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Yang-Mills action from minimally coupled bosons on \(\mathbb R^ 4\) and on the four-dimensional Moyal plane. (English) Zbl 1076.58023
Summary: We consider bosons on (Euclidean) \(\mathbb R^4\) 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 \(\mathbb R^4_{\theta}\). 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 \(\mathbb R^4\) still holds if one replaces all pointwise products by the noncommutative Moyal product.

MSC:
58J52 Determinants and determinant bundles, analytic torsion
58J42 Noncommutative global analysis, noncommutative residues
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81T13 Yang-Mills and other gauge theories in quantum field theory
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