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Journal of Physics Conference Series

The Problem of Time in Quantum Cosmology: A Decoherent Histories View

T. Christodoulakis, Petros Wallden

Published Feb 1, 2011• 4 citations
Publisher

Abstract

The problem of time in quantum gravity arises due to the diffeomorphisms invariance of the theory and appears via the Hamiltonian constraint, in the canonical quantizations. There is a need for a description where one can ask some timeless questions that still encode some sense of temporality. The decoherent histories approach to quantum theory, already at the kinematical level admits an internal time. Several alternative proposals for resolving the problem of time via the decoherent histories, exist, and in this contribution we focus on one particular and examine how it manifests itself at some simple cosmological models.

DOI
10.1088/1742-6596/283/1/012041

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International Journal of Modern Physics A•Aug 10, 2002•0 citations
The problem of time in quantum gravity arises due to the diffeomorphisms invariance of the theory and appears via the Hamiltonian constraint, in the canonical quantizations. There is a need for a description where one can ask some timeless questions that still encode some sense of temporality. The decoherent histories approach to quantum theory, already at the kinematical level admits an internal time. Several alternative proposals for resolving the problem of time via the decoherent histories, exist, and in this contribution we focus on one particular and examine how it manifests itself at some simple cosmological models.
The development of quantum cosmology, in which Stephen Hawking played a crucial role, has frequently encountered substantial conceptual and technical difficulties related to the problem of time in quantum gravity and to general issues concerning the foundations of quantum theory. In this contribution to Stephen's 60th Birthday Conference, I describe some recent work in which the decoherent histories approach to quantum theory is used to quantize simple cosmological models and perhaps shed some light on some of these difficulties.
One of the problems of quantum cosmology follows from the fact that the Hamiltonian H of classical general relativity equals zero. Quantizing canonically in the Schrodinger picture, the Schrodinger equation for the wave function Ψ of the universe is therefore the so-called Wheeler-DeWitt equation (where Ĥ is the operator version of H) and we have no quantum dynamics. In particular, it follows directly that, if Ȓ is the operator representing the radius of the universe, where <Ȓ>ψ = Ψ<|Ȓ|>Ψ, for every Ψ. The universe described by (1) does not expand. But worse still is the fact that there is no time development, contrary to our experience. Like a particle in an eigenstate of zero energy, the state of the universe does not change. The still universe resulting from (1) is a manifestation of the more general “problem of time” afflicting canonical quantum gravity. Despite rare claims to the contrary, most consider it to be a serious difficulty for quantum cosmology.
The problem of time in quantum gravity is often presented as a consequence of applying quantum theory to general relativity. These notes adopt a broader perspective. We first examine how time enters classical mechanics, relativity, quantum theory, and quantum field theory, distinguishing three principal aspects: causal order, temporal duration, and the present. This analysis shows that the conceptual difficulties associated with time do not arise only when gravity is quantised. Tensions between causal structure, clocks, observables, and measurement are already present in the theories from which quantum gravity is constructed. We then survey responses to the problem of time across the principal approaches to quantum gravity. These include canonical and Wheeler-DeWitt quantisation, path-integral and histories formulations, perturbative and background-dependent approaches, causal sets, and twistor theory. The programmes are compared according to whether they retain an external time, render time problematic through quantisation, or incorporate causal or temporal structure into their foundations. Particular emphasis is placed on questions arising from quantum foundations and relativistic quantum information. We conclude by examining linearised gravity and weak-gravity quantum systems. This regime shows that the problem of time is not confined to Planck-scale physics, but arises already in attempts to provide a predictive and operationally meaningful quantum description of spacetime near flat geometry.
The unsatisfactory status of the interpretation of the wave function of the Universe in canonical quantum gravity is reviewed. The ``naive interpretation'' obtained by straightforwardly applying the standard interpretive rules to the canonical quantization of general relativity is manifestly unacceptable; the ``WKB interpretation'' has only a limited domain of applicability; and the ``conditional probability interpretation'' requires one to pick out a ``preferred time variable'' (or preferred class of such variables) from among the dynamical variables. Evidence against the possibility of using a dynamical variable to play the role of ``time'' in the conditional probability interpretation is provided by the fact (proven here) that in ordinary Schr\"odinger quantum mechanics for a system with a Hamiltonian bounded from below, no dynamical variable can correlate monotonically with the Schr\"odinger time parameter t, and thus the role of t in the interpretation of Schr\"odinger quantum mechanics cannot be replaced by that of a dynamical variable. We also argue that the interpretive problems of quantum gravity are not alleviated by the incorporation of observers into the theory. Faced with these difficulties, we seek a formulation of canonical quantum gravity in which an appropriate nondynamical time parameter is present. By analogy with a parametrized form of ordinary Schr\"odinger quantum mechanics, we make a proposal for such a formulation. A specific proposal considered in detail yields a theory which corresponds at the classical level to general relativity with an arbitrary, unspecified cosmological constant. In minisuperspace models, this proposal yields a quantum theory with satisfactory interpretive properties, although it is unlikely that this theory will admit sufficiently many observables for general spacetimes. Nevertheless, we feel that the approach suggested here is worthy of further investigation.
In this review we present the problem of time in quantum physics, including a short history of the problem and the known objections about considering time as a quantum observable. The need to deal with time as an observable is elaborated through some unresolved problems. The lack of a consistent theory of time is currently hindering the formulation of a full-fledged theory of quantum gravity. It is argued that the proposal set forth by several authors of considering an intrinsic measurement of quantum time, besides having the conventional external time, is compelling. Recently several suggestions have been put forward to revive the proposal of Page and Wootters (1983), elaborating and resolving some of the main ambiguities of the original proposal and opening new scope for understanding its content. The approach followed in these new contributions exposes the need to go beyond the limitations enforced by the conventional approach of quantum physics. The attitude of covariant loop quantum gravity, in which it is called to completely ignore time, is also discussed. This review could be a step forward in an endeavour to reform our outlook of the unification of the theory of relativity and quantum physics by furnishing the conceptual ground needed for this goal. Intentionally, some technical details are avoided since we aim to present the approaches to resolve the problem in a simple way with the clearest possible outlook. These can be looked up in the original references provided.
Inspired by the central equation of canonical quantum gravity, the timeless Wheeler-DeWitt equation, we investigate timeless properties for quantum cosmological models. In particular we are interested in the following question: What is the probability of the system entering certain regions of its configuration space without reference to time at all? For the classical case this is a property of the entire system trajectories and can be formulated using a phase space probability distribution 1. In the quantum case we propose and investigate a probability which we obtain by adapting the decoherent histories approach to quantum theory 2 to the timeless case and by using the induced inner product on the physical state space of quantum cosmological models 3. For the example of the Klein-Gordon equation we have investigated the case of space–like surfaces and our results agree with the standard operator methods 4. For a more general model we consider a semiclassical approximation as well as the effect of an environment 1. When the histories are decoherent, the probabilities approximately coincide with the classical case, with the phase space probability distribution replaced by the Wigner function of the quantum state. The influence functional of the environment respects the reparametrization invariance of the quantum cosmological Wheeler-DeWitt equation.