonical Quantum Gravity: Constrain Before Quantising ........ 44 4.1.1 BasicIdeas .............................. 44 4.1.2 ProblemsWiththeFormalism . 45 4.2 The Internal Schr¨odinger Interpretation . ........ 46 4.2.1 TheMainIdeas ............................ 46 4.2.2 The Main Advantages of the Scheme . 48 4.2.3 AMinisuperspaceModel . 49 4.2.4 Mean Extrinsic Curvature Time . 51 4.2.5 TheMajorProblems ......................... 52 4.3 Matter Clocks and Reference Fluids . ... 59 4.3.1 TheBasicIdeas............................ 59 4.3.2 TheGaussianReferenceFluid . 60 4.3.3 AdvantagesandProblems . 61 4.4 UnimodularGravity ............................. 62 5 IDENTIFYTIMEAFTERQUANTISATION 63 5.1 Canonical Quantum Gravity: Quantise Before Constraining........ 63 5.1.1 The Canonical Commutation Relations for Gravity . .... 64 5.1.2 TheImpositionoftheConstraints. 65 5.1.3 ProblemswiththeDiracApproach . 66 5.1.4 Representations on Functionals Ψ[g] ................ 67 5.1.5 The Wheeler-DeWitt Equation . 68 5.1.6 A Minisuperspace Example . 69 5.2 The Klein-Gordon Interpretation for Quantum Gravity . ........ 71 5.2.1 The Analogue of a Point Particle Moving in a Curved Spacetime . 71 5.2.2 Applying the Idea to Quantum Gravity . 73 5.3 ThirdQuantisation .............................. 75 5.4 The Semiclassical Approximation to Quantum Gravity . ....... 77 5.4.1 TheEarlyIdeas............................ 77 5.4.2 The WKB Approximation to Pure Quantum Gravity . 78 5.4.3 Semiclassical Quantum Gravity and the Problem of Time ..... 80 2 5.4.4 TheMajorProblems ......................... 82 5.5 DecoherenceofWKBSolutions . 84 5.5.1 The Main Idea in Conventional Quantum Theory . 84 5.5.2 Applications to Quantum Cosmology . 87 6 TIMELESS INTERPRETATIONS OF QUANTUM GRAVITY 88 6.1 The Na¨ıve Schr¨odinger Interpretation . ....... 88 6.2 The Conditional Probability Interpretation . ........ 92 6.2.1 TheMainIdeas ............................ 92 6.2.2 Conditional Probabilities in Conventional Quantum Theory . 93 6.2.3 The Timeless Extension . 94 6.3 The Consistent Histories Interpretation . ....... 95 6.3.1 Preamble ............................... 96 6.3.2 Consistent Histories in Conventional Quantum Theory ...... 97 6.3.3 The Application to Quantum Gravity . 100 6.3.4 ProblemsWiththeFormalism . 102 6.4 The Frozen Formalism: Evolving Constants of Motion . ....... 104 7 CONCLUSIONS 107 3 1 INTRODUCTION 1.1 Preamble These notes are based on a course of lectures given at the NATO Advanced Summer Institute “Recent Problems in Mathematical Physics”, Salamanca, June 15–27, 1992. The notes reflect part of an extensive investigation with Karel Kuchaˇrinto the problem of time in quantum gravity. An excellent recent review is Kuchaˇr(1992b), to which the present article is complementary to some extent. In particular, my presentation is slanted towards the more conceptual aspects of the problem and, as this a set of lecture notes (rather than a review paper proper), I have also included a fairly substantial technical introduction to the canonical theory of general relativity. However, there is inevitably a strong overlap with many portions of Kuchaˇr’s paper, and I am grateful to him for permission to include this material plus a number of ideas that have emerged in our joint discussions. Therefore, the credit for any good features in the present account should be shared between us; the credit for the mistakes I claim for myself alone. 1.2 Preliminary Remarks The problem of ‘time’ is one of the deepest issues that must be addressed in the search for a coherent theory of quantum gravity. The major conceptual problems with which it is closely connected include: the status of the concept of probability and the extent to which it is conserved; • the status of the associated concepts of causality and unitarity; • the time-honoured debate about whether quantum gravity should be approached • via a canonical,or a covariant, quantisation scheme; the extent to which spacetime is a meaningful concept; • the extent to which classical geometrical concepts can, or should, be maintained • in the quantum theory; the way in which our classical world emerged from some primordial quantum event • at the big-bang; the whole question of the interpretation of quantum theory and, in particular, the • domain of applicability of the conventional Copenhagen view. The prime source of the problem of time in quantum gravity is the invariance of classical general relativity under the group Diff( ) of diffeomorphisms of the spacetime M 4 manifold . This stands against the simple Newtonian picture of a fixed time param- eter, andM tends to produce quantisation schemes that apparently lack any fundamental notion of time at all. From this perspective, the heart of the problem is contained in the following questions: 1. How should the notion of time be re-introduced into the quantum theory of gravity? 2. In particular, should attempts to identify time be made at the classical level, i.e., before quantisation, or should the theory be quantised first? 3. Can ‘time’ still be regarded as a fundamental concept in a quantum theory of gravity, or is its status purely phenomenological? If the concept of time is not fundamental, should it be replaced by something that is: for example, the idea of a history of a system, or process, or an ordering structure that is more general than that afforded by the conventional idea of time? 4. If ‘time’ is only an approximate concept, how reliable is the rest of the quantum- mechanical formalism in those regimes where the normal notion of time is not applicable? In particular, how closely tied to the concept of time is the idea of probability? This is especially relevant in those approaches to quantum gravity in which the notion of time emerges only after the theory has been quantised. In addition to these questions—which apply to quantum gravity in general—there is also the partly independent issue of the applicability of the concept of time (and, indeed, of quantum theory in general) in the context of quantum cosmology. Of particular relevance here are questions of (i) the status of the Copenhagen interpretation of quantum theory (with its emphasis on the role of measurements); and (ii) the way in which our present classical universe, including perhaps the notion of time, emerged from the quantum origination event. A key ingredient in all these questions is the realisation that the notion of time used in conventional quantum theory is grounded firmly in Newtonian physics. Newtonian time is a fixed structure, external to the system: a concept that is manifestly incompatible with diffeomorphism-invariance and also with the idea of constructing a quantum theory of a truly closed system (such as the universe itself). Most approaches to the problem of time in quantum gravity 3 seek to address this central issue by identifying an internal time which is defined in terms of the system itself, using either the gravitational field or the matter variables that describe the material content of the universe. The various schemes differ in the way such an identification is made and the point in the procedure at which it is invoked. Some of
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