Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization remains subtle due to the intrinsic ambiguity of biorthogonal eigenspaces. Since left and right eigenvectors are not related by Hermitian conjugation, the associated Berry connection is generally nonunique, leading to complex geometric phases and ambiguously defined holonomies. Here, we formulate a covariant geometric framework for non-Hermitian quantum systems based on the metric structure of the underlying Hilbert space. We show that, in the quantum regime with continuous state evolution, the conventional Berry connection and the associated Berry holonomy over closed parameter-space loops can be consistently defined only in the pseudo-Hermitian limit, where the spectrum is real. For generic non-Hermitian Hamiltonians with complex spectra, the relevant geometric object is instead the Aharonov-Anandan holonomy associated with cyclic evolution in projective Hilbert space. Within the pseudo-Hermitian regime, we construct a unique Hermitian Berry connection that is covariant under arbitrary GL(N,C) frame transformations and reduces to the standard Berry connection in the Hermitian limit. The resulting formalism separates the intrinsic geometry of the Hamiltonian eigenspace from contributions arising from the parameter dependence of the Hilbert-space metric, revealing that the conventional biorthogonal formulation generally mixes these distinct geometric effects. Consequently, geometric phases, synthetic gauge fields, and topological characteristics commonly attributed to non-Hermitian eigenspace geometry may, in part, originate from the underlying metric structure. Our framework therefore provides a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.
Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization remains subtle due to the intrinsic ambiguity of biorthogonal eigenspaces. Since left and right eigenvectors are not related by Hermitian conjugation, the associated Berry connection is generally nonunique, leading to complex geometric phases and ambiguously defined holonomies. Here, we formulate a covariant geometric framework for non-Hermitian quantum systems based on the metric structure of the underlying Hilbert space. We show that, in the quantum regime with continuous state evolution, the conventional Berry connection and the associated Berry holonomy over closed parameter-space loops can be consistently defined only in the pseudo-Hermitian limit, where the spectrum is real. For generic non-Hermitian Hamiltonians with complex spectra, the relevant geometric object is instead the Aharonov-Anandan holonomy associated with cyclic evolution in projective Hilbert space. Within the pseudo-Hermitian regime, we construct a unique Hermitian Berry connection that is covariant under arbitrary GL(N,C) frame transformations and reduces to the standard Berry connection in the Hermitian limit. The resulting formalism separates the intrinsic geometry of the Hamiltonian eigenspace from contributions arising from the parameter dependence of the Hilbert-space metric, revealing that the conventional biorthogonal formulation generally mixes these distinct geometric effects. Consequently, geometric phases, synthetic gauge fields, and topological characteristics commonly attributed to non-Hermitian eigenspace geometry may, in part, originate from the underlying metric structure. Our framework therefore provides a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.
Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization remains subtle due to the intrinsic ambiguity of biorthogonal eigenspaces. Since left and right eigenvectors are not related by Hermitian conjugation, the associated Berry connection is generally nonunique, leading to complex geometric phases and ambiguously defined holonomies. Here we formulate a covariant geometric framework for non-Hermitian quantum systems based on the metric structure of the underlying Hilbert space. We show that, in the quantum regime with continuous state evolution, the conventional Berry connection and the associated Berry holonomy over closed parameter-space loops can be consistently defined only in the pseudo-Hermitian limit, where the spectrum is real. For generic non-Hermitian Hamiltonians with complex spectra, the relevant geometric object is instead the Aharonov--Anandan holonomy associated with cyclic evolution in projective Hilbert space. Within the pseudo-Hermitian regime, we construct a unique Hermitian Berry connection that is covariant under arbitrary
GL(N,C) frame transformations and reduces to the standard Berry connection in the Hermitian limit. The resulting formalism separates the intrinsic geometry of the Hamiltonian eigenspace from contributions arising from the parameter dependence of the Hilbert-space metric, revealing that the conventional biorthogonal formulation generally mixes these distinct geometric effects. Consequently, geometric phases, synthetic gauge fields, and topological characteristics commonly attributed to non-Hermitian eigenspace geometry may, in part, originate from the underlying metric structure. Our framework therefore provides a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.
Geometric phases form a rich subject in mathematical physics, lying at the intersection of differential geometry, functional analysis, and topology. This combination of ideas can make the subject difficult to approach for advanced undergraduate students, even though it is very rewarding. The aim of this thesis is to give a self-contained introduction to the mathematical framework behind geometric phases. Assuming only basic differential geometry, we develop the theory of vector bundles, connections, curvature, and holonomy, and use it to describe the Berry and Wilczek–Zee phases as holonomies of natural connections on complex vector bundles. We also discuss the Aharonov–Bohm effect as an example of a flat connection with globally nontrivial holonomy.
This paper investigates a geometric framework for the gravitational Aharonov-Bohm effect in four-dimensional spacetime, demonstrating how spacetime curvature induces nonlocal quantum phase shifts within field-free regions. By constructing vector bundles on spacetime manifolds equipped with Levi-Civita connections, we derive the holonomy transformations for parallel-transported quantum states. Under the Newtonian approximation, metric decomposition into Minkowski background plus scalar potential perturbations reveals through the linearized Einstein field equations that the gravitationally induced phase shift is mathematically isomorphic to its electromagnetic counterpart. These results establish the quantum observability of gravitational gauge structures and provide theoretical support for experimental verification via atom interferometry.
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