Symplectic Quantum Structure

D. J. Simms, L. Castel

Dariusz Chruściński

Dariusz Chruściński

Hoshang Heydari

We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard Kähler framework, we introduce an extension in which the symplectic structure is allowed to couple to a metric-affine background geometry, leading to a deformation of the Hamiltonian flow on the state space. We show that, under suitable conditions, the deformed structure remains symplectic and defines a well-posed Hamiltonian system. The formulation reduces to standard Schrödinger dynamics in the limit where the geometric deformation vanishes. Explicit analytical examples are constructed to illustrate the effect of the deformation. In particular, curvature-dependent deformations lead to a rescaling of Hamiltonian flows, while torsion-induced contributions produce direction-dependent corrections. In addition, geometric phases acquire corrections determined by the deformed symplectic structure. These results provide a mathematically consistent framework for exploring geometric modifications of quantum evolution induced by background curvature and affine structure.

Piotr Dulian, Adam Sawicki

In this work we explore the implications of applying the formalism of symplectic geometry to quantum mechanics, particularly focusing on many-particle systems. We extend the concept of a symplectic indicator of entanglement, originally introduced by Sawicki et al., Commun. Math. Phys. 305, 441 (2011), to these complex systems. Specifically, we demonstrate that the restriction of the symplectic structure to manifolds comprising all states characterized by isospectral reduced one-particle density matrices ${M}_{\ensuremath{\mu}(\ensuremath{\psi})}^{0}$ exhibits degeneracy for nonseparable states. We prove that the degree of degeneracy at any given state $|\ensuremath{\varphi}\ensuremath{\rangle}\ensuremath{\in}{M}_{\ensuremath{\mu}(\ensuremath{\psi})}^{0}$ corresponds to the degree of degeneracy of the symplectic form $\ensuremath{\omega}$ when restricted to the manifold of states that are locally unitary equivalent with $|\ensuremath{\varphi}\ensuremath{\rangle}$. Additionally, we provide a physical interpretation of this symplectic indicator of entanglement, articulating it as an inherent ambiguity within the associated classical dynamical framework. Our findings underscore the pivotal role of symplectic geometry in elucidating entanglement properties in quantum mechanics and suggest avenues for further exploration into the geometric structures underlying quantum state spaces.