A Quantum is a Complex Structure on Classical Phase Space

TL;DR

This paper explores how the concept of a quantum depends on the choice of complex-differentiable structure on classical phase space, highlighting the role of duality transformations and observer dependence.

Contribution

It analyzes the impact of different complex structures on classical phase space in defining quantum mechanics, emphasizing the observer-dependent nature of quantum notions.

Findings

Quantum notions depend on the chosen complex structure.

Duality transformations relate different quantum descriptions.

Complex structures on phase space are not unique.

Abstract

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space.