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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.16174 |
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| _version_ | 1866908803477274624 |
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| author | Barvinsky, Andrei O. Kalugin, Alexey E. |
| author_facet | Barvinsky, Andrei O. Kalugin, Alexey E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Notes on peculiarities of Schwinger--DeWitt technique: one-loop double poles, total-derivative terms and determinant anomalies Barvinsky, Andrei O. Kalugin, Alexey E. High Energy Physics - Theory 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. |
| title | Notes on peculiarities of Schwinger--DeWitt technique: one-loop double poles, total-derivative terms and determinant anomalies |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2408.16174 |