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arXiv:2104.10151

On the Problem of Time(s) in Quantum Mechanics and Quantum Gravity: recent integrating developments and outlook

M. Bauer, Cesar Augusto Aguillon

Published Apr 20, 2021• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2104.10151

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Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.

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.

Background:In quantum mechanics energy is an observable, whereas time is a parameter. Pauli's theorem and the problem of time in canonical quantum gravity reflect this asymmetry.Material and methods:We define the time operator

\hat{t}=\hat{x}/c

and form, with the relativistic energy

\hat{E}_R

, Einstein's constant

\kappa

and the time

\hat{t}_0

in the rest system, operators of gravitational-field and curvature type. Commutation relations, wave equations and spectra are derived in the time and in the energy representation.Results:The commutator

[\hat{t},\hat{E}_R]=\tfrac{i\hbar}{c}\hat{v}_x

is bounded with norm

\hbar

and is not a multiple of the identity, so Pauli's theorem is not contradicted. The sum

c\,\hat{t}+\kappa\hat{E}_R

is unitarily equivalent to the time operator and has a continuous spectrum with explicit eigenfunctions. A postulated field equation,

\hat{G}_0+\hat{\Lambda}=\kappa\hat{E}_R

, has the structure of Einstein's equation; the term

\hat{\Lambda}

is an operator with continuous spectrum, the Ricci scalar operator contains the energy, and for small velocities the equation becomes the Schr\"odinger equation of a particle in a homogeneous field.Conclusion:Based on the postulated field equation, a quantization of the gravitational-field and curvature terms is derived: they become operators with explicit commutation relations and spectra. The metric tensor of Einstein's field equation has not been quantized.\par

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 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