Yang-mills Theories on Noncommutative Space-time

We describe some recent progress in our understanding of Yang-Mills theories formulated on noncommutative spaces and in particular how to formulate the standard model on such spaces The idea that space-time might be noncommutative at short distances is not new but it was taken very seriously recently because noncommutative coordinates were found in a specific limit of string theory. This is nevertheless not the only motivation to study Yang-Mills theories on noncommu-tative spaces. In the early days of quantum field theories, it was thought that a fundamental cutoff might be useful to regularize the infinities appearing in these theories. Nowadays it is understood that gauge theories describing the strong and electroweak interactions are renormalizable and thus infinities cancel, but it might still be useful to have a fundamental cutoff to make sense of a quantum theory of gravity, whatever this might be. A more pragmatic approach is that space-time could simply be noncom-mutative at short distances in which case one has to understand how the standard model can emerge as a low energy model of a Yang-Mills theory formulated on a noncommutative space-time. The simplest noncommutative relations one can study are [ˆ x µ , ˆ x ν ] ≡ ˆ x µ ˆ x ν − ˆ x ν ˆ x µ = iθ µν , θ µν ∈ C. Postulating such relations implies that Lorentz covariance is explicitly broken. These relations also imply uncertainty relations for space-time coordinates which are a reminiscence of the famous Heisenberg uncertainty relations for momentum and space coordinates. Note that θ µν is a dimensional full quantity, dim(θ µν)=mass −2. If this mass scale is large enough, θ µν can be used as an expansion parameter like in quantum mechanics. We adopt the usual convention: a variable or function with a hat is a noncommutative one.