arXiv subjects

Philipp A. Hoehn

Publications and source records attributed to Philipp A. Hoehn.

34 records · Page 2Linked to original sources

arXiv2022

Internal quantum reference frames for finite Abelian groups

Employing internal quantum systems as reference frames is a crucial concept in quantum gravity, gauge theories and quantum foundations whenever external relata are unavailable. In this work, we give a comprehensive and self-contained treatment of such quantum reference frames (QRFs) for the case when the underlying configuration space is a finite Abelian group, significantly extending our previous work (Quantum 5, 530 (2021)). The simplicity of this setup admits a fully rigorous quantum information-theoretic analysis, while maintaining sufficient structure for exploring many of the conceptual and structural questions also pertinent to more complicated setups. We exploit this to derive several important structures of constraint quantization with quantum information-theoretic methods and to reveal the relation between different approaches to QRF covariance. In particular, we characterize the "physical Hilbert space" -- the arena of the "perspective-neutral" approach -- as the maximal subspace that admits frame-independent descriptions of purifications of states. We then demonstrate the kinematical equivalence and, surprising, dynamical inequivalence of the "perspective-neutral" and the "alignability" approach to QRFs. While the former admits unitaries generating transitions between arbitrary subsystem relations, the latter, remarkably, admits no such dynamics when requiring symmetry-preservation. We illustrate these findings by example of interacting discrete particles, including how dynamics can be described "relative to one of the subsystems".

quant-ph↗

arXiv2022

Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance

We describe a completely general and fully non-perturbative framework for constructing dynamical reference frames in generally covariant theories, and for understanding the gauge-invariant observables that they yield. Our approach makes use of a 'universal dressing space', which contains as a subset every possible dynamical frame. We describe examples of such frames, including matter frames, a popular construction based on boundary-anchored geodesics and one using minimal surfaces -- but our formalism does not depend on the existence of a boundary. The class of observables we construct generalises and unifies the dressed and relational approaches to constructing gravitational observables, including single-integral and canonical power-series constructions. All these (possibly gravitationally charged) relational observables describe physics in a precise sense relative to the dynamical frame and respect a notion of 'relational' locality based on the relationships between fields. By using 'relational atlases', i.e. collections of dynamical frames glued together by field-dependent maps (which are relational observables too), we can construct relationally local observables throughout spacetime. This further establishes a framework for dynamical frame covariance that permits us to change between arbitrary relational frame perspectives. Relational locality obeys many desirable properties: we prove that it satisfies microcausality in the bulk (in tension with previous work done mainly in a perturbative setting which we comment on), and show that it permits a relational version of local bulk dynamics. Relational locality is therefore arguably more physically meaningful than the ordinary notion of locality. Thus, our formalism -- which we argue to be an updated, gauge-invariant version of general covariance -- refutes the commonly claimed non-existence of local gravitational bulk physics.

hep-th↗

arXiv2022

Quantum Relativity of Subsystems

One of the most basic notions in physics is the partitioning of a system into subsystems, and the study of correlations among its parts. In this work, we explore these notions in the context of quantum reference frame (QRF) covariance, in which this partitioning is subject to a symmetry constraint. We demonstrate that different reference frame perspectives induce different sets of subsystem observable algebras, which leads to a gauge-invariant, frame-dependent notion of subsystems and entanglement. We further demonstrate that subalgebras which commute before imposing the symmetry constraint can translate into non-commuting algebras in a given QRF perspective after symmetry imposition. Such a QRF perspective does not inherit the distinction between subsystems in terms of the corresponding tensor factorizability of the kinematical Hilbert space and observable algebra. Since the condition for this to occur is contingent on the choice of QRF, the notion of subsystem locality is frame-dependent.

quant-ph↗

arXiv2022

Edge modes as reference frames and boundary actions from post-selection

We introduce a general framework realizing edge modes in (classical) gauge field theory as dynamical reference frames, an often suggested interpretation that we make entirely explicit. We focus on a bounded region $M$ with a co-dimension one time-like boundary $Γ$, which we embed in a global spacetime. Taking as input a variational principle at the global level, we develop a systematic formalism inducing consistent variational principles (and in particular, boundary actions) for the subregion $M$. This relies on a post-selection procedure on $Γ$, which isolates the subsector of the global theory compatible with a general choice of gauge-invariant boundary conditions for the dynamics in $M$. Crucially, the latter relate the configuration fields on $Γ$ to a dynamical frame field carrying information about the spacetime complement of $M$; as such, they may be equivalently interpreted as frame-dressed or relational observables. Generically, the external frame field keeps an imprint on the ensuing dynamics for subregion $M$, where it materializes itself as a local field on the time-like boundary $Γ$; in other words, an edge mode. We identify boundary symmetries as frame reorientations and show that they divide into three types, depending on the boundary conditions, that affect the physical status of the edge modes. Our construction relies on the covariant phase space formalism, and is in principle applicable to any gauge (field) theory. We illustrate it on three standard examples: Maxwell, Abelian Chern-Simons and non-Abelian Yang-Mills theories. In complement, we also analyze a mechanical toy-model to connect our work with recent efforts on (quantum) reference frames.

hep-th↗

arXiv2021

Perspective-neutral approach to quantum frame covariance for general symmetry groups

In the absence of external relata, internal quantum reference frames (QRFs) appear widely in the literature on quantum gravity, gauge theories and quantum foundations. Here, we extend the perspective-neutral approach to QRF covariance to general unimodular Lie groups. This is a framework that links internal QRF perspectives via a manifestly gauge-invariant Hilbert space in the form of "quantum coordinate transformations", and we clarify how it is a quantum extension of special covariance. We model the QRF orientations as coherent states which give rise to a covariant POVM, furnishing a consistent probability interpretation and encompassing non-ideal QRFs whose orientations are not perfectly distinguishable. We generalize the construction of relational observables, establish a variety of their algebraic properties and equip them with a transparent conditional probability interpretation. We import the distinction between gauge transformations and physical symmetries from gauge theories and identify the latter as QRF reorientations. The "quantum coordinate maps" into an internal QRF perspective are constructed via a conditioning on the QRF's orientation, generalizing the Page-Wootters formalism and a symmetry reduction procedure. We find two types of QRF transformations: gauge induced "quantum coordinate transformations" as passive unitary changes of description and symmetry induced active changes of relational observables from one QRF to another. We reveal new effects: (i) QRFs with non-trivial orientation isotropy groups can only resolve isotropy-group-invariant properties of other subsystems; (ii) in the absence of symmetries, the internal perspective Hilbert space "rotates" through the kinematical subsystem Hilbert space as the QRF changes orientation. Finally, we invoke the symmetries to generalize the quantum relativity of subsystems before comparing with other approaches. [Abridged]

quant-ph↗

arXiv2021

Quantum reference frame transformations as symmetries and the paradox of the third particle

In a quantum world, reference frames are ultimately quantum systems too -- but what does it mean to "jump into the perspective of a quantum particle"? In this work, we show that quantum reference frame (QRF) transformations appear naturally as symmetries of simple physical systems. This allows us to rederive and generalize known QRF transformations within an alternative, operationally transparent framework, and to shed new light on their structure and interpretation. We give an explicit description of the observables that are measurable by agents constrained by such quantum symmetries, and apply our results to a puzzle known as the `paradox of the third particle'. We argue that it can be reduced to the question of how to relationally embed fewer into more particles, and give a thorough physical and algebraic analysis of this question. This leads us to a generalization of the partial trace (`relational trace') which arguably resolves the paradox, and it uncovers important structures of constraint quantization within a simple quantum information setting, such as relational observables which are key in this resolution. While we restrict our attention to finite Abelian groups for transparency and mathematical rigor, the intuitive physical appeal of our results makes us expect that they remain valid in more general situations.

quant-ph↗

arXiv2021

The Trinity of Relational Quantum Dynamics

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) Schrödinger 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. We develop a quantization procedure for relational Dirac observables using covariant POVMs which encompass non-ideal clocks. 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 Kuchař'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 Kuchař'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.

quant-ph↗

arXiv2020

Equivalence of approaches to relational quantum dynamics in relativistic settings

We have previously shown (arXiv:1912.00033) 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. We ascribe the time according to the clock subsystem to a POVM which is covariant with respect to 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.

gr-qc↗

arXiv2018

An operational approach to spacetime symmetries: Lorentz transformations from quantum communication

In most approaches to fundamental physics, spacetime symmetries are postulated a priori and then explicitly implemented in the theory. This includes Lorentz covariance in quantum field theory and diffeomorphism invariance in quantum gravity, which are seen as fundamental principles to which the final theory has to be adjusted. In this paper, we suggest, within a much simpler setting, that this kind of reasoning can actually be reversed, by taking an operational approach inspired by quantum information theory. We consider observers in distinct laboratories, with local physics described by the laws of abstract quantum theory, and without presupposing a particular spacetime structure. We ask what information-theoretic effort the observers have to spend to synchronize their descriptions of local physics. If there are "enough" observables that can be measured universally on several different quantum systems, we show that the observers' descriptions are related by an element of the orthochronous Lorentz group O^+(3,1), together with a global scaling factor. Not only does this operational approach predict the Lorentz transformations, but it also accurately describes the behavior of relativistic Stern-Gerlach devices in the WKB approximation, and it correctly predicts that quantum systems carry Lorentz group representations of different spin. This result thus hints at a novel information-theoretic perspective on spacetime.

quant-ph↗

arXiv2017

Can chaos be observed in quantum gravity?

Full general relativity is almost certainly 'chaotic'. We argue that this entails a notion of nonintegrability: a generic general relativistic model, at least when coupled to cosmologically interesting matter, likely possesses neither differentiable Dirac observables nor a reduced phase space. It follows that the standard notion of observable has to be extended to include non-differentiable or even discontinuous generalized observables. These cannot carry Poisson-algebraic structures and do not admit a standard quantization; one thus faces a quantum representation problem of gravitational observables. This has deep consequences for a quantum theory of gravity, which we investigate in a simple model for a system with Hamiltonian constraint that fails to be completely integrable. We show that basing the quantization on standard topology precludes a semiclassical limit and can even prohibit any solutions to the quantum constraints. Our proposed solution to this problem is to refine topology such that a complete set of Dirac observables becomes continuous. In the toy model, it turns out that a refinement to a polymer-type topology, as e.g. used in loop gravity, is sufficient. Basing quantization of the toy model on this finer topology, we find a complete set of quantum Dirac observables and a suitable semiclassical limit. This strategy is applicable to realistic candidate theories of quantum gravity and thereby suggests a solution to a long-standing problem which implies ramifications for the very concept of quantization. Our work reveals a qualitatively novel facet of chaos in physics and opens up a new avenue of research on chaos in gravity which hints at deep insights into the structure of quantum gravity.

gr-qc↗

arXiv2015

Chaos, Dirac observables and constraint quantization

There is good evidence that full general relativity is non-integrable or even chaotic. We point out the severe repercussions: differentiable Dirac observables and a reduced phase space do not exist in non-integrable constrained systems and are thus unlikely to occur in a generic general relativistic context. Instead, gauge invariant quantities generally become discontinuous, thus not admitting Poisson-algebraic structures and posing serious challenges to a quantization. Non-integrability also renders the paradigm of relational dynamics cumbersome, thereby straining common interpretations of the dynamics. We illustrate these conceptual and technical challenges with simple toy models. In particular, we exhibit reparametrization invariant models which fail to be integrable and, as a consequence, can either not be quantized with standard methods or lead to sick quantum theories without a semiclassical limit. These troubles are qualitatively distinct from semiclassical subtleties in unconstrained quantum chaos and can be directly traced back to the scarcity of Dirac observables. As a possible resolution, we propose to change the method of quantization by refining the configuration space topology until the generalized observables become continuous in the new topology and can acquire a quantum representation. This leads to the polymer quantization method underlying loop quantum cosmology and gravity. Remarkably, the polymer quantum theory circumvents the problems of the quantization with smooth topology, indicating that non-integrability and chaos, while a challenge, may not be a fundamental obstruction for quantum gravity.

gr-qc↗

arXiv2014

Canonical linearized Regge Calculus: counting lattice gravitons with Pachner moves

We afford a systematic and comprehensive account of the canonical dynamics of 4D Regge Calculus perturbatively expanded to linear order around a flat background. To this end, we consider the Pachner moves which generate the most basic and general simplicial evolution scheme. The linearized regime features a vertex displacement (`diffeomorphism') symmetry for which we derive an abelian constraint algebra. This permits to identify gauge invariant `lattice gravitons' as propagating curvature degrees of freedom. The Pachner moves admit a simple method to explicitly count the gauge and `graviton' degrees of freedom on an evolving triangulated hypersurface and we clarify the distinct role of each move in the dynamics. It is shown that the 1-4 move generates four `lapse and shift' variables and four conjugate vertex displacement generators; the 2-3 move generates a `graviton'; the 3-2 move removes one `graviton' and produces the only non-trivial equation of motion; and the 4-1 move removes four `lapse and shift' variables and trivializes the four conjugate symmetry generators. It is further shown that the Pachner moves preserve the vertex displacement generators. These results may provide new impetus for exploring `graviton dynamics' in discrete quantum gravity models.

gr-qc↗

arXiv2014

Classification of constraints and degrees of freedom for quadratic discrete actions

We provide a comprehensive classification of constraints and degrees of freedom for variational discrete systems governed by quadratic actions. This classification is based on the different types of null vectors of the Lagrangian two-form and employs the canonical formalism developed in arXiv:1303.4294 [math-ph] (J. Math. Phys. 54, 093505 (2013)) and arXiv:1401.6062 [gr-qc] (J. Math. Phys. 55, 083508 (2014)). The analysis is carried out in both the classical and quantum theory and applies to systems with both temporally varying or constant discretization. In particular, it is shown explicitly how changes in the discretization, e.g. resulting from canonical coarse graining or refining operations or an evolving background geometry, change the dynamical content of the system. It is demonstrated how, on a temporally varying discretization, constraints, Dirac observables, symmetries, reduced phase spaces and physical Hilbert spaces become spacetime region dependent. These results are relevant for free field theory on an evolving lattice and linearized discrete gravity models.

math-ph↗

arXiv2014

Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces

A temporally varying discretization often features in discrete gravitational systems and appears in lattice field theory models subject to a coarse graining or refining dynamics. To better understand such discretization changing dynamics in the quantum theory, an according formalism for constrained variational discrete systems is constructed. While the present manuscript focuses on global evolution moves and, for simplicity, restricts to Euclidean configuration spaces, a companion article discusses local evolution moves. In order to link the covariant and canonical picture, the dynamics of the quantum states is generated by propagators which satisfy the canonical constraints and are constructed using the action and group averaging projectors. This projector formalism offers a systematic method for tracing and regularizing divergences in the resulting state sums. Non-trivial coarse graining evolution moves lead to non-unitary, and thus irreversible, projections of physical Hilbert spaces and Dirac observables such that these concepts become evolution move dependent on temporally varying discretizations. The formalism is illustrated in a toy model mimicking a `creation from nothing'. Subtleties arising when applying such a formalism to quantum gravity models are discussed.

gr-qc↗

arXiv2011

Effective relational dynamics

We provide a synopsis of an effective approach to the problem of time in the semiclassical regime. The essential features of this new approach to evaluating relational quantum dynamics in constrained systems are illustrated by means of a simple toy model.

gr-qc↗

arXiv2011

Canonical formalism for simplicial gravity

We summarise a recently introduced general canonical formulation of discrete systems which is fully equivalent to the covariant formalism. This framework can handle varying phase space dimensions and is applied to simplicial gravity in particular.

gr-qc↗