Collective excitations of many-body electron systems can carry internal structure, tied to the quantum geometry of the Hilbert space in which they are embedded. This has been shown explicitly for particle-hole-like excitations, which carry a ``quantum geometric dipole'' (QGD) that is essentially an electric dipole moment associated with the state. We demonstrate in this work that this property can be formulated in a generic way, which does not require wave functions expressed in terms of single particle-hole states. Our formulation exploits the density matrix associated with a branch of excitations that evolves continuously with its momentum
K, from which one may extract single-particle states allowing a construction of the QGD. We demonstrate the formulation using the single-mode approximation for excited states of two quantum Hall systems: the first for an integrally filled Landau level, and the second for a fractional quantum Hall state at filling factor $\ensuremath{\nu}=1/m$, with
m an odd integer. In both cases we obtain the same result for the QGD, which can be attributed to the translational invariance assumed of the system. Our study demonstrates that the QGD is an intrinsic property of collective modes which is valid beyond approximations one might make for their wave functions.
We provide an extended account of the recent statistical mechanical theory of gauge invariance against operator shifting in quantum many-body systems [J. Müller and M. Schmidt, Phys. Rev. E 114, L022104 (2026)10.1103/dzz3-m28k]. The gauge transformation is enacted by a shifting superoperator that displaces the fundamental position and momentum degrees of freedom. The shifting superoperator constitutes a map between Hilbert space operators, and it features Lie algebra commutator structure. Averages of general observables remain invariant under the shifting both in and out of thermal equilibrium, as well as in ground states. The gauge invariance induces exact sum rules that interconnect global observables and associated locally resolved correlation functions. In particular we describe the resulting one-body force, hyperforce, product, and two-body sum rules. We relate the shifting superoperator to a previously formulated quantum canonical transformation and present the generalization of quantum shifting to multicomponent systems. The gauge theory respects fundamental fermionic and bosonic particle properties, as we demonstrate by proving the compatibility of operator shifting and exchange symmetry. We formulate the quantum version of hyperdensity functional theory to provide formal access to hyperforces as well as to general averaged quantum observables via universal density functionals. For time-dependent situations, we describe quantum dynamical gauge invariance and prove exact dynamical sum rules for nonequilibrium situations, as generated by Hamiltonian time dependence. We argue for the fundamental status of statistical mechanical gauge invariance based on the compliance of the underlying geometry with canonical quantization according to Dirac's correspondence principle. Analogies and differences of the quantum mechanical sum rules with their classical counterparts remain indicative of the respective levels of description.
Description [31 Essential] Part III of a Three-Paper Trilogy on Connection-Based Observables A wide range of physical phenomena — including the Aharonov–Bohm effect, Berry phase, Wilson loops, and energy transport in the 0-Sphere model — demonstrate that experimentally observable quantities often depend not on local field values or curvature alone, but on effects accumulated along trajectories. This work articulates a unifying principle: physical observables are fundamentally line integrals of connections, or equivalently, holonomies and accumulated geometric phases defined along histories. By examining canonical examples from quantum mechanics and gauge theory — the Aharonov–Bohm effect, Berry phase, and Wilson loops — alongside the explicit thermal dynamics of the 0-Sphere two-kernel system, we argue that line integrals constitute a natural and universal class of fundamental observables across quantum, gauge, and gravitational domains. Local tensors, field strengths, and curvature are shown to play a secondary role, encoding infinitesimal or loop-level properties of parallel transport, while the operationally meaningful content resides in accumulated effects along physically realized trajectories. The Aharonov–Bohm effect is reinterpreted as holonomy without curvature: charged particles acquire measurable phase shifts even when propagating through regions where the electromagnetic field tensor vanishes, with the phase difference determined by the line integral of the vector potential along trajectories. Within the 0-Sphere model, this is further understood as arising from finite path-length differences between spatially separated trajectories, producing temporal phase shifts that accumulate from the electron's internal phase structure. Berry phase is presented as the primacy of adiabatic histories: geometric phase accumulation along closed paths in parameter space, insensitive to traversal rate and irreducible to instantaneous Hamiltonian properties. Wilson loops exemplify gauge-invariant line integrals in non-Abelian gauge theories, capturing global information inaccessible through local field strengths alone. The 0-Sphere two-kernel system provides an explicit realization of these abstract principles. Energy transport between localized kernels
A and
B is driven by a thermal gradient
∇T=E0sinθ, which can be expressed as the gradient of a scalar potential
ϕ=Te2−Te1. The line integral
∫ABA⋅dr along any trajectory connecting the kernels depends only on the endpoint values
ϕ(rend)−ϕ(rstart), illustrating that physically relevant information is encoded in boundary values of scalar potentials rather than microscopic trajectory details. This concrete manifestation demonstrates how abstract geometric principles — line integrals along trajectories as primary observables — manifest as operationally meaningful forces and phases grounded in explicit physical mechanisms. This perspective does not modify existing theories or their empirical predictions. Instead, it clarifies their logical hierarchy by placing histories, connections, and line-integral observables at the foundational level, with locality and curvature emerging as effective descriptors. In doing so, the work provides a coherent conceptual bridge linking interference phenomena, gauge invariance, and gravitational dynamics under a single geometric principle centered on line integrals as fundamental observables. Preceding Work - Part I: This paper builds upon the foundation established in Geometric Structure of Spinorial Phase Accumulation along Thermal Geodesics (Zenodo, 2025, https://doi.org/10.5281/zenodo.18067760), which demonstrated why line integrals are the physically appropriate geometric objects in the 0-Sphere framework. That foundational work showed that energy transport between two localized kernels
A and
B defines a radiative gradient with only one effective spatial degree of freedom, constraining the photon-sphere energy centroid to move along a one-dimensional trajectory. As a consequence, physical quantities associated with energy transport — such as phase accumulation or internal state evolution — are defined only along this trajectory, making line integrals the natural mathematical objects rather than surface or volume integrals. Importantly, Part I grounded the abstract notion of line integrals in a concrete physical process: the actual motion of energy along a defined path parametrized by proper time, with the
SU(2) spinorial connection arising from Thomas precession along thermal geodesics. The separation between external bosonic motion (
π rotation for one-way traversal) and internal spinorial phase accumulation (
2π for one-way,
4π for round trip) was clarified, establishing the geometric correspondence with Poincaré semicircular geodesics. Preceding Work - Part II: The connection-based perspective was further developed in From Curvature to Connection: Revisiting the Geometric Origin of Conservation Laws (Zenodo, 2026, https://doi.org/10.5281/zenodo.18135855), which demonstrated how conservation laws themselves emerge as geometric consistency conditions associated with accumulated connection and proper-time structure along thermal geodesics. That work reinterpreted the stress–energy tensor
Tμν as a derived descriptor encoding the macroscopic manifestation of underlying geometric consistency, rather than as a fundamental causal source. The Einstein field equation was reformulated as an a posteriori consistency condition — a geometric checksum — verifying that the accumulated connection-based structure underlying energy transport remains self-consistent. Critically, Part II identified a common limitation shared by all previous connection-based approaches to gravitational theory. Trilogy Structure: Together, these three papers constitute a comprehensive argument establishing line integrals of connections along histories as the fundamental observables in physics: Part I establishes why line integrals are physically appropriate, Part II shows how conservation laws emerge from this structure, and Part III (the present work) demonstrates the universality of this principle across quantum, gauge, and gravitational domains — including the Aharonov–Bohm effect as holonomy without curvature, Berry phase as a history-dependent observable, Wilson loops as gauge-invariant line integrals, and 0-Sphere thermal dynamics as a concrete realization. Keywords Line integrals Holonomy Aharonov–Bohm effect Berry phase Wilson loops Connection-based observables Geometric phase The 0-Sphere model Thermal geodesics Gauge theory History-dependent observables Curvature as secondary descriptor Quantum interference Energy transport
This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.
Revised Abstract (Version 1.1 – Geometric Clarification) This preprint introduces field-depth (w) as an intrinsic sequential resonance coordinate associated with spatial structure, providing a geometric basis for key quantum phenomena. In this framework, the wavefunction is expressed as Ψ(x, y, z; w), where w does not represent an additional spatial dimension but an sequential coherence parameter governing the stability of standing-wave configurations within a three-dimensional spatial manifold. A localized particle state is interpreted not as a projection from a higher-dimensional space, but as the observable manifestation of a resonance-stabilized nodal structure whose internal coherence configuration is dynamically fixed. This formulation yields direct geometric interpretations of several quantum-mechanical features that are typically treated as abstract or axiomatic: • Wave–particle duality emerges naturally: quantum entities are extended resonance structures, while particle detections correspond to stabilized spatial envelopes of internally coherent states. • Spin arises as a phase-orientation condition within the resonance structure, producing the known 4π periodicity of spin-½ systems without requiring additional spatial geometry. • Quantum entanglement is understood as shared internal resonance configuration, allowing correlated outcomes without invoking superluminal signaling or geometric nonlocality. This model does not modify the Schrödinger or Dirac equations; rather, it clarifies the geometric and coherence-theoretic context in which their solutions are realized. The approach remains consistent with the SU(2) fiber-bundle structure of spinor fields and with experimental tests of quantum correlations. A forthcoming Version 2 will further refine the geometric interpretation by explicitly distinguishing spatial dimensionality from sequential coherence structure and by developing the implications of this clarification for electron stability, charge distribution, and resonance-mediated interactions.
Exact equilibrium constraints in quantum many-body systems, such as sum rules and Ward identities, can be hard to expose because local currents are distributional and equal-time commutators may contain anomaly-like contact terms. We present a simple geometric organization of the recently identified quantum shift symmetry. The local shifting superoperator is viewed as a statistical-gauge connection acting on observables, so hyperforces are covariant derivatives and force balance becomes a thermal Ward identity expressed with the Kubo-Mori inner product. For compact smearings we prove bulk flatness: smeared shift generators close on the Lie bracket of vector fields with no bulk Schwinger term. This closure yields an iterative, all-orders Ward-Bianchi hierarchy of equal-time constraints that closes on a finite set of force insertions and a bilinear force-gradient kernel, leading to general antisymmetry relations and positivity bounds. We fix the microscopic generator by quantizing the diffeomorphism moment map, where Weyl and half-density quantization agree and give a self-adjoint operator without bulk ordering anomalies. Although the bulk connection is flat, global topology remains through holonomy, which links naturally to twisted boundary conditions and flux threading. Background electromagnetic fields deform the algebra through explicit density-weighted curvature insertions rather than a bulk central extension. As an application we treat fractional quantum Hall fluids on a torus, connect the deformation to the guiding-center GMP-W_{∞} algebra, and verify the key identities with exact diagonalization.