Geometric Deformation of Quantum Mechanics

TL;DR

This paper introduces Curved Quantum Mechanics, a geometric framework where quantum states are represented on a curved manifold influenced by gravity, leading to a transition from quantum to classical behavior based on system mass.

Contribution

It develops a new geometric approach to quantum mechanics using an infinite-dimensional Kähler manifold, linking curvature to gravity and system mass, and explores the resulting dynamics.

Findings

Exact solutions for complex projective and hyperbolic spaces.

Identification of a bifurcation at a critical curvature value.

Demonstration of quantum-classical transition with increasing mass.

Abstract

We develop a novel approach to Quantum Mechanics that we call Curved Quantum Mechanics. We introduce an infinite-dimensional K\"ahler manifold ${\cal M}$, that we call the state manifold, such that the cotangent space $T_z^*{\cal M}$ is a Hilbert space. In this approach, a state of a quantum system is described by a point in the cotangent bundle $T^*{\cal M}$, that is, by a point $z\in{\cal M}$ in the state manifold and a one-form $\psi\in T^*_z{\cal M}$. The quantum dynamics is described by an infinite-dimensional Hamiltonian system on the state manifold with a magnetic field $H$, which reduces to the Schr\"odinger equation for zero curvature and reduces to the equations of geodesics for zero magnetic field. The curvature of the state manifold is determined by gravity, that is, by the mass/energy of the system, so that for microscopic systems the manifold is flat and for macroscopic systems it is strongly curved, which prohibits Schr\"odinger cat type states. We solved the dynamical equations exactly for the complex projective space and the complex hyperbolic space and show that in the case of negative curvature there is a bifurcation at a critical value of the curvature. This means that for small mass all modes are in the quantum regime with the unitary periodic dynamics and for large mass there are classical modes, with not a periodic but rather an exponential time evolution leading to a collapse of the state vector.