q-Legendre transformation: partition functions and quantization of the Boltzmann constant

In this paper we construct a q-analogue of the Legendre transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wavefunctions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton-Jacobi equation in mechanics to the q-Legendre transformation in thermodynamics. It is shown that this requires a non-commutative analogue of the Planck-Boltzmann constants (ℎ and kB) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the quasithermodynamic parameter corresponding to the Boltzmann constant with an infinite collection of generators of the so-called epoche (bracketing) algebra. The latter is an infinite-dimensional non-commutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of 'deformations of deformations' of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way that the trees with a single leaf correspond to the observables of the limiting thermodynamic system.