Noncommutative gauge field theories: a no-go theorem

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) algebra only admits the irreducible n×n matrix-representation. Hence the gauge fields are in n×n matrix form, while the matter fields 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 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.