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Derivation of the geometrical phase

Arno Böhm, Luis J. Boya, Brian K. Kendrick

The geometric (Berry) phase is shown to have its origin in the nontrivial geometry of the fiber bundle: Hilbert space\ensuremath{\rightarrow}space of states. The nontrivial geometry comes simply from the scalar product in Hilbert space. A comprehensive treatment of the geometrical phase is presented, bringing together the various ideas in the literature.

Physical Review AFeb 1, 199193 citations

Differential Geometry of Quantum Theory Toward an Emergent Hilbert-Space Formulation on a Coherence Manifold

Borros Arneth

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 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.

Jun 11, 20260 citations

Invariance of the geometrical phase under time-dependent unitary transformations

Brian K. Kendrick

The effects of time-dependent unitary transformations on the geometrical phase are investigated in the general non adiabatic setting. A detailed study considering both paths of an interference experiment shows that even though the geometry of the fibre bundle: Hilbert space to space of states changes. The measured relative phase, geometrical phase and dynamical phase are all invariant under these transformations.

Journal of Physics A Mathematical and GeneralFeb 21, 19926 citations

The non-adiabatic classical geometric phase and its bundle-theoretic interpretation

Gavriel Segre

A bundle theoretic interpretation of the non-adiabatic classical geometric phase is given in the Koopman representation of dynamics

arXiv:math/0403189Mar 10, 20041 citation

Remarks on the Geometric Nature of Quantum Phase Space

Róbert Hermann

Several geometric remarks on the foundations of quantum mechanics are presented. For example, quantum phase space is the tangent bundle to quantum configuration space.

Journal of Mathematical PhysicsNov 1, 19659 citations

A note on geometric phases

Franco Salmistraro

The usual definition of a geometric phase involves particular conditions on the behaviour of the state vector of a quantum system. We show that they can be removed in some cases by appealing directly to the Schrödinger equation. In this way we obtain an extension of a family of geometric phases previously introduced by us.

Europhysics Letters (EPL)Jun 15, 19980 citations

The Quantum Phase

Abstract

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The Quantum Phase

Abstract

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