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Heat kernel: Proper-time method, Fock–Schwinger gauge, path integral, and Wilson line

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Abstract

This paper is devoted to the proper-time method and describes a model case that reflects the subtleties of constructing the heat kernel, is easily extended to more general cases (curved space, manifold with a boundary), and contains two interrelated parts: an asymptotic expansion and a path integral representation. We discuss the significance of gauge conditions and the role of ordered exponentials in detail, derive a new nonrecursive formula for the Seeley–DeWitt coefficients on the diagonal, and show the equivalence of two main approaches using the exponential formula.

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Acknowledgments

The authors are grateful to A. G. Pronko for the discussion of the text.

Funding

This research was supported by a grant from the Russian Science Foundation (Project No. 18-11-00297).

A. V. Ivanov is a winner of the Young Russian Mathematician award and thanks its sponsors and jury.

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Authors and Affiliations

  1. St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

    A. V. Ivanov

  2. St. Petersburg National Research University of Information Technologies, Mechanics and Optics, St. Petersburg, Russia

    N. V. Kharuk

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Correspondence to A. V. Ivanov.

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Ivanov, A.V., Kharuk, N.V. Heat kernel: Proper-time method, Fock–Schwinger gauge, path integral, and Wilson line. Theor Math Phys 205, 1456–1472 (2020). https://doi.org/10.1134/S0040577920110057

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