Yang–Mills action from minimally coupled bosons on R4 and on the four-dimensional Moyal plane

Abstract

We consider bosons on (Euclidean) R4 that are minimally coupled to an external Yang–Mills field. We compute the logarithmically divergent part of the cutoff regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang–Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative four-dimensional Moyal plane Rθ4. We also include a detailed comparison of our cutoff regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space R4 still holds if one replaces all pointwise products by the noncommutative Moyal product.