Quantization, deformations, and new homological and categorical methods in mathematical physics

The purpose of this announcement is to describe "brackets" which generate both classical (Poisson) and quantum (commutator) brackets.

The principal step in deformation quantisation is an introduction of a "deformation parameter" q which labels different quantum pictures and for a special value (like q<0) the classical one. Similarly, our construction is based on an introduction by means of the Fourier transform of new variables (s,x,y) such that (x,y) is the Fourier dual to (q,p) and s is the Fourier dual to the Planck constant ħ. It appeared that points (s,x,y) are elements of the Heisenberg group Hⁿ. It is known since the works of von Neumann that the Heisenberg picture of quantum mechanics is generated by infinite dimensional non-commutative irreducible unitary representations of the Heisenberg group Hⁿ. But one-dimensional (commutative!) unitary representations of Hⁿ are often unemployed. It is shown within the presented framework that these one-dimensional representations contain classical dynamics exactly in the same way as the infinite-dimensional ones contain quantum dynamics. We present a p-mechanical version of brackets and a dynamical equation generated by them. Our considerations are illustrated by a simple example of the harmonic oscillator. More involved examples allowing to mix quantum and classic components within one system will be presented elsewhere.

In this case the deformation parameter is not an external object for the deformed system. Rather both (the parameter and the system itself) are incorporated in a single structure -- the Heisenberg (undeformed) group in this example. It is an interesting question if such unification is possible for other established examples of quantum deformations.

Reference/Link:

• http://arXiv.org/abs/math-ph/0007030