Geometry and Structure of Quantum Phase Space

Chen Zeqian, Sun Mu, Zhi Yin

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.

E. R. Caianiello, Gabriella Di Genova

Eva-Maria Graefe, Michael Höning, Hans Jürgen Korsch

We investigate the classical limit of non-Hermitian quantum dynamics arising from a coherent state approximation, and show that the resulting classical phase space dynamics can be described by generalised "canonical" equations of motion, for both conservative and dissipative motion. The dynamical equations combine a symplectic flow associated with the Hermitian part of the Hamiltonian with a metric gradient flow associated with the anti-Hermitian part of the Hamiltonian. We derive this structure of the classical limit of quantum systems in the case of a Euclidean phase space geometry. As an example we show that the classical dynamics of a damped and driven oscillator can be linked to a non-Hermitian quantum system, and investigate the quantum classical correspondence. Furthermore, we present an example of an angular momentum system whose classical phase space is spherical and show that the generalised canonical structure persists for this nontrivial phase space geometry.

M. Moshińsky

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.

James Thomas Wheeler