This paper is a duplicated. Updated and complete version available at https://doi.org/10.5281/zenodo.21591128" I determine the behaviour, under a change of internal quantum reference frame (QRF), of the geometry and gauge structure obtained from the mixed-state quantum geometric tensor (msQGT) of a reduced density matrix ρObs. The setting is the Relational–Informational Model (RIM), in which the emergent metric and gauge field are the real and imaginary sectors of the msQGT of an observer’s state, defined relationally within a Wheeler–DeWitt universe; I take that identification as given and ask only how the resulting structure transforms between internal observers. Since ρObs is a subsystem state conditioned on a QRF, it inherits the frame relativity of subsystems established by the perspective-neutral QRF programme, which I make precise. In the ideal regime, where the frame-change map is a unitary, the observer’s spectral degeneracy pattern—which fixes the emergent gauge group—and the topological charges of the Uhlmann curvature are frame-invariant, while the metric and gauge potentials are frame-covariant; I verify the topological invariance numerically for spin-1/2 and spin-1 monopoles, and show that it holds only for smooth, single-valued frame changes (singular, Dirac-string changes inject charge). In the physical regime, where the relevant constraint generator is bounded below, the factorisation condition of Ali Ahmad et al. fails in some frames, so that whether ρObs—and hence the emergent gauge content—is even definable becomes frame-relative. The new element relative to existing QRF results is that the msQGT endows the QRF-relative reduced state with an information-geometric (Uhlmann/Yang–Mills) structure whose integer-valued topological charges constitute a new class of QRF invariant, distinct from—and complementary to—the known continuous resource invariant (the conserved entanglement-plus-coherence sum), which carries no gauge structure. The upshot is a sharp division: the objective content of the emergent structure is topological, its metric/gauge representation is perspectival, and in the physical regime even the observer’s existence as a subsystem is frame-relative.
This paper is a duplicated. Updated and complete version available at https://doi.org/10.5281/zenodo.21591128" I determine the behaviour, under a change of internal quantum reference frame (QRF), of the geometry and gauge structure obtained from the mixed-state quantum geometric tensor (msQGT) of a reduced density matrix ρObs. The setting is the Relational–Informational Model (RIM), in which the emergent metric and gauge field are the real and imaginary sectors of the msQGT of an observer’s state, defined relationally within a Wheeler–DeWitt universe; I take that identification as given and ask only how the resulting structure transforms between internal observers. Since ρObs is a subsystem state conditioned on a QRF, it inherits the frame relativity of subsystems established by the perspective-neutral QRF programme, which I make precise. In the ideal regime, where the frame-change map is a unitary, the observer’s spectral degeneracy pattern—which fixes the emergent gauge group—and the topological charges of the Uhlmann curvature are frame-invariant, while the metric and gauge potentials are frame-covariant; I verify the topological invariance numerically for spin-1/2 and spin-1 monopoles, and show that it holds only for smooth, single-valued frame changes (singular, Dirac-string changes inject charge). In the physical regime, where the relevant constraint generator is bounded below, the factorisation condition of Ali Ahmad et al. fails in some frames, so that whether ρObs—and hence the emergent gauge content—is even definable becomes frame-relative. The new element relative to existing QRF results is that the msQGT endows the QRF-relative reduced state with an information-geometric (Uhlmann/Yang–Mills) structure whose integer-valued topological charges constitute a new class of QRF invariant, distinct from—and complementary to—the known continuous resource invariant (the conserved entanglement-plus-coherence sum), which carries no gauge structure. The upshot is a sharp division: the objective content of the emergent structure is topological, its metric/gauge representation is perspectival, and in the physical regime even the observer’s existence as a subsystem is frame-relative.
"This record is a duplicate. The updated and complete version available at https://doi.org/10.5281/zenodo.19842968" Within the Relational-Informational Model (RIM), where gauge fields emerge from the Uhlmann curvature of the observer’s reduced density matrix ρObs, I show that the number of fermion generations is determined by the topology of the degeneracy locus—the set of spacetime points where eigenvalues of ρObs coincide. By the von Neumann–Wigner non-crossing theorem, pairwise eigenvalue de- generacy in a Hermitian family requires three independent real conditions (codi- mension 3). In four-dimensional spacetime, each degeneracy condition therefore defines a one-dimensional submanifold—a degeneracy string. For the SU(3) sector of RIM, the spectrum of ρObs contains three eigenvalues {p1,p2,p3}, admitting ex- actly 3 2 = 3 independent pairwise degeneracies. Consequently, three topologically distinct degeneracy strings exist, each carrying a distinct sector of the Uhlmann holonomy. I identify these three sectors with the three fermion generations.An explicit qutrit toy model demonstrates that a closed loop in parameter space encircling all three degeneracy points produces a Z3 monodromy—a cyclic permuta- tion of eigenvalues—whose braid structure provides an information-geometric origin for the CKM mixing matrix and CP violation. The SU(2) sector contributes no additional generations (since 2 2 = 1) but instead generates the isospin doublet structure within each generation. This result constitutes a sharp, falsifiable prediction: the number of fermion generations is exactly three, as a topological consequence of pairwise degeneracies in a three-level quantum system embedded in four-dimensional spacetime. The prediction is consistent with precision electroweak data (Nν = 2.984 ±0.008 from LEP) and current LHC bounds on fourth-generation quarks.
A reduced density matrix conditioned on an internal quantum reference frame (QRF) carries, through its mixed-state quantum geometric tensor, a Fisher–Bures metric and an Uhlmann gauge field. We determine which parts of this structure are frame independent, separating passive frame changes (reparametrizations and observer-local unitaries) from active ones (a change of the internal frame itself) within the perspective-neutral framework. Under passive changes the spectrum and the eigenbundle Chern numbers are invariant for every smooth single-valued transformation, the metric and the Uhlmann holonomy only for reparametrizations and constant unitaries. Under an active change between ideal frames the observer’s marginal obeys an exact shifted-overlap law for every relational state; its charge distribution is frame invariant, which gives the entanglement plus-coherence invariant of Cepollaro et al. in closed form. For a product relative state the law reduces to a partial twirl (relativization), under which the metric can only contract. For a two level observer the geometry is then graded by one inter-frame coherence |κ|: the Bures area is π|κ|, the Chern numbers survive for every |κ| > 0, and the equatorial Uhlmann Z2 holonomy only for |κ| > √3/2, the known thermal critical point reached here by coherence instead of temperature. For bounded generators the observer’s definability is itself frame relative.
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).
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.
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]