Quantum mechanics is formulated in terms of complex phase and action-dependent amplitudes. In this work, quantum phase is identified with the holonomy of a geometric connection, and Planck's constant appears as the conversion scale between geometric transport and physical action. This provides a geometric formulation of quantum kinematics in which phase accumulation, noncommutativity, and quantization arise naturally from connection geometry and topology. This paper is Part I of a series which develops a geometric framework for quantum mechanics based on holonomy and transport structure. Subsequent papers develop operator representations and dynamical structure. This Zenodo record establishes priority and archival timestamp. Public release via arXiv is in progress.
Quantum mechanics is formulated in terms of complex phase and action-dependent amplitudes. In this work, quantum phase is identified with the holonomy of a geometric connection, and Planck's constant appears as the conversion scale between geometric transport and physical action. This provides a geometric formulation of quantum kinematics in which phase accumulation, noncommutativity, and quantization arise naturally from connection geometry and topology. This paper is Part I of a series which develops a geometric framework for quantum mechanics based on holonomy and transport structure. Subsequent papers develop operator representations and dynamical structure. This Zenodo record establishes priority and archival timestamp. Public release via arXiv is in progress.
This paper examines measurement and representational limits within a geometric formulation of quantum mechanics based on connection geometry and holonomy. Measurement is interpreted as projection onto invariant quantities associated with comparison of states under parallel transport. This projection preserves invariant phase relations while discarding non-invariant information. As a result, measurement outcomes appear discrete and probabilistic at the representational level. Correlations between subsystems arise from shared geometric structure and exhibit nonlocal consistency without introducing additional dynamical mechanisms. The analysis clarifies how measurement and classical behavior can be understood within a geometric account of quantum kinematics.
Quantum mechanics is traditionally formulated in terms of complex Hilbert spaces, wavefunctions, and linear evolution equations. While extraordinarily successful, this framework leaves open several conceptual questions concerning the geometric origin of the wavefunction, the meaning of quantum phase, and the status of Hilbert space as a fundamental structure. In this work, we investigate a coherence-geometric reformulation in which the fundamental object is a coherence coordinate , where represents radial coherence and represents phase coherence associated with geometric transport. We show that the standard structures of quantum mechanics admit a natural interpretation within a differential-geometric framework. The wavefunction emerges from coherence coordinates, probability density is generated by radial coherence, probability flow is generated by phase coherence, and Hilbert spaces appear as spaces of square-integrable functions defined on a coherence manifold. The resulting formulation provides a unified geometric language connecting quantum mechanics, geometric phases, fiber bundles, holonomies, gauge theory, and differential geometry
Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.
We extend the sphere-on-sphere geometric quantization mechanism of the Operator-Derived Dimension (ODD) framework to derive the full kinematical structure of quantum mechanics. Starting from the topological origin of Planck’s constant h as a winding invariant of the S1 intersection between the five-dimensional bulk and the four-dimensional causal horizon, we show that the symplectic form, canonical commutation relations, phase, spin, and the path-integral formulation arise uniquely and necessarily from the same geometry. Quantization is therefore not postulated but enforced by topology, while operator algebra emerges from the transverse slicing dynamics. This places the foundations of quantum mechanics on geometric and topological necessity rather than axiomatic assumption.
This paper examines the geometric structure underlying amplitudes, interference, and probability in quantum mechanics. Building on the interpretation of quantum phase as holonomy, amplitudes are analyzed in terms of parallel transport in complex line bundles. Interference is interpreted as arising from the composition of alternative transport paths, and probability emerges from the invariant comparison of amplitudes under phase transformations. Under general assumptions of linearity and phase invariance, the squared-norm structure associated with probability is shown to arise naturally from consistency conditions on amplitude composition. The analysis focuses on kinematic structure and does not assume specific dynamical models.
Abstract Standard quantum mechanics represents global phase with a single U(1) structure and typically treats phase transport as geometrically trivial unless additional external structure is introduced. In multi-carrier phase algebras, however, closed phase evolution can exhibit intrinsic holonomy, implying a nontrivial internal connection and curvature on the phase fiber. This article isolates the geometric junction at which internal (phase-fiber) curvature becomes operationally accessible as an effective curvature on the base manifold governing spatial transport. The mechanism is formulated in connection-theoretic terms: an internal transport law induces an effective base-level connection acting on multi-component amplitudes, whose curvature controls measurable departures from flat propagation. We highlight three observable channels—loop-dependent phase transport, interferometric shifts, and semiclassical deflections—through which internal holonomy can be tested without assuming a curved metric a priori. The resulting framework provides a compact geometric interpretation of how spatial curvature may emerge as a projection of internal phase holonomy, and it delineates the conditions under which this projection becomes experimentally distinguishable.