The quantum switch is known to probe order-sensitive operator structure, including commutators, incompatibility measures, and unitary invariants \citep{Chi13,Ore12,Pro14,Gao22,Aza25}. Related indefinite-order protocols can read out loop-like geometric phases \citep{Zha20,Yin23,Bar24}. This paper develops the gauge-holonomy case. Let
Uγ1 and
Uγ2 be path-ordered holonomies of two loops based at a common point
x0 and represented unitarily on a target Hilbert space. For a standard two-branch quantum switch, with one branch applying
U then
V and the other applying
V then
U, the lower off-diagonal control coherence is\[\chi=\Tr\!\left(K(U,V)\rho\right),\qquadK(U,V)=U^\dagger V^\dagger U V.\]Thus the switch reads out a based holonomy group commutator. Scalar Abelian holonomies and, more generally, commuting holonomies are order-invisible in the bare switch. Abelian parameter sensitivity can occur only through relative encoding or reference structure. Non-Abelian holonomies are directly visible through control-qubit quadratures. The finite-loop statement is exact and gauge covariant: under a common base-point transformation,
K↦gKg−1, while traces and covariantly prepared target expectations are gauge invariant. The local small-loop reduction is separate and perturbative. For small based loops with logarithms $X=\Log U$ and $Y=\Log V$,
logK(U,V)=[X,Y]−21[X+Y,[X,Y]]+terms of degree ≥4.For loops contained in a coordinate patch,
X begins with the area contraction of the curvature, followed by loop-shape-dependent covariant-derivative and path-ordering corrections. The leading local contribution to
logK is therefore a curvature-commutator term, but only in this controlled small-loop regime. A constructive gauge-loop realization, an
SU(2) signal model, a lattice-gauge analogue, and an experimental readout model are given. The result is an operator-level and gauge-theoretic proposal for using switch interferometry to read out non-Abelian holonomy order interference.
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
Abstract: This work develops a unified and gauge-invariant framework in which quantum interference is described as a measurement of relative holonomy between two coherent evolutions in a combined configuration space X = M × Λ. Both the Aharonov–Bohm (Abelian) and Wilczek–Zee (non-Abelian) effects appear as limits of a single Wilson-loop observable. A block connection is introduced on the direct-sum bundle E = H ⊕ (T X ⊗ ℂ), with structure group U(n) × O(p,q), thereby unifying Berry parallel transport and Levi-Civita geometry within one connection. The formalism ensures gauge and diffeomorphism covariance, analytic well-posedness under back-reaction, and establishes conditions for reconstructing the spacetime metric from holonomy data. The resulting theory provides a mathematically consistent and experimentally testable synthesis of geometric phase, interference, and spacetime connection. Potential physical implementations include mesoscopic rings, cold-atom systems, and superconducting circuits. Keywords: geometric phase; holonomy; Aharonov–Bohm effect; Berry phase; Wilczek–Zee connection; differential geometry; quantum interference; gauge theory.
Cylindrical Dirac modes underlie constructions in rotating QCD matter, boost-invariant Dirac-field quantization in heavy-ion physics, and high-energy twisted-particle scattering. The corresponding complete spinor frames can be regarded as alternative bases, but their equivalence does not determine the global behavior of eigenlines selected by conserved observables. Within the positive-energy doublet of the free massive Dirac Hamiltonian, we compare three conserved resolutions:
K, which couples spin to transverse momentum;
Km, a mass-dependent operator derived from the transverse Dirac Hamiltonian; and helicity. On the common regular domain away from the momentum axis, explicit smooth, single-valued
SU(2) transformations relate all three splittings. Although globally
SU(2)-equivalent on this common domain, they exhibit three distinct global extension behaviors. The
K projectors have azimuth-dependent polar limits and do not extend continuously to the axis. For nonzero mass, the
Km projectors extend smoothly over the enclosed momentum ball and define Chern-trivial eigenlines. The helicity projectors are smooth on every nonzero momentum sphere, but their eigenlines carry opposite unit Chern numbers and cannot extend through the enclosed origin. The parent positive-energy Dirac connection Abelianizes exactly in the
K eigenlines on fixed-azimuth meridians, whereas its full three-dimensional curvature has noncommuting components. We obtain the azimuthal Wilson loop in closed form and derive the exact conversion probability between the two
K branches under purely geometric positive-energy transport.
We develop a coherence-geometric framework in which transport processes on projective state spaces generate spectral determinant structures and geometric state-counting relations. The construction begins from a normalized complex coherence variable and its extension to a three-component coherence manifold whose physical state space is naturally identified with a complex projective geometry. Closed coherence cycles define holonomy operators analogous to Wilson loops in gauge theory. The corresponding spectral determinant generates a partition function whose cycle invariant is determined by the accessible volume of the underlying coherence manifold. This geometric state-counting principle establishes a direct connection between projective geometry, spectral theory, and coherence transport. A fundamental spectral invariant emerges from the equality between spectral and geometric partition structures and subsequently generates a hierarchy of higher-order coherence quantities. The resulting framework exhibits close mathematical connections to geometric quantum mechanics, Kähler geometry, geometric quantization, spectral geometry, and non-Abelian gauge theory. Rather than modifying established gauge theories, the formalism is proposed as a possible deeper coherence-geometric layer whose effective transport structures reproduce familiar holonomy properties while suggesting a common geometric origin for certain spectral hierarchies appearing in theoretical physics.
We investigate the geometry of mixed quantum states near rank-changing points, showing that these singularities function as effective geometric defects. The Uhlmann connection is well-defined on the full-rank sector of the density-matrix manifold, while rank-deficient states form singular boundary strata where the bundle structure degenerates. By restricting to a punctured state manifold that excludes the singular set, we obtain a well-defined gauge structure and identify an asymptotically robust invariant: the Uhlmann holonomy around noncontractible loops encircling the defect on a restricted two-dimensional punctured submanifold. In an exactly solvable qutrit model, a restricted submanifold emerges on which the connection is locally flat yet carries nontrivial monodromy, analogous to flat connections with Aharonov--Bohm-type transport. The holonomy depends only on the ratios of the vanishing eigenvalues under frozen radial dependence of the eigenbasis geometry and a fixed angular loop. In contrast, the Uhlmann curvature may diverge path-dependently when eigenvalues shrink with distinct powers, with a leading spectral-prefactor scaling law, establishing that the holonomy survives as a universal asymptotic invariant while the curvature remains non-universal. Within the effective SU(2) defect sector, the conjugacy class of the holonomy, equivalently the Wilson loop variable, provides a continuous, non-quantized classification of the asymptotic monodromy surrounding the rank-deficient defect. This non-quantization does not imply a lack of robustness: the asymptotic holonomy is an invariant of the restricted punctured submanifold and is insensitive to smooth deformations of the loop or the radial profile within the fixed spectral-ratio sector.