Yang-Mills action from minimally coupled bosons on R^4 and on the 4D Moyal plane

TL;DR

This paper calculates the divergent part of the quantum effective action for bosons coupled to Yang-Mills fields on R^4 and extends the analysis to the noncommutative Moyal plane, confirming the proportionality to the Yang-Mills action.

Contribution

It introduces a pseudodifferential operator approach to analyze quantum effects on noncommutative space, extending known results from commutative to noncommutative geometry.

Findings

Divergent part of effective action proportional to Yang-Mills action

Method applicable to both commutative and noncommutative spaces

Regularization techniques compared and validated

Abstract

We consider bosons on Euclidean R^4 that are minimally coupled to an external Yang-Mills field. We compute the logarithmically divergent part of the cut-off 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 (also known as noncommutative flat space). We also include a detailed comparison of our cut-off 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 R^4 still holds if one replaces all pointwise products by the noncommutative Moyal product.