Quantization of Contact Manifolds and Thermodynamics

Details Quantization of Contact Manifolds and Thermodynamics Quantization of Contact Manifolds and Thermodynamics

2007-03-20

arxiv.org/abs/math-ph/0703061v2

AnnalsPhys.323:768-782,2008

Physics

Mathematical Physics

Additional references; typos fixed; a clarifying remark added

Scientific paper

10.1016/j.aop.2007.05.001

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.

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