Quantum mechanics on the projective Hilbert space has a natural geometric structure: the Fubini–Study metric and the symplectic form combine into a Kähler geometry. The Schrödinger equation is Hamiltonian flow on this symplectic manifold.
The paper on geometric quantum mechanics in a symplectic framework (arXiv: 2603.22354) extends this by coupling the quantum system to a metric-affine background geometry — allowing curvature and torsion to deform the symplectic structure.
The deformation is controlled: the symplectic form remains nondegenerate (the flow is still Hamiltonian), and when the geometric deformation vanishes, standard Schrödinger dynamics is recovered. But with curvature, the Hamiltonian flow is rescaled — time runs faster or slower depending on the local geometry. With torsion, the flow acquires directional corrections — the system is pushed sideways by the twisting of the background.
Geometric phases — the Berry phase acquired by a quantum state transported around a loop — pick up corrections from the deformed symplectic structure. The phase depends not just on the Hamiltonian and the path, but on the geometry of the space in which the path is drawn.
The through-claim: quantum dynamics is sensitive to the geometry of its configuration space in a computable way. Curvature rescales time; torsion deflects trajectories; both modify geometric phases. The standard Schrödinger equation assumes flat projective space; deforming the geometry deforms the equation, and the deformations have explicit physical signatures.
2603.22354. Mathematical physics / geometric quantum mechanics / symplectic geometry / metric-affine geometry / Berry phase.