We discuss the construction of relational observables in time-reparametrization invariant quantum mechanics, and we argue that their physical interpretation can be understood in terms of conditional probabilities, which are defined from the solutions of the quantum constraint equation in a generalization of the Page-Wootters formalism. In this regard, we show how conditional expectation values of worldline tensor fields are related to quantum averages of suitably defined relational observables. We also comment on how the dynamics of these observables can be related to a notion of quantum reference frames. After presenting the general formalism, we analyze a recollapsing cosmological model, for which we construct unitarily evolving quantum relational observables. We conclude with some remarks about the relevance of these results for the construction and interpretation of diffeomorphism-invariant operators in quantum gravity.
We discuss the construction of relational observables in time-reparametrization invariant quantum mechanics, and we argue that their physical interpretation can be understood in terms of conditional probabilities, which are defined from the solutions of the quantum constraint equation in a generalization of the Page-Wootters formalism. In this regard, we show how conditional expectation values of worldline tensor fields are related to quantum averages of suitably defined relational observables. We also comment on how the dynamics of these observables can be related to a notion of quantum reference frames. After presenting the general formalism, we analyze a recollapsing cosmological model, for which we construct unitarily evolving quantum relational observables. We conclude with some remarks about the relevance of these results for the construction and interpretation of diffeomorphism-invariant operators in quantum gravity.
We formulate quantum reference frames within the Dirac quantization of gauge theories, where physical states are obtained from a kinematical Hilbert space Hkin by group averaging. We assume the gauge group G is unimodular, and we define a quantum reference frame as a system of coherent states associated with a group GR of frame orientations acting on a tensor factor of Hkin. In direct analogy with the Faddeev-Popov procedure in path integrals, we show that the physical inner product can be written in terms of a choice of frame orientation with an appropriate operator measure, but that the physical Hilbert space is independent of this choice of frame. This yields a generalization of the Page-Wootters reduction map used to define conditional states and conditional probabilities of measurement outcomes given a frame orientation. These probabilities correspond to the expectation values of relational Dirac observables obtained from generalizations of previously considered relativization maps. Our main result is that the map that implements a change of frame is unitary, even when G does not act as a tensor-product representation across a factorization of Hkin. The formalism therefore accommodates interactions between reference frames and the systems described relative to them. We apply this formalism to temporal frames, showing that a unitary Schrödinger equation emerges for an interacting clock and system, and that a previously reported time-nonlocal Schrödinger equation is recovered in an appropriate limit. With this, problems of non-unitarity are avoided by the use of the gauge-fixing measure in the physical inner product. We also discuss the case of interacting spatial frames, where the known correspondence between coherence in one frame and entanglement in another is shown to depend on the coupling.
The problem of time in quantum gravity calls for a relational solution. Using quantum reduction maps, we establish a previously unknown equivalence between three approaches to relational quantum dynamics: (1) relational observables in the clock-neutral picture of Dirac quantization, (2) Page and Wootters' (PW) Schr\"odinger picture formalism, and (3) the relational Heisenberg picture obtained via symmetry reduction. Constituting three faces of the same dynamics, we call this equivalence the trinity. In the process, we develop a quantization procedure for relational Dirac observables using covariant positive operator-valued measures which encompass nonideal clocks and resolve the nonmonotonicity issue of realistic quantum clocks reported by Unruh and Wald. The quantum reduction maps reveal this procedure as the quantum analog of gauge-invariantly extending gauge-fixed quantities. We establish algebraic properties of these relational observables. We extend a recent ``clock-neutral'' approach to changing temporal reference frames, transforming relational observables and states, and demonstrate a clock dependent temporal nonlocality effect. We show that Kucha\ifmmode \check{r}\else \v{r}\fi{}'s criticism, alleging that the conditional probabilities of the PW formalism violate the constraint, is incorrect. They are a quantum analog of a gauge-fixed description of a gauge-invariant quantity and equivalent to the manifestly gauge-invariant evaluation of relational observables in the physical inner product. The trinity furthermore resolves a previously reported normalization ambiguity and clarifies the role of entanglement in the PW formalism. The trinity finally permits us to resolve Kucha\ifmmode \check{r}\else \v{r}\fi{}'s criticism that the PW formalism yields wrong propagators by showing how conditional probabilities of relational observables give the correct transition probabilities. Unlike previous proposals, our resolution does not invoke approximations, ideal clocks or ancilla systems, is manifestly gauge invariant, and easily extends to an arbitrary number of conditionings.
Diffeomorphism invariance is a central feature of general relativity. Without external reference structures, matter and geometry must be specified relationally, with respect to internal subsystems serving as reference frames. In quantum gravity, these reference systems must themselves be treated as quantum, motivating the use of quantum reference frames. In this work, we address how such a relational description could be formulated within linearized quantum gravity. To this purpose, we introduce quantum reference fields, i.e.,sets of four dynamical scalar fields whose stress-energy tensors enter the gravitational constraints. These fields extend the notion of quantum reference frames to local field-theoretic reference systems, allowing matter and gravitational degrees of freedom to be described relationally with respect to physical quantum systems whose states cannot be sharply peaked on definite field configurations. By generalizing the perspective-neutral construction of quantum reference frames, we show that relational, gauge invariant observables admit reduced descriptions in the perspective of each quantum reference field, and we derive the unitary transformations relating them. The resulting unitary maps implement local quantum coordinate changes between different internal perspectives, and act on the linearized gravitational field with an analogous structure to a linearized diffeomorphism, but with the classical gauge parameter replaced by a physical quantum field. Finally, we construct a relational von Neumann-type measurement scheme, showing how the corresponding reduced observables can be accessed operationally from the perspective of a quantum reference field.
Abstract We discuss a systematic way in which a relational dynamics can be established relative to periodic clocks both in the classical and quantum theories, emphasising the parallels between them. We show that: (1) classical and quantum relational observables that encode the value of a quantity relative to a periodic clock are only invariant along the gauge orbits generated by the Hamiltonian constraint if the quantity itself is periodic, and otherwise the observables are only transiently invariant per clock cycle (this implies, in particular, that counting winding numbers does not lead to invariant observables relative to the periodic clock); (2) the quantum relational observables can be obtained from a partial group averaging procedure over a single clock cycle; (3) there is an equivalence (‘trinity’) between the quantum theories based on the quantum relational observables of the clock-neutral picture of Dirac quantisation, the relational Schrodinger picture of the Page–Wootters(PW) formalism, and the relational Heisenberg picture that follows from quantum deparameterisation, all three taken relative to periodic clocks (implying that the dynamics in all three is necessarily periodic); (4) in the context of periodic clocks, the original PW definition of conditional probabilities fails for systems that have a continuous energy spectrum and, using the equivalence between the PW and the clock-neutral, gauge-invariant formalism, must be suitably updated. Finally, we show how a system evolving periodically with respect to a periodic clock can evolve monotonically with respect to an aperiodic clock, without inconsistency. The presentation is illustrated by several examples, and we conclude with a brief comparison to other approaches in the literature that also deal with relational descriptions of periodic clocks.
We combine the ``evolving constants'' approach to the construction of observables in canonical quantum gravity with the Page-Wootters formulation of quantum mechanics with a relational time for generally covariant systems. This overcomes the objections levied by Kucha\ifmmode \check{r}\else \v{r}\fi{} against the latter formalism. The construction is formulated entirely in terms of Dirac observables, avoiding in all cases the physical observation of quantities that do not belong in the physical Hilbert space. We work out explicitly the example of the parametrized particle, including the calculation of the propagator. The resulting theory also predicts a fundamental mechanism of decoherence.
It is mandatory to know how to operationally define and translate a reference frame into mathematics, in order that a physical interpretation of theory calculations in terms of observational data is possible. The situation is particularly challenging for gauge systems such as General Relativity where spacetime coordinates are subject to spacetime diffeomorphisms considered as gauge transformations turning coordinates into non-observables. This motivates the idea of operationally defined (material) reference frames which specify coordinates in terms of matter or geometry reference fields leading to the concept of relational observables, relational reference frames and gauge reduction. Upon quantisation, all fields become operator valued distributions. Now new conceptual and technical questions arise such as: Should one reduce before or after quantisation and how are the reference fields quantised respectively in either route? Is a reference frame itself subject to quantisation and how are different quantum reference frames related? How does the gauge reduction fit into this, i.e. how can it be that a certain reference field is considered a non-observable in one reference frame and an observable in another which upon quantisation even displays fluctuations? How precisely are gauge dependent fields interpreted in terms of the relational observables in a given reference frame? What is the relative dynamics, e.g. how exactly are physical Hamiltonians of two relational reference frames related? The present conceptual work addresses these and related questions in a non-perturbative field theory context of sufficient generality to cover General Relativity coupled to standard matter. A central role is played by the concept of the relational reference frame transformation (RRFT) for which a general formula is derived and its properties are explored.