How to switch between relational quantum clocks

Abstract

Every clock is a physical system and thereby ultimately quantum. A naturally arising question is thus how to describe time evolution relative to quantum clocks and, specifically, how the dynamics relative to different quantum clocks are related. This is a particularly pressing issue in view of the multiple choice facet of the problem of time in quantum gravity, which posits that there is no distinguished choice of internal clock in generic general relativistic systems and that different choices lead to inequivalent quantum theories. Exploiting a recent unifying approach to switching quantum reference systems [Vanrietvelde et al 2020 Quantum 4 225; Vanrietvelde et al 2018 arXiv:1809.05093[quant-ph])], we exhibit a systematic method for switching between different clock choices in the quantum theory. We illustrate it by means of the parametrized particle, which, like gravity, features a Hamiltonian constraint. We explicitly switch between the quantum evolution relative to the non-relativistic time variable and that relative to the particle’s position, which requires carefully regularizing the zero-modes in the so-called time-of-arrival observable. While this toy model is simple, our approach is general and, in particular, directly amenable to quantum cosmology. It proceeds by systematically linking the reduced quantum theories relative to different clock choices via the clock-choice-neutral Dirac quantized theory, in analogy to coordinate changes on a manifold. This method suggests a new perspective on the multiple choice problem, indicating that it is rather a multiple choice feature of the complete relational quantum theory, taken as the conjunction of Dirac quantized and quantum deparametrized theories. Precisely this conjunction permits one to consistently switch between different temporal reference systems, which is a prerequisite for a quantum notion of general covariance. Finally, we show that quantum uncertainties generically lead to a discontinuity in the relational dynamics when switching clocks, in contrast to the classical case.

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

“…Accordingly, choosing a dynamical reference system amounts to removing redundancy from the description. As such, we may interpret the redundancy-containing description (in both the classical and quantum theory) as a perspectiveneutral description of physics, i.e., as a global description of physics prior to having chosen a reference system, from whose perspective the remaining degrees of freedom are to be described [7,30,31,71,72]. This perspective-neutral structure is thus proposed as the reference-frame-neutral structure for dynamical (i.e., non-idealized) reference systems.…”

“…In this article we focus purely on temporal diffeomorphisms and thus on temporal reference frames/systems, or simply clocks. In this case, we refer to the perspective-neutral structure as a clock-(choice-)neutral structure [7,30,31], which we briefly review here in both the classical and quantum theory. It is a description of the physics, prior to having chosen a temporal reference system relative to which the dynamics of the remaining degrees of freedom are to be described.…”

“…Being of odd dimension dim P kin − 1, C is also not a phase space. A proper phase space description can be obtained, e.g., through phase space reduction by gauge-fixing [7,30,31,38,58]. Given a choice of clock function T, we may consider the gaugefixing condition T = const, which may be valid only locally on C. Since F f ,T (τ ) is constant along each orbit generated by C H for each value of τ , we do not lose any information about the relational dynamics by restricting to T = const and leaving τ free.…”

“…(i) a Dirac quantization scheme, wherein relational observables are constructed that encode correlations between evolving and clock degrees of freedom [1,2,4,, (ii) the Page-Wootters formalism, which defines a relational dynamics in terms of conditional probabilities for clock and evolving degrees of freedom [7,25,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57], and (iii) classical or quantum deparametrizations, which result in a reduced quantum theory that only treats the evolving degrees of freedom as quantum [1,2,7,10,30,31,[58][59][60][61][62][63][64][65].…”

“…The transformations between (i)-(iii) of the trinity also allowed us to address the multiple choice problem in Höhn et al [7] by extending a previous method for changing temporal reference frames, i.e., clocks, in the quantum theory [30,31,47] (see also [32][33][34]69]). The resolution to the problem lies in part in realizing that a solution to the Wheeler-DeWitt equation encodes the relations between all subsystems, including the relations between subsystems employed as clocks to track the dynamics of other subsystems; there are multiple choices of clocks, each of which can be used to define dynamics.…”