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
The Aharonov–Bohm (AB) effect demonstrates that a charged particle can acquire an observable phase shift in a region where the local magnetic field identically vanishes provided the configuration space is multiply connected and the electromagnetic connection exhibits non-trivial holonomy. We present a gauge-covariant physico-mathematical framework that systematically disentangles three regimes: (i) the standard ideal AB effect governed by the holonomy of a flat but globally inexact connection on ΩR=R2∖DR with DR={r≤R}; (ii) the path-integral decomposition into homotopy sectors indexed by the winding number, which is constructed on the universal cover; and (iii) non-topological corrections arising from evanescent wave-function penetration when the solenoid is modeled as a finite radial barrier. The ideal AB phase is taken as the reference topological holonomy, while the finite-barrier extension developed here separates the measured phase, under explicit weak-penetration assumptions, as Δφtotal=ΔφABhol+δφfinite+O(Teff2). Here, ΔφABhol=qΦ/ħ is the standard topological holonomy, whereas δφfinite=O(e−2κℓb) is a small dynamical, microphysics-dependent correction controlled by the barrier height, particle energy, and effective forbidden length ℓb. Within Laskin’s fractional quantum mechanics, we use the fractional extension as a diagnostic test: the topological AB phase is independent of the Lévy index α, whereas the finite-barrier correction acquires an anomalous, α-dependent penetration length. The numerical material provides gauge-covariant diagnostics: a Peierls-substituted split-operator scheme, a sector-resolved Monte Carlo path integral benchmarked against the exact Aharonov–Bohm propagator of Gerry and Singh, a WKB/transfer-matrix finite-barrier estimate tracking δφfinite→0, and Berry-phase computations used as consistency checks.
Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms. They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schrödinger Picture. These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality. For particles passing through nonvanishing magnetic or electric fields they lead to cancellations of Aharonov-Bohm phases at the observation point, generalizing earlier semiclassical experimental observations (of Werner & Brill) to delocalized (spread-out) quantum states. This leads to a correction of previously unnoticed sign-errors in the literature, and to a natural explanation of the deeper reason why certain time-dependent semiclassical arguments are consistent with static results in purely quantal Aharonov-Bohm configurations. These nonlocalities also provide a remedy for misleading results propagating in the literature (concerning an uncritical use of Dirac phase factors, that persists since the time of Feynman's work on path integrals). They are shown to conspire in such a way as to exactly cancel the instantaneous Aharonov-Bohm phase and recover Relativistic Causality in earlier " paradoxes" (such as the van Kampen thought-experiment), and to also complete Peshkin's discussion of the electric Aharonov-Bohm effect in a causal manner. The present formulation offers a direct way to address time-dependent single- vs double-slit experiments and the associated causal issues -- issues that have recently attracted attention, with respect to the inability of current theories to address them.
The Aharonov-Bohm (AB) effect, wherein an electron acquires a phase shift in a field-free region due to electromagnetic potentials, poses a profound challenge to the ontology of quantum mechanics and gauge theories. This paper demonstrates that gauge-invariant explanations, which attribute the phase to measurable quantities, rely on nonlocal and discontinuous mechanism and fail to account for its continuous accumulation along the electron's path—a process evident in the generalized AB effect, where a time-varying magnetic flux induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a new analysis spanning quantum mechanics and quantum electrodynamics, I argue that the electromagnetic potentials
Aμ, fixed in the Lorenz gauge, emerge as the fundamental physical reality, offering a local, relativistically consistent account of the phase’s generation. This exclusion of gauge-invariant paradigms reverberates across gauge theories: it redefines the Higgs mechanism, favoring dynamic potentials over static invariants, and extends to general relativity, where gravitational potentials
gμν may anchor spacetime’s substantival reality via a gravitational AB effect. By unraveling the AB phase’s continuous generation—locally mediated by potentials—this study not only addresses a long-standing conundrum but also bridges electromagnetic, gravitational, and Yang-Mills frameworks, offering a unified potential-centric perspective with novel implications for physics and philosophy.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential
A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics
A is in some sense more ``fundamental'' than
B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant
B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the
B field in the region inaccessible to the particle.
The standard semiclassical treatment of the Aharonov-Bohm (AB) effect, which predicts no observable phase shift for open electron paths in field-free regions due to gauge dependence, is shown to be incomplete. By fully quantizing both the probe electron and the electromagnetic source within quantum electrodynamics (QED), we derive an effective action that decomposes the AB phase into two equal contributions: a gauge-dependent probe phase from the source's potential acting on the electron, and a gauge-invariant source phase arising from the electron's quantum backreaction on the source via virtual-photon exchange. For open paths, where the probe phase vanishes under gauge choice, the source phase persists as a measurable part of the closed-loop value, isolated through a polarization--magnetization decomposition that separates its invariant bulk from a gauge-dependent boundary term. This uncovers a leading-order QED correction to nonrelativistic quantum mechanics, resolves longstanding debates on the AB phase's origin and locality, and refines gauge principles by revealing hidden gauge invariants in formally gauge-dependent quantities. While challenging to test directly, a conceptual scheme involving phase transfer to auxiliary qubits illustrates its potential observability, emphasizing its novelty beyond conventional closed-loop paradigms.