Construction of quantum Dirac observables and the emergence of WKB time

Abstract

We describe a method of construction of gauge-invariant operators (Dirac observables or "evolving constants of motion") from the knowledge of the eigenstates of the gauge generator of time-reparametrisation invariant mechanical systems. These invariant operators evolve unitarily with respect to an arbitrarily chosen time variable. We emphasise that the dynamics is relational, both in the classical and quantum theories. In this framework, we show how the "emergent WKB time" often employed in quantum cosmology arises from a weak-coupling expansion of invariant transition amplitudes, and we illustrate an example of singularity avoidance in a vacuum Bianchi I (Kasner) model. * Gauge conditions of the form given in (5) are sufficient for our purposes, although more general gauge conditions are possible [12].† This corresponds to the statement that invariant extensions are obtained by writing gauge-fixed quantities in an arbitrary gauge.

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

“…11 We also refer the reader to the recent work [38], which we became aware of while completing this manuscript. It extends some of the results of Höhn et al [7] as well, though using the different formalism developed in Chataignier [37]. It also does not employ covariant clocks in the case of quadratic clock Hamiltonians.…”

“…The quantization of relational observables is non-trivial, especially because Equation (4) may not be globally defined on C, and depends very much on the properties of the chosen clock. Steps toward systematically quantizing relational Dirac observables have been undertaken (e.g., in [7,17,20,37,38]) and part of this article is devoted to further developing them for a class of relativistic models.…”

“…Such a group averaging is known as the G-twirl operation and we denote it G as in the last line of Equation (31). G-twirl operations have previously been mostly studied in the context of spatial quantum reference frames, e.g., see [130][131][132], but have also appeared in some constructions of quantum Dirac observables (e.g., see [7,11,37,38,106]) 12 . As discussed in Höhn et al [7], this G-twirl constitutes the quantum analog of a gauge-invariant extension of a gauge-fixed quantity.…”

“…By contrast, let us now exhibit what form of conditional probabilities the covariant clock POVM E T of section 4.2 gives rise to. We now insert e C (τ ) = σ |τ , σ τ , σ | and, as before, e x into the conditional probability Equation (37). The crucial difference between the covariant clock POVM E T and the clock operatort (which is covariant with respect toĈ σ , but notĈ H ) is that the denominator of Equation ( 37) is equal to the physical inner product in the former case (see Corollary 2) but not in the latter 22 .…”

“…While completing this manuscript, we became aware of Chataignier [38], which independently extends some results of Höhn et al [7] on the conditional probability interpretation of relational observables and their equivalence with the Page-Wootters formalism into a more general setting. However, a different formalism [37] is used in Chataignier [38], which does not employ covariant clock POVMs and therefore the two works complement one another.…”