Abstract Standard quantum mechanics encodes phase degrees of freedom through a single U(1) structure, implicitly assuming a flat phase geometry in which global phase transport is isometric and any curvature must be introduced externally, typically via spacetime metrics or interaction potentials. In this work, we show that replacing the U(1) phase algebra with the minimal cyclic algebra defined by j3=−1 (C₃) leads to an intrinsically curved phase space without invoking any external geometric input. The multi-carrier structure of C₃ generates nontrivial holonomy under closed phase evolution, producing residual phase components that cannot be eliminated by gauge transformations. This holonomy naturally defines an intrinsic connection whose curvature arises solely from the algebraic transition rules between phase carriers. Within this framework, we construct a C₃-generalized Laplace operator and analyze its Green function, demonstrating phase-mediated propagation that departs from flat-space behavior. When embedded into a C₃-generalized Schrödinger equation, curvature effects emerge dynamically through phase gradients, without the explicit introduction of gravitational or geometric potentials. In the weak-field limit, an effective Newtonian potential is recovered from hidden-phase gradients, providing a direct link to classical physics. The results reframe curvature as an emergent property of multi-carrier phase geometry and offer a unified perspective connecting quantum phase structure, measurement, and emergent spacetime features.
Abstract We introduce a phase-geometric framework in which time evolution is governed by a covariant derivative defined over a multi-carrier phase space. The associated connection induces a non-abelian curvature whose holonomy generates measurable phase-transport effects. Within this unified geometric setting, the quantum–classical transition, phase-channel mixing, and time-transport anomalies arise from a single underlying mechanism. In contrast to conventional approaches, where gauge structures are imposed externally, the connection emerges intrinsically from the phase geometry itself. The framework yields calculable thresholds for classicalization and leads to novel, testable predictions based on curvature-induced phase holonomy.
Liouville pathways provide the conceptual foundation for interpreting multidimensional spectroscopies and are commonly represented as independent contributions to the nonlinear response. In open quantum systems, however, the effective reduced Liouvillian need not preserve this observational pathway decomposition. Environmental interactions can transport amplitude between pathway sectors during the free-evolution intervals, generating measurable signatures in the nonlinear spectroscopic response.} We develop a geometric framework in which pathway transport is governed by a Liouvillian connection, its associated curvature, and the resulting observational holonomy. The framework applies to open quantum systems in which the environment selects a pointer basis distinct from the observational basis used to construct the spectroscopic response. This basis incompatibility induces transport among Liouville pathways, generating characteristic spectral distortions and a nontrivial Liouvillian curvature. Using a Duhamel expansion of the Liouvillian propagator, we derive a reconstruction procedure that identifies the transport operators responsible for the observed redistribution of pathway weight, accurate throughout the full range of basis misalignment. This perspective reframes spectral features as determined not only by which pathways exist but by how amplitude is transported among them. Spectral distortions, peak shifts, and otherwise-forbidden pathway contributions become geometric signatures of a curved Liouville-space manifold rather than phenomenological broadening corrections, identifying pathway geometry as a complementary layer of organization in nonlinear spectroscopy.
This paper establishes a geometric bridge between two operational results concerning gravity-mediated entanglement and phase-based field structure. First, prior work showed that entanglement in minimal gravity-mediated interferometric models arises if and only if the interaction phases contain a non-factorable plaquette invariant. This invariant represents an oriented loop phase that cannot be decomposed into local contributions and therefore serves as a necessary and sufficient condition for entanglement generation. Second, the Harmonic Code framework describes persistent physical effects as loop-structured phase geometry, in which closed phase loops encode enduring information about curvature and field structure. The present work unifies these viewpoints by demonstrating that the plaquette phase emerges naturally as a curvature-induced loop holonomy. An effective phase-gradient field is introduced, and the oriented loop phase is defined as Π = ∮ k · dℓ = ∬ (∇ × k) · dA, showing that the non-factorable phase invariant is directly generated by the curl of the phase-gradient field. A minimal rotational field model is used to derive the relation Π = 2εA, where A is the enclosed area and ε is the curl parameter. Numerical simulations verify the key geometric properties expected of a true holonomy: Linear scaling with enclosed area Shape independence at fixed area Sign reversal under orientation flip Convergence under discretization refinement The loop phase Π is then mapped to standard two-qubit entanglement measures. For the separable input state |++⟩, the concurrence and negativity are shown to depend only on Π: C(Π) = |sin(Π/2)| N(Π) = C/2 These results demonstrate that the non-factorable phase structure required for gravity-mediated entanglement has a direct geometric origin as a curvature-induced loop holonomy. The paper therefore provides a unified operational statement: Curvature generates loop phases, and those loop phases are exactly the structures responsible for gravity-mediated entanglement. This establishes a concrete geometric mechanism linking spacetime curvature, phase holonomy, and quantum entanglement, and offers a clear, experimentally testable framework for probing quantum aspects of gravity.
Quantum mechanics is traditionally formulated on complex Hilbert spaces, whereas gauge theories are formulated on fiber bundles equipped with connections and curvature. Despite the central role of geometry in modern physics, the Hilbert-space structure of quantum theory is usually postulated rather than derived. In this work, we investigate a differential-geometric formulation in which quantum states are represented as points on a coherence manifold characterized by a radial coherence coordinate and a holonomy-induced phase coordinate. The resulting geometry naturally admits the structure of a fiber bundle whose base space is associated with a coherence lattice and whose fiber corresponds to a phase circle. Within this framework, Hilbert spaces emerge as function spaces defined on the coherence manifold, while Schrödinger dynamics arise from the Laplace–Beltrami operator associated with the underlying geometry. The construction unifies concepts from geometric quantum mechanics, fiber-bundle gauge theory, holonomy, and coherence geometry within a common differential-geometric framework.
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