Relativity of $\hbar$

TL;DR

This paper explores the geometric structure of quantum mechanics by linking coherent states to complex structures on phase space, revealing how dualities relate to non-holomorphic transformations and the local nature of quantum coherence.

Contribution

It establishes a geometric framework connecting coherent states with complex structures on phase space, providing a new perspective on dualities in quantum mechanics.

Findings

Coherent states correspond to complex structures on phase space.

Duality transformations are non-holomorphic changes of coordinates.

Quantum coherence is a local property on classical phase space.

Abstract

Looking for a quantum-mechanical implementation of duality, we formulate a relation between coherent states and complex-differentiable structures on classical phase space ${\cal C}$. A necessary and sufficient condition for the existence of locally-defined coherent states is the existence of an almost complex structure on ${\cal C}$. A necessary and sufficient condition for globally-defined coherent states is a complex structure on ${\cal C}$. The picture of quantum mechanics that emerges is conceptually close to that of a geometric manifold covered by local coordinate charts. Instead of the latter, quantum mechanics has local coherent states. A change of coordinates on ${\cal C}$ may or may not be holomorphic. Correspondingly, a transformation between quantum-mechanical states may or may not preserve coherence. Those that do not preserve coherence are duality transformations. A duality appears as the possibility of giving two or more, apparently different, descriptions of the same quantum-mechanical phenomenon. Coherence becomes a local property on classical phase space. Observers on ${\cal C}$ not connected by means of a holomorphic change of coordinates need not, and in general will not, agree on what is a semiclassical effect vs. what is a strong quantum effect