Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2308.12912

Matter relative to quantum hypersurfaces

Philipp A. Höhn, Andrea Russo, Alexander R. H. Smith

Published Aug 24, 2023• 0 citations
arXivHTMLPDF

Abstract

We explore the canonical description of a scalar field as a parameterized field theory on an extended phase space that includes additional embedding fields that characterize spacetime hypersurfaces $\mathsf{X}$ relative to which the scalar field is described. This theory is quantized via the Dirac prescription and physical states of the theory are used to define conditional wave functionals $|ψ_ϕ[\mathsf{X}]\rangle$ interpreted as the state of the field relative to the hypersurface $\mathsf{X}$, thereby extending the Page-Wootters formalism to quantum field theory. It is shown that this conditional wave functional satisfies the Tomonaga-Schwinger equation, thus demonstrating the formal equivalence between this extended Page-Wootters formalism and standard quantum field theory. We also construct relational Dirac observables and define a quantum deparameterization of the physical Hilbert space leading to a relational Heisenberg picture, which are both shown to be unitarily equivalent to the Page-Wootters formalism. Moreover, by treating hypersurfaces as quantum reference frames, we extend recently developed quantum frame transformations to changes between classical and nonclassical hypersurfaces. This allows us to exhibit the transformation properties of a quantum field under a larger class of transformations, which leads to a frame-dependent particle creation effect.

DOI
10.48550/arxiv.2308.12912

Related papers

Matter relative to quantum hypersurfaces

Philipp A. Höhn, Andrea Russo, Alexander R. H. Smith

We explore the canonical description of a scalar field in 1 + 1 dimensional Minkowski space as a parametrized field theory on an extended phase space that includes additional embedding fields that characterize spacetime hypersurfaces X relative to which the scalar field is described. This theory is quantized via the Dirac prescription and physical states of the theory are used to define conditional wave functionals | ψ ϕ [ X ] ⟩ interpreted as the state of the field relative to the hypersurface X , thereby extending the Page-Wootters formalism to quantum field theory. It is shown that this conditional wave functional satisfies the Tomonaga-Schwinger equation, thus demonstrating the formal equivalence between this extended Page-Wootters formalism and standard quantum field theory. We also construct relational Dirac observables and define a quantum deparametrization of the physical Hilbert space leading to a relational Heisenberg picture, which are both shown to be unitarily equivalent to the Page-Wootters formalism. Moreover, by treating hypersurfaces as quantum reference frames, we extend recently developed quantum frame transformations to changes between classical and nonclassical hypersurfaces. This allows us to exhibit the transformation properties of a quantum field under a larger class of transformations, which leads to a frame-dependent particle creation effect. Published by the American Physical Society 2024

Physical review. D/Physical review. D.•May 6, 2024•15 citations

Interacting quantum reference frames

Leonardo Chataignier, Alexander R. H. Smith

We formulate quantum reference frames within the Dirac quantization of gauge theories, where physical states are obtained from a kinematical Hilbert space $\mathcal{H}_{\rm kin}$ 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 $G_R$ of frame orientations acting on a tensor factor of $\mathcal{H}_{\rm kin}$. 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 $\mathcal{H}_{\rm kin}$. 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.

quant-pharXiv:2610.28732•Oct 22, 2026•0 citations

Spacetime Quantum Reference Frames and superpositions of proper times

Flaminia Giacomini

In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.

arXiv:2101.11628•Quantum•Jul 22, 2021•74 citations

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

Philipp A. Höhn, Alexander R. H. Smith, Maximilian P. E. Lock

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.

Frontiers in Physics•Jun 18, 2021•79 citations

Relational observables, reference frames, and conditional probabilities

Leonardo Chataignier

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.

arXiv:2006.05526•Physical review. D/Physical review. D.•Jan 19, 2021•26 citations

Quantum Reference Fields Transformations in Linearized Quantum Gravity

Lin-Qing Chen, Flaminia Giacomini

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.

gr-qcCross-listedarXiv:2606.09344•Jun 9, 2026
We explore the canonical description of a scalar field as a parameterized field theory on an extended phase space that includes additional embedding fields that characterize spacetime hypersurfaces X\mathsf{X}X relative to which the scalar field is described. This theory is quantized via the Dirac prescription and physical states of the theory are used to define conditional wave functionals ∣ψϕ[X]⟩|ψ_ϕ[\mathsf{X}]\rangle∣ψϕ​[X]⟩ interpreted as the state of the field relative to the hypersurface X\mathsf{X}X, thereby extending the Page-Wootters formalism to quantum field theory. It is shown that this conditional wave functional satisfies the Tomonaga-Schwinger equation, thus demonstrating the formal equivalence between this extended Page-Wootters formalism and standard quantum field theory. We also construct relational Dirac observables and define a quantum deparameterization of the physical Hilbert space leading to a relational Heisenberg picture, which are both shown to be unitarily equivalent to the Page-Wootters formalism. Moreover, by treating hypersurfaces as quantum reference frames, we extend recently developed quantum frame transformations to changes between classical and nonclassical hypersurfaces. This allows us to exhibit the transformation properties of a quantum field under a larger class of transformations, which leads to a frame-dependent particle creation effect.
We explore the canonical description of a scalar field in 1 + 1 dimensional Minkowski space as a parametrized field theory on an extended phase space that includes additional embedding fields that characterize spacetime hypersurfaces X relative to which the scalar field is described. This theory is quantized via the Dirac prescription and physical states of the theory are used to define conditional wave functionals | ψ ϕ [ X ] ⟩ interpreted as the state of the field relative to the hypersurface X , thereby extending the Page-Wootters formalism to quantum field theory. It is shown that this conditional wave functional satisfies the Tomonaga-Schwinger equation, thus demonstrating the formal equivalence between this extended Page-Wootters formalism and standard quantum field theory. We also construct relational Dirac observables and define a quantum deparametrization of the physical Hilbert space leading to a relational Heisenberg picture, which are both shown to be unitarily equivalent to the Page-Wootters formalism. Moreover, by treating hypersurfaces as quantum reference frames, we extend recently developed quantum frame transformations to changes between classical and nonclassical hypersurfaces. This allows us to exhibit the transformation properties of a quantum field under a larger class of transformations, which leads to a frame-dependent particle creation effect. Published by the American Physical Society 2024
We formulate quantum reference frames within the Dirac quantization of gauge theories, where physical states are obtained from a kinematical Hilbert space Hkin\mathcal{H}_{\rm kin}Hkin​ by group averaging. We assume the gauge group GGG is unimodular, and we define a quantum reference frame as a system of coherent states associated with a group GRG_RGR​ of frame orientations acting on a tensor factor of Hkin\mathcal{H}_{\rm kin}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 GGG does not act as a tensor-product representation across a factorization of Hkin\mathcal{H}_{\rm kin}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.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
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.
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.
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.