Notes on peculiarities of Schwinger--DeWitt technique: one-loop double poles, total-derivative terms and determinant anomalies

Abstract

We discuss peculiarities of the Schwinger--DeWitt technique for quantum effective action, associated with the origin of dimensionally regularized double-pole divergences of the one-loop functional determinant for massive Proca model in a curved spacetime. These divergences have the form of the total-derivative term generated by integration by parts in the functional trace of the heat kernel for the Proca vector field operator. Because of the nonminimal structure of second-order derivatives in this operator, its vector field heat kernel has a nontrivial form, involving the convolution of the scalar d'Alembertian Green's function with its heat kernel. Moreover, its asymptotic expansion is very different from the universal predictions of Gilkey-Seeley heat kernel theory because the Proca operator violates one of the basic assumptions of this theory -- the nondegeneracy of the principal symbol of an elliptic operator. This modification of the asymptotic expansion explains the origin of double-pole total-derivative terms. Another hypostasis of such terms is in the problem of multiplicative determinant anomalies -- lack of factorization of the functional determinant of a product of differential operators into the product of their individual determinants. We demonstrate that this anomaly should have the form of total-derivative terms and check this statement by calculating divergent parts of functional determinants for products of minimal and nonminimal second-order differential operators in curved spacetime.

arXiv comment12 pages

References in corpus (9)

• Heat kernel expansion: user's manual
• Agravity
• Gravity, Non-Commutative Geometry and the Wodzicki Residue
• Zeta-function regularization, the multiplicative anomaly and the Wodzicki residue
• A Note on the Wodzicki Residue
• Regularization schemes and the multiplicative anomaly
• Heat kernel expansion for higher order minimal and nonminimal operators
• Functional Determinant of the Massive Laplace Operator and the Multiplicative Anomaly
• Universality and ambiguity in fermionic effective actions

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