The Geometry of Quantum Mechanics

TL;DR

This paper introduces a geometric 'dequantisation' method that transforms quantum mechanics into classical theories, potentially revealing multiple classical limits beyond the standard  limit, with implications for understanding quantum-classical relationships.

Contribution

It proposes a novel geometric 'dequantisation' process applicable to any quantum theory, expanding the concept of classical limits beyond traditional approaches.

Findings

Provides a geometric procedure for dequantising quantum mechanics.

Shows that multiple classical limits can be obtained, including the standard  limit.

Suggests a framework linking quantum theories with diverse classical counterparts.

Abstract

A recent notion in theoretical physics is that not all quantum theories arise from quantising a classical system. Also, a given quantum model may possess more than just one classical limit. These facts find strong evidence in string duality and M-theory, and it has been suggested that they should also have a counterpart in quantum mechanics. In view of these developments we propose "dequantisation", a mechanism to render a quantum theory classical. Specifically, we present a geometric procedure to "dequantise" a given quantum mechanics (regardless of its classical origin, if any) to possibly different classical limits, whose quantisation gives back the original quantum theory. The standard classical limit $\hbar\to 0$ arises as a particular case of our approach.