Distortion of the Poisson Bracket by the Noncommutative Planck Constants

Abstract

In this paper we introduce a kind of "noncommutative neighbourhood" of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a "distortion" of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical shadow.

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

“…Let us now go back to the analogy between the semiclassical quantum mechanics and the quasithermodynamic statistical physics discussed in the introduction. This paper is a natural continuation of [1]. It is important to stress, that in order to construct a reasonable q-analogue of a classical theory (mechanics or thermodynamics), it is not enough just to apply the q-analysis (replacing derivatives with q-derivatives, factorials with q-factorials, etc.).…”

“…It is necessary to introduce the Planck-Boltzmann constants → 0 or k B → 0 into the theory first. The paper [1] is focused on → 0, and starts with an investigation of a q-analogue of the Weyl quantization map in quantum mechanics. In the non-q-deformed case this is just a symmetrization map linking the classical coordinates x and momenta p, with the quantized coordinates and momenta x and p. Trying to construct a reasonable q-analogue of such symmetrization map in case q is a 2s × 2s matrix of formal variables (2) defining the braidings (3) on the phase space (where s is the number of degrees of freedom), one realizes that the Planck constant should acquire indices, → i,j , i, j ∈ [2s], and should have same commutation properties as the q-commutator…”

“…Denote the set of all such collections as S q ( x). Then the q-analogue of a solution of the Hamilton-Jacobi equation can be perceived as an infinite sequence S (0) , S (1) , S (2) , . .…”

“…It turns out, that the consequences of these investigations seem to go much deeper than one would expect and indicate a necessity to replace the fundamental constants and k B with infinite collections of quantities { Γ } Γ and {(k B ) Γ } Γ , respectively. These collections correspond to the generators of the epoché algebra introduced in [1], and are structured in a slightly more complicated way than infinite dimensional matrices (the Heisenberg's quantization). The index Γ varies over a set of finite leaf-labelled planar binary trees of different sizes, while the labelling set is just the set of symbols corresponding to the degrees of freedom of the limiting physical system.…”

“…In one of our recent papers [1], among other things, we were interested in motivating a q-analogue of the Weyl quantization map W in quantum mechanics. Recall, that the map W can be described as follows.…”