We propose a cohomological modelling schema of quantum state spaces and their connectivity structures in relation to the formulation of global geometric phase phenomena. In the course of this schema, we introduce the notion of Hermitian differential line sheaves or unitary rays and classify their gauge equivalence classes in terms of a global differential invariant given by the de Rham cohomology class of the curvature. Furthermore, we formulate and interpret physically the curvature recognition integrality theorem for unitary rays. Using this recognition theorem, we define the notion of a quantum spectral beam and show that it has an affine space structure with structure group given by the characters of the fundamental group.
We propose a cohomological modelling schema of quantum state spaces and their connectivity structures in relation to the formulation of global geometric phase phenomena. In the course of this schema, we introduce the notion of Hermitian differential line sheaves or unitary rays and classify their gauge equivalence classes in terms of a global differential invariant given by the de Rham cohomology class of the curvature. Furthermore, we formulate and interpret physically the curvature recognition integrality theorem for unitary rays. Using this recognition theorem, we define the notion of a quantum spectral beam and show that it has an affine space structure with structure group given by the characters of the fundamental group.
The geometric phase in quantum mechanics was originally elucidated in the context of the adiabatic theorem. This and other limitations were later removed. We show how this phase can be presented as the simplest functional of smooth open curves of unit vectors in Hilbert space, invariant under the two groups of local phase changes and monotonic reparametrisations. This viewpoint clarifies the role of the early Pancharatnam criterion for two Hilbert space vectors to be `in phase', in the later geometric phase context. Noncyclic evolution is handled in a very direct manner. Connections to the Bargmann invariants of quantum mechanics are immediate and can be exploited. A class of geometric phases arises in connection with unitary representations of Lie groups. In this case a surprisingly complete analysis of the structure of the geometric phase is possible, motivated by the Wigner-Eckart theorem. The interplay between geometric phase ideas and Lie group unitary representations is extraordinarily rich; the differential geometric structures of Lie groups and coset spaces play very important roles and allow classification of all possible geometric phases associated with a given Lie group. Examples involving SU(2), SU(3) and the metaplectic group Mp(2) are presented.
We develop a unified geometric framework in which the quantum geometric tensor (QGT) simultaneously governs both the metrology of gauge-invariant curvature and the symmetry classification of random-matrix ensembles. The symmetric Fubini–Study sector, generated by the Jordan product of symmetric logarithmic derivatives (SLDs), determines the quantum Fisher information and therefore the information gain associated with closed-loop curvature acquisition. The antisymmetric Berry sector, classified by the de Rham cohomology class of the SLD connection, determines whether the accumulated holonomy phase is linearly observable. We show that time-reversal symmetry acts as a topological gate on the readable Berry channel: the holonomy survives only in the unitary Wigner–Dyson class (GUE), where the cohomology class is nontrivial. This converts the previously phenomenological time-reversal “gate factor” into a cohomological indicator. In the symplectic class (GSE), the non-Abelian obstruction is identified with the Jordan associator curvature and appears operationally as a Holevo incompatibility penalty. The resulting cost-efficiency functional is shown to be genuinely U-shaped across the Dyson classes, with a strict minimum at GUE arising from the non-monotonic topology of the readable holonomy sector. We further establish a bridge between the QGT and thermodynamic fluctuation geometry, derive curvature-statistics scaling laws from Wigner level repulsion, and analyze the deformation of the Kähler structure under decoherence through the Bures metric and mean Uhlmann curvature. The framework unifies Berry geometry, quantum metrology, random-matrix universality, and information geometry into a single cohomological description of curvature acquisition.
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.
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
Crystallographic groups are conventionally studied in real space to characterize crystal symmetries. Recent work has recognized that when these symmetries are realized projectively, momentum space inherently accommodates nonsymmorphic symmetries, thereby evoking the concept of \textit{momentum-space crystallographic groups} (MCGs). Here, we reveal that the cohomology of MCGs encodes fundamental data of crystalline topological band structures. Specifically, the collection of second cohomology groups, H2(ΓF,Z), for all MCGs ΓF, provides an exhaustive classification of Abelian crystalline topological insulators, serving as an effective approximation to the full crystalline topological classification. Meanwhile, the third cohomology groups H3(ΓF,Z) across all MCGs exhaustively classify all possible twistings of point-group actions on the Brillouin torus, essential data for twisted equivariant K-theory. Furthermore, we establish the isomorphism Hn+1(ΓF,Z)≅Hn(ΓF,F(RFd,U(1))) for $ n\ge 1$, where F(RFd,U(1)) denotes the space of continuous U(1)-valued functions on the dD momentum space RFd. The case n=1 yields a complete set of topological invariants formulated in purely algebraic terms, which differs fundamentally from the conventional formulation in terms of differential forms. The case n=2, analogously, provides a fully algebraic description for all such twistings. Thus, the cohomological theory of MCGs serves as a key technical framework for analyzing crystalline topological phases within the general setting of projective symmetry.
A complete theory for the line bundle structure in quantum mechanics and quantum field theory is given. This includes a general method for constructing curvature terms and flat connection terms. The necessary and sufficient condition for the existence of the integrality condition is obtained. The role of torsion parts in the first homology group of the configuration space is clarified. A possible extension to the higher dimensional vector bundle and its physical meanings are considered, too. Finally many physically interesting applications are given to illustrate our theory_ In particular, the local and global anomalies and other related topics including Berry's phase are discussed.