Quantization of Contact Manifolds and Thermodynamics

TL;DR

This paper explores the quantization of contact geometry in thermodynamics, linking classical thermodynamic variables with quantum algebra, and demonstrates its application to geometrical optics and odd-dimensional spheres.

Contribution

It introduces a novel quantum algebra for contact geometry, extending classical thermodynamics and geometrical optics into a quantum framework.

Findings

Quantum contact algebra is associative but non-trivial on constants.

The approach correctly models the transition from geometrical to wave optics.

Quantum contact geometry is exemplified on odd-dimensional spheres.

Abstract

The physical variables of classical thermodynamics occur in conjugate pairs such as pressure/volume, entropy/temperature, chemical potential/particle number. Nevertheless, and unlike in classical mechanics, there are an odd number of such thermodynamic co-ordinates. We review the formulation of thermodynamics and geometrical optics in terms of contact geometry. The Lagrange bracket provides a generalization of canonical commutation relations. Then we explore the quantization of this algebra by analogy to the quantization of mechanics. The quantum contact algebra is associative, but the constant functions are not represented by multiples of the identity: a reflection of the classical fact that Lagrange brackets satisfy the Jacobi identity but not the Leibnitz identity for derivations. We verify that this `quantization' describes correctly the passage from geometrical to wave optics as well. As an example, we work out the quantum contact geometry of odd-dimensional spheres.