Noncommutative Gauge Field Theories: A No-Go Theorem

TL;DR

This paper proves a no-go theorem for noncommutative gauge theories, showing strict limitations on gauge algebra representations and matter field transformations, which impacts the formulation of noncommutative Standard Models.

Contribution

It establishes a no-go theorem that constrains the structure of noncommutative gauge theories, particularly regarding gauge algebra representations and matter field charges.

Findings

The gauge algebra admits only irreducible n by n matrix representations.

Matter fields can only be in fundamental, adjoint, or singlet states.

Matter fields can transform under at most two noncommutative gauge group factors.

Abstract

Studying the general structure of the noncommutative (NC) local groups, we prove a no-go theorem for NC gauge theories. According to this theorem, the closure condition of the gauge algebra implies that: 1) the local NC $u(n)$ {\it algebra} only admits the irreducible n by n matrix-representation. Hence the gauge fields are in n by n matrix form, while the matter fields {\it can only be} in fundamental, adjoint or singlet states; 2) for any gauge group consisting of several simple-group factors, the matter fields can transform nontrivially under {\it at most two} NC group factors. In other words, the matter fields cannot carry more than two NC gauge group charges. This no-go theorem imposes strong restrictions on the NC version of the Standard Model and in resolving the standing problem of charge quantization in noncommutative QED.