Abstract
We have previously shown that three approaches to relational quantum dynamics—relational Dirac observables, the Page-Wootters formalism and quantum deparametrizations—are equivalent. Here we show that this “trinity” of relational quantum dynamics holds in relativistic settings per frequency superselection sector. Time according to a clock subsystem is defined via a positive operator-valued measure (POVM) that is covariant with respect to the group generated by its (quadratic) Hamiltonian. This differs from the usual choice of a self-adjoint clock observable conjugate to the clock momentum. It also resolves Kuchař's criticism that the Page-Wootters formalism yields incorrect localization probabilities for the relativistic particle when conditioning on a Minkowski time operator. We show that conditioning instead on the covariant clock POVM results in a Newton-Wigner type localization probability commonly used in relativistic quantum mechanics. By establishing the equivalence mentioned above, we also assign a consistent conditional-probability interpretation to relational observables and deparametrizations. Finally, we expand a recent method of changing temporal reference frames, and show how to transform states and observables frequency-sector-wise. We use this method to discuss an indirect clock self-reference effect and explore the state and temporal frame-dependence of the task of comparing and synchronizing different quantum clocks.
Highlights
In general relativity, time plays a different role than in classical and quantum mechanics, or quantum field theory on a Minkowski background
As noted in the introduction, we showed in Höhn et al [7] that three formulations of relational quantum dynamics, namely (i) quantum relational Dirac observables, (ii) the relational Schrödinger picture of the Page-Wootters formalism, and (iii) the relational Heisenberg picture obtained through quantum deparametrization, are equivalent for models described by the Hamiltonian constraint in Equation (9) when the clock Hamiltonian has a continuous, non-degenerate spectrum; the three formulations form a trinity of relational quantum dynamics
In this work we demonstrated the equivalence of three distinct approaches to relational quantum dynamics—relational Dirac observables, the Page-Wootters formalism, and quantum deparametrizations—for models described by a Hamiltonian constraint in which the momentum of the system being employed as a clock appears quadratically
Summary
Time plays a different role than in classical and quantum mechanics, or quantum field theory on a Minkowski background. This contrasts with the usual approach of employing an operator conjugate to the clock momentum (i.e., the Minkowski time operator in the case of a relativistic particle) This covariant clock POVM is instrumental in our resolution of Kuchar’s criticism (a) that the Page-Wootters formalism yields wrong localization probabilities for relativistic systems. We apply these temporal frame change maps to explore an indirect clock self-reference and the temporal frame and state dependence of comparing and synchronizing readings of different quantum clocks 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 PageWootters formalism into a more general setting.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.