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Schwinger-DeWitt expansion for the heat kernel of nonminimal operators in causal theories

A. O. Barvinsky*, A. E. Kalugin†, and W. Wachowski‡

  • *Contact author: barvin@td.lpi.ru
  • †Contact author: kalugin.ae@phystech.edu
  • ‡Contact author: vladvakh@gmail.com
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Phys. Rev. D 112, 076032 – Published 27 October, 2025

DOI: https://doi.org/10.1103/1njq-346g

Abstract

We suggest a systematic calculational scheme for heat kernels of covariant nonminimal operators in causal theories whose characteristic surfaces are null with respect to a generic metric. The calculational formalism is based on a pseudodifferential operator calculus that allows one to build a linear operator map from the heat kernel of the minimal operator to the nonminimal one. This map is realized as a local expansion in powers of spacetime curvature, dimensional background fields, and their covariant derivatives with the coefficients—the functions of the Synge world function and its derivatives. Finiteness of these functions, determined by multiple proper time integrals, is achieved by a special subtraction procedure, which is an important part of the calculational scheme. We illustrate this technique on the examples of the vector Proca model and the vector field operator with a nondegenerate principal symbol. We also discuss smoothness properties of heat kernels of nonminimal operators in connection with the nondegenerate nature of their operator symbols.

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References (32)

  1. B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965).
  2. P. B. Gilkey, The spectral geometry of a Riemannian manifold, J. Diff. Geom. 10, 601 (1975).
  3. P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (CRC Press, Boca Raton, Florida, 1995).
  4. A. O. Barvinsky and G. A. Vilkovisky, The generalized Schwinger–DeWitt technique in gauge theories and quantum gravity, Phys. Rep. 119, 1 (1985).
  5. A. O. Barvinsky, Heat kernel expansion in the background field formalism, Scholarpedia 10, 31644 (2015).
  6. D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rep. 388, 279 (2003).
  7. V. P. Gusynin, Seeley–Gilkey coefficients for fourth-order operators on a Riemannian manifold, Nucl. Phys. B333, 296 (1990).
  8. A. O. Barvinsky and W. Wachowski, Heat kernel expansion for higher order minimal and nonminimal operators, Phys. Rev. D 105, 065013 (2022).
  9. A. O. Barvinsky, A. V. Kurov, and W. Wachowski, Commutator technique for the heat kernel of minimal higher derivative operators, Phys. Rev. D 110, 085023 (2023).
  10. I. Jack and H. Osborn, Background field calculations in curved spacetime (I). General formalism and application to scalar fields, Nucl. Phys. B234, 331 (1984).
  11. V. P. Gusynin, E. V. Gorbar, and V. V. Romankov, Heat kernel expansion for nonminimal differential operations and manifolds with torsion, Nucl. Phys. B362, 449 (1991).
  12. A. Barvinsky, G. Camargo, A. Kalugin, N. Ohta, and I. Shapiro, Local term in the anomaly-induced action of Weyl quantum gravity, Phys. Rev. D 108, 086018 (2023).
  13. A. O. Barvinsky and A. E. Kalugin, Notes on peculiarities of the Schwinger-DeWitt technique: One-loop double poles, total-derivative terms, and determinant anomalies, Phys. Rev. D 110, 105007 (2024).
  14. A. O. Barvinsky, A. V. Kurov, and S. M. Sibiryakov, Beta functions of (3+1)-dimensional projectable Hořava gravity, Phys. Rev. D 105, 044009 (2022).
  15. V. P. Gusynin, Asymptotics of the heat kernel for nonminimal differential operators, Ukrainian Mathematical Journal 43, 1432 (1991).
  16. V. P. Gusynin and E. V. Gorbar, Local heat kernel asymptotics for nonminimal differential operators, Phys. Lett. B 270, 29 (1991).
  17. P. B. Gilkey, T. P. Branson, and S. A. Fulling, Heat equation asymptotics of nonminimal operators on differential forms, J. Math. Phys. (N.Y.) 32, 2089 (1991).
  18. T. P. Branson, P. B. Gilkey, and A. Pierzchalski, Heat equation asymptotics of elliptic operators with non-scalar leading symbol, Mathematische Nachrichten 166, 207 (1994).
  19. E. Guendelman, A. Leonidov, V. Nechitailo, and D. Owen, On the heat kernel in covariant background gauge, Phys. Lett. B 324, 160 (1994).
  20. S. Alexandrov and D. Vassilevich, Heat kernel for nonminimal operators on a Kähler manifold, J. Math. Phys. (N.Y.) 37, 5715 (1996).
  21. I. Avramidi and T. Branson, Heat kernel asymptotics of operators with non-Laplace principal part, Rev. Math. Phys. 13, 847 (2001).
  22. I. G. Avramidi, Gauged gravity via spectral asymptotics of non-Laplace type operators, J. High Energy Phys. 07 (2004) 030.
  23. I. Moss and D. Toms, Invariants of the heat equation for non-minimal operators, J. Phys. A 47, 215401 (2014).
  24. B. Iochum and T. Masson, Heat trace for Laplace type operators with non-scalar symbols, J. Geom. Phys. 116, 90 (2017).
  25. D. Grasso and S. Kuzenko, Effective actions in supersymmetric gauge theories: Heat kernels for non-minimal operators, J. High Energy Phys. 06 (2023) 120.
  26. A. O. Barvinsky and G. A. Vilkovisky, Covariant perturbation theory (II). Second order in the curvature. General algorithms, Nucl. Phys. B333, 471 (1990).
  27. A. O. Barvinsky and V. F. Mukhanov, New nonlocal effective action, Phys. Rev. D 66, 065007 (2002).
  28. A. O. Barvinsky, Y. V. Gusev, V. F. Mukhanov, and D. V. Nesterov, Nonperturbative late time asymptotics for the heat kernel in gravity theory, Phys. Rev. D 68, 105003 (2003).
  29. R. Endo, Gauge dependence of the gravitational conformal anomaly for the electromagnetic field, Prog. Theor. Phys. 71, 1366 (1984).
  30. E. Fradkin and G. Vilkovisky, On the renormalization of quantum field theory in curved space-time, Lett. Nuovo Cimento 19, 47 (1977).
  31. A. O. Barvinsky and G. A. Vilkovisky, The effective action in quantum field theory: Two-loop approximation, in Quantum Field Theory and Quantum Statistics. Essays in Honour of the Sixtieth Birthday of E. S. Fradkin (Hilger, Bristol, 1987), Vol. 1, pp. 245–275.
  32. https://rscf.ru/en/project/23-12-00051/.

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Phys. Rev. D 112, 076032– Published 27 October, 2025
  • Received 11 August 2025
  • Accepted 7 October 2025
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