Yang-Mills theory for non-semisimple groups

TL;DR

This paper explores Yang-Mills theories for non-semisimple groups, revealing that they can have more gauge fields than generators, which may have significant physical implications.

Contribution

It demonstrates that non-semisimple groups can have additional Yang-Mills fields and analyzes a specific two-generator non-commutative example.

Findings

Non-semisimple groups can have more gauge fields than generators.

Additional fields influence gauge transformations of original fields.

Detailed study of a two-generator non-commutative group.

Abstract

For semisimple groups, possibly multiplied by U(1)'s, the number of Yang-Mills gauge fields is equal to the number of generators of the group. In this paper, it is shown that, for non-semisimple groups, the number of Yang-Mills fields can be larger. These additional Yang-Mills fields are not irrelevant because they appear in the gauge transformations of the original Yang-Mills fields. Such non-semisimple Yang-Mills theories may lead to physical consequences worth studying. The non-semisimple group with only two generators that do not commute is studied in detail.