We propose a relational bundle-geometric formulation of the gravitational path integral by invoking the new tool of quantum reference frames (QRFs), which in gravity are gauge-covariant coordinate systems constructed from the available field content. Formulated in terms of relational (frame-dressed) observables, this yields a manifestly gauge-invariant path integral without ghosts and anomalies, and in which observables and their correlators are local to a frame. While eliminating the need for gauge fixing, it is equivalent to Faddeev-Popov versions in which the QRF is gauge-fixed, recovering certain previous proposals. A key feature is its covariance under QRF changes: it is a perspective-neutral path integral which encodes all internal QRF perspectives and the transformations between them. This leads to several qualitative predictions: local correlators and time evolution of relational observables in one QRF perspective become fuzzy in another, and a new spectrum of relational vacua arises. Comprised of frame-dependent no-boundary and asymptotic ground states, a vacuum from one perspective appears generally excited in another. Finally, we construct gauge-invariant, yet frame-dependent effective actions by coupling sources exclusively to relational observables, setting the stage for a relational definition of renormalization.
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6 pages + appendix, 6 figures, comments welcome; v2: added missing "in gravity" to arXiv title
We propose a relational bundle-geometric formulation of the gravitational path integral by invoking the new tool of quantum reference frames (QRFs), which in gravity are gauge-covariant coordinate systems constructed from the available field content. Formulated in terms of relational (frame-dressed) observables, this yields a manifestly gauge-invariant path integral without ghosts and anomalies, and in which observables and their correlators are local to a frame. While eliminating the need for gauge fixing, it is equivalent to Faddeev-Popov versions in which the QRF is gauge-fixed, recovering certain previous proposals. A key feature is its covariance under QRF changes: it is a perspective-neutral path integral which encodes all internal QRF perspectives and the transformations between them. This leads to several qualitative predictions: local correlators and time evolution of relational observables in one QRF perspective become fuzzy in another, and a new spectrum of relational vacua arises. Comprised of frame-dependent no-boundary and asymptotic ground states, a vacuum from one perspective appears generally excited in another. Finally, we construct gauge-invariant, yet frame-dependent effective actions by coupling sources exclusively to relational observables, setting the stage for a relational definition of renormalization.
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]
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.
Given the importance of quantum reference frames (QRFs) to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of QRFs, which is valid in both fields. Here we continue with such a unifying approach, begun in [Quantum 4, 225 (2020)], whose key ingredient is a symmetry principle, which enforces physics to be relational. Thanks to gauge related redundancies, this leads to a perspective-neutral structure which contains all frame choices at once and via which frame perspectives can be consistently switched. Formulated in the language of constrained systems, the perspective-neutral structure is the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. QRF changes thus amount to a gauge transformation. We show that they take the form of `quantum coordinate changes'. We illustrate this in a general mechanical model, namely the relational N -body problem in 3D space with rotational and translational symmetry. This model is especially interesting because it features the Gribov problem so that globally valid gauge fixing conditions, and hence relational frame perspectives, are absent. The constraint surface is topologically non-trivial and foliated by 3-, 5- and 6-dimensional gauge orbits, where the lower dimensional orbits are a set of measure zero. The N -body problem also does not admit globally valid canonically conjugate pairs of Dirac observables. These challenges notwithstanding, we exhibit how one can construct the QRF transformations for the 3-body problem. Our construction also sheds new light on the generic inequivalence of Dirac and reduced quantization through its interplay with QRF perspectives.
This paper is the thirty-first in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of independently existing objects. It is the third panel — the capstone — of the eighth grouping, a foundational-physics trilogy. Its subject is spacetime localization: where and when a record or system is — its position, orientation, relative phase, and time. The claim is that localization is real but not authoritatively global. There is no Global Frame: no physically privileged reference system that fixes absolute location and orientation, a privileged global simultaneity, a classical frame standing outside quantum theory, and one physically privileged, globally valid perspectival gauge-fixing. Localization is relational: positions and times are defined relative to a declared quantum reference frame (QRF), itself a quantum system that may be superposed and entangled with what it localizes (Giacomini, Castro-Ruiz, & Brukner, 2019; Loveridge, Miyadera, & Busch, 2018). What keeps this objective rather than subjective is a strict typing of cross-frame comparison: a single gauge-invariant relational (Dirac) observable has one invariant prediction, its perspectival representations intertwined by a frame-change map (case A); distinct relational observables form a relationally consistent web (case B); and where two frames share no common refinement their claims are declared incomparable (case C). "Objective" means invariance of the physical predictions under the declared symmetry group and the frame-change groupoid over the admissible frame family, to a declared tolerance — never frame-independence in an absolute sense. The flagship R-A is a dynamics-independent kinematic existence witness: a finite-dimensional discrete-ring model exhibits one physical state and two admissible ideal QRFs under which the localization profile and the entanglement structure differ — a sharp product relative to one particle; an entangled state with a delocalized, mixed marginal relative to another — while the distribution of the same relational observable is preserved. R-B splits into a domain covariance on shared refinements (ideal QRFs compose and agree on shared Dirac predictions) and a model-dependent global-perspective obstruction: where a globally valid relational gauge-fixing fails, as in known Gribov-type models (Vanrietvelde, Höhn, & Giacomini, 2023), no single perspective covers the whole physical state space — which is neither the same as, nor a consequence of, the absence of a privileged frame, and does not preclude a perspective-neutral description. R-C places Paper 8's relational time as the time-translation special case, with its finite-resolution-clock, conditional-covariance, and record–clock qualifications, separating the retired claim (no global clock) from the deferred one (relativistic foliation with dynamical geometry). R-D identifies and transports the frame-representable localization sector of certain Paper-30 grains, partially discharging that paper's grain-coherence deferral while leaving interaction, fragmentation, redundancy, and history-family structure grain-relative. This completes the trilogy's dismantling of the Global Bookkeeper while keeping geometry fixed: the quantum-mechanics–gravity interface is prepared, not entered. Localization is frame-relative but objective: no absolute frame, no globally gauge-fixed perspective claimed beyond what models establish, and no anarchy. ALAC Paper 31. Third panel (capstone) of the eighth grouping — a foundational-physics trilogy (29 Public Quantum Records -> 30 Classicality Without a Global Classicalizer -> 31 Spacetime Localization Without a Global Frame). Part of the Consistency-Constrained Information History (Akashic Ledger Architecture Cosmology) series. Presents no new quantum mechanics; results are interpretive (L2) on imported physics (L1).
We develop the kinematical and operator-algebraic foundations of a groupoid-based relational quantum field theory (RQFT) on curved spacetimes. Indeed, the usual group-based quantum reference frame (QRF) formalism is not directly suited to generic curved Lorentzian backgrounds as global symmetry groups are typically absent or too small. We formulate a notion of QRF for a continuous groupoid. This yields a groupoid relativization map and relational observables. We show that a localization limit recovers the ordinary non-relational description. We construct canonical sharp groupoid QRFs, which form the groupoidal counterpart of the ideal group QRFs based on L2(G). We further prove that the groupoid QRF construction reduces to the standard operational QRF formalism for locally compact groups. The action groupoid QRFs are torsor QRFs only for specific classes of fields of positive operator-valued measures (POVMs), and the torsor relativization map only applies to constant operator fields of system observables.
We review the foundations of RQFT in Minkowski spacetime. We prove new results that further link RQFT to Wightman QFT. We show that covariant POVMs are μ-continuous with respect to quasi-invariant σ-finite positive Borel measures μ. Thus, relational quantum fields can be understood as the smearing of pointwise-defined kernels with respect to the QRF's statistics. We develop RQFT in curved spacetime, where we argue that the correct replacement for the Poincaré group is the Poincaré groupoid of the spacetime. We also indicate how the framework extends further to internal gauge symmetry and relational gauge-covariant quantum field theory via Atiyah groupoids, providing a first step towards formulating a relational quantum Yang-Mills field theory.
Abstract We construct relational observables in group field theory (GFT) in terms of covariant positive operator-valued measures (POVMs), using techniques developed in the context of quantum reference frames (QRFs). We focus on matter QRFs; this can be generalized to other types of frames within the same POVM-based framework. The resulting family of relational observables provides a covariant framework to extract localized observables from GFT, which is typically defined in a perspective-neutral way. Then, we compare this formalism with previous proposals for relational observables in GFT. We find that our QRF-based relational observables overcome the intrinsic limitations of previous proposals while reproducing the same continuum limit results concerning expectation values of the number and volume operators on coherent states. Nonetheless, there can be important differences for more complex operators, as well as for other types of GFT states. Finally, we also use a specific class of POVMs to show how to project states and operators from the more general perspective-neutral GFT Fock space to a perspective-dependent one where a scalar matter field plays the role of a relational clock.