Deformation Quantization for Heisenberg Supergroup

TL;DR

This paper develops a non-formal deformation framework for the Heisenberg supergroup actions, introducing C*-superalgebras and a Universal Deformation Formula, with applications to noncommutative quantum field theory.

Contribution

It introduces a new deformation method for supergroup actions using Weyl ordered Kirillov's orbit method and defines C*-superalgebras compatible with this deformation.

Findings

Constructed a non-formal deformation machinery for Heisenberg supergroup actions.

Developed a Universal Deformation Formula for R^{m|n} actions.

Applied the framework to interpret renormalizability in noncommutative quantum field theory.

Abstract

We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R^d. However, the method used here differs from Rieffel's one: we obtain a Universal Deformation Formula for the actions of R^{m|n} as a byproduct of Weyl ordered Kirillov's orbit method adapted to the graded setting. To do so, we have to introduce the notion of C*-superalgebra, which is compatible with the deformation, and which can be seen as corresponding to noncommutative superspaces. We also use this construction to interpret the renormalizability of a noncommutative Quantum Field Theory.