Reformulation of Quantum Theory

TL;DR

This paper reformulates quantum mechanics using symplectic geometry, emphasizing a Hamiltonian perspective on the Hilbert space and extending the framework to arbitrary symplectic manifolds, moving away from linear structures.

Contribution

It introduces a symplectic geometric reformulation of quantum theory that generalizes the standard Hilbert space approach to arbitrary symplectic manifolds.

Findings

Reveals quantum mechanics as a Hamiltonian system on symplectic manifolds

Generalizes quantum structures beyond Hilbert spaces

Provides a geometric framework applicable to diverse symplectic manifolds

Abstract

We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert space as a linear real manifold equipped with its canonical symplectic form and restricting only to the expectation-value functions of Hermitian operators. Then in this framework, we reformulate the structure of quantum mechanics in the language of symplectic manifolds and avoid linear structure of Hilbert space in such a way that the results can be stated for an arbitrary symplectic manifold.