This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.
This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.
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.
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.
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 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.
For giving a geometric interpretation of the observable-geometric phase with application to quantum computation, operator geometry on a finite-dimensional complex Hilbert space was developed by Chen (2020), i.e., geometry over the operator-principal G-bundle of the unitary operator group based on the observable space. In this paper, we further develop operator geometry on an infinite-dimensional separable complex Hilbert space. To this end, we use the theory of unbounded operators for describing the differential structure of infinite-dimensional observable spaces and defining quantum connections and quantum parallel transportation. Then, we introduce the notion of the observable-geometric phase for infinite-dimensional quantum systems and give its geometric interpretation in terms of operator geometry. Finally, we present a topological index for Floquet topological phases and prove a local formula for it in terms of the observable-geometric phases, which are measurable in the physical sense.