Field theories on spaces with linear fuzziness

TL;DR

This paper develops field theory tools on noncommutative spaces where position operators follow Lie algebra relations, showing these models can be free from ultraviolet divergences when based on compact groups like SO(3).

Contribution

It introduces calculation methods for field theories on Lie algebra noncommutative spaces and analyzes their divergence properties, especially for compact groups.

Findings

Models with compact Lie groups are free from UV divergences.

Derived key tools for field calculations on Lie algebra noncommutative spaces.

Detailed example with SO(3)/SU(2) illustrates the approach.

Abstract

A noncommutative space is considered the position operators of which satisfy the commutativity relations of a Lie algebra. The basic tools for calculation on this space, including the product of the fields, inner product and the proper measure for integration are derived. Some general aspects of perturbative field theory calculations on this space are also discussed. Among the features of such models is that they are free from ultraviolet divergences (and hence free from UV/IR mixing as well), if the group is compact. The example of the group SO(3) or SU(2) is investigated in more detail.