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

We consider bosons on Euclidean R^4 that are minimally coupled to an externalYang-Mills field. We compute the logarithmically divergent part of the cut-offregularized quantum effective action of this system. We confirm the knownresult that this term is proportional to the Yang-Mills action. We use pseudodifferential operator methods throughout to prepare the groundfor a generalization of our calculation to the noncommutative four-dimensionalMoyal plane (also known as noncommutative flat space). We also include adetailed comparison of our cut-off regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique ofasymptotic expansion in the momentum variable with operator theoretic argumentsin order to keep separated quantum from noncommutativity effects. We show thatthe result from the commutative space R^4 still holds if one replaces allpointwise products by the noncommutative Moyal product.Comment: 37 pages, v2 contains an improved treatment of the theta function in Appendix A.