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arXiv:2408.08953 · Physical review. D/Physical review. D.

Phase time and Ehrenfest’s theorem in relativistic quantum mechanics and quantum gravity

Leonardo Chataignier

Published Dec 23, 2024• 2 citations
arXivHTMLPDF

Abstract

We revisit the concept of phase time, which has been previously proposed as a solution to the problem of time in quantum gravity. Concretely, we show how the geometry of configuration space together with the phase of the wave function of the Universe can, under certain conditions, lead to the definition of a positive-definite squared norm from which probabilities can be defined. The norm is conserved under time evolution, and we obtain a version of Ehrenfest's theorem that is analogous to the one in ordinary quantum mechanics. We address how the present approach avoids some difficulties that were encountered in previous attempts at defining an Ehrenfest theorem relative to phase time in the context of relativistic quantum mechanics and canonical quantum gravity, and how it is connected to a notion of gauge fixing the inner product. We conclude with a brief outlook.

DOI
10.1103/physrevd.110.126023

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We revisit the concept of phase time, which has been previously proposed as a solution to the problem of time in quantum gravity. Concretely, we show how the geometry of configuration space together with the phase of the wave function of the Universe can, under certain conditions, lead to the definition of a positive-definite squared norm from which probabilities can be defined. The norm is conserved under time evolution, and we obtain a version of Ehrenfest's theorem that is analogous to the one in ordinary quantum mechanics. We address how the present approach avoids some difficulties that were encountered in previous attempts at defining an Ehrenfest theorem relative to phase time in the context of relativistic quantum mechanics and canonical quantum gravity, and how it is connected to a notion of gauge fixing the inner product. We conclude with a brief outlook.
The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of the Ehrenfest theorem for quantum gravity field theory has remained elusive, while its validation poses challenging conceptual questions. In fact, this presupposes a number of minimum requirements, which include the prescription of quantum Hamiltonian operator, the definition of scalar product, and the identification of dynamical evolution parameter. In this paper, it is proven that the target can be established in the framework of the manifestly covariant quantum gravity theory (CQG theory). This follows as a consequence of its peculiar canonical Hamiltonian structure and the commutator-bracket algebra that characterizes its representation and probabilistic interpretation. The theoretical proof of the theorem for CQG theory permits to elucidate the connection existing between quantum operator variables of gravitational field and the corresponding expectation values to be interpreted as dynamical physical observables set in the background metric space-time.
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.
The problem of time in the quantization of gravity arises from the fact that time in Schrödinger’s equation is a parameter. This sets time apart from the spatial coordinates, represented by operators in quantum mechanics (QM). Thus “time” in QM and “time” in general relativity (GR) are seen as mutually incompatible notions. The introduction of a dynamical time operator in relativistic quantum mechanics (RQM), that follows from the canonical quantization of special relativity and that in the Heisenberg picture is also a function of the parameter [Formula: see text] (identified as the laboratory time), prompts to examine whether it can help to solve the disfunction referred to above. In particular, its application to the conditional interpretation of time in the canonical quantization approach to quantum gravity is developed.
Canonical quantization applied to closed systems leads to static equations, the Wheeler-deWitt equation in Quantum Gravity and the time independent Schrödinger equation in Quantum Mechanics. How to restore time is the Problem of Time(s). Integrating developments are: a) entanglement of a microscopic system with its classical environment accords it a time evolution description, the time dependent Schrödinger equation, where t is the laboratory time measured by clocks; b) canonical quantization of Special Relativity yields both the Dirac Hamiltonian and a self adjoint "time" operator, restoring to position and time the equivalent footing accorded to energy and momentum in Relativistic Quantum Mechanics. It introduces an intrinsic time property τ associated with the mass of the system, and a basis additional to the usual configuration, momentum and energy basis. As a generator of momentum displacements and consequently of energy, it invalidates Pauli's objection to the existence of a time operator. It furthermore complies with the requirements to condition the other observables in the conditional interpretation of QG. As Pauli's objection figures explicit or implicitly in most current developments of QM and QG, its invalidation opens to research the effect of this new two times perspective on such developments.
The problem of time in the quantization of gravity arises from the fact that time in Schroedinger's equation is a parameter. This sets time apart from the spatial coordinates, represented by operators in quantum mechanics (QM). Thus "time" in QM and "time" in General Relativity (GR) are seen as mutually incompatible notions. The introduction of a dy- namical time operator in relativistic quantum mechanics (RQM), that in the Heisenberg representation is also a function of the parameter t (iden- tifed as the laboratory time), prompts to examine whether it can help to solve the disfunction referred to above. In particular, its application to the conditional interpretation of the canonical quantization approach toquantum gravity is developed. 1
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.