Geometrical Formulation of Quantum Mechanics

TL;DR

This paper presents a geometric reformulation of quantum mechanics using the language of symplectic and Kähler manifolds, offering new insights and potential pathways for generalizations beyond the standard framework.

Contribution

It introduces a geometric framework for quantum mechanics based on symplectic and Riemannian structures, enabling new perspectives and generalizations of the theory.

Findings

States are represented by points on a symplectic manifold.

Quantum observables correspond to real-valued functions on this manifold.

The Schrödinger evolution is described by symplectic flows generated by Hamiltonian functions.

Abstract

States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schr\"odinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features---such as uncertainties and state vector reductions---which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric---a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.