- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Covariant extrinsic curvature expansion of the nonlocal effective action for a massless scalar field on a manifold with boundary

A. Boasso, C. D. Fosco, B. C. Guntsche, and F. D. Mazzitelli

PDF

Phys. Rev. D 114, 025012 – Published 15 July, 2026

DOI: https://doi.org/10.1103/yg33-72z3

Abstract

We study the nonlocal effective action of a massless scalar field defined on a flat manifold with a curved boundary. Using a heat-kernel approach, we derive a covariant expansion of the nonlocal contribution to quadratic order in the extrinsic curvature tensor. Our construction provides a geometric framework that both reproduces earlier results obtained for Monge-patch embeddings and extends them to more general surfaces that need not admit a global Monge-patch description. The expansion is valid in the regime where gradients of the extrinsic curvature dominate over nonlinear curvature effects. As an application, we compute the particle-creation rate for an oscillating deformed ring in 2+1 dimensions and an oscillating deformed sphere in 3+1 dimensions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (48)

  1. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 1982), ISBN [Amazon][WorldCat], [Amazon][WorldCat].
  2. Leonard E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2009), 8, ISBN [Amazon][WorldCat], [Amazon][WorldCat], [Amazon][WorldCat].
  3. Robert M. Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics, Chicago Lectures in Physics (University of Chicago Press, Chicago, IL, 1995). ISBN [Amazon][WorldCat].
  4. Iosif L. Buchbinder and Ilya Shapiro, Introduction to Quantum Field Theory with Applications to Quantum Gravity, Oxford Graduate Texts (Oxford University Press, New York, 2021), 3, ISBN [Amazon][WorldCat], [Amazon][WorldCat].
  5. Esteban A. Calzetta and Bei-Lok B. Hu, Nonequilibrium Quantum Field Theory (Oxford University Press, New York, 2009), ISBN [Amazon][WorldCat], [Amazon][WorldCat], [Amazon][WorldCat], [Amazon][WorldCat], [Amazon][WorldCat].
  6. Bryce S. DeWitt, Quantum field theory in curved spacetime, Phys. Rep. 19, 295 (1975).
  7. D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rep. 388, 279 (2003).
  8. Klaus Kirsten, Spectral Functions in Mathematics and Physics (Chapman and Hall/CRC, Boca Raton, 2001), ISBN [Amazon][WorldCat].
  9. I. G. Avramidi, A covariant technique for the calculation of the one-loop effective Action, Nucl. Phys. B355, 712 (1991).
  10. G. A. Vilkovisky, The Gospel according to DeWitt, Quantum Theory of Gravity: Essays in Honor of the 60th Birthday of Bryce S. DeWitt, edited by S. M. Christensen (Adam Hilger, Bristol, 1984).
  11. A. O. Barvinsky and G. A. Vilkovisky, Beyond the Schwinger-Dewitt technique: Converting loops into trees and in-in currents, Nucl. Phys. B282, 163 (1987).
  12. A. O. Barvinsky and G. A. Vilkovisky, Covariant perturbation theory. 2: Second order in the curvature, General algorithms, Nucl. Phys. B333, 471 (1990).
  13. A. O. Barvinsky, Yu. V. Gusev, G. A. Vilkovisky, and V. V. Zhytnikov, The one loop effective action and trace anomaly in four-dimensions, Nucl. Phys. B439, 561 (1995).
  14. A. O. Barvinsky, Yu. V. Gusev, G. A. Vilkovisky, and V. V. Zhytnikov, The basis of nonlocal curvature invariants in quantum gravity theory. Third order, J. Math. Phys. (N.Y.) 35, 3525 (1994).
  15. A. G. Mirzabekian and G. A. Vilkovisky, Particle creation in the effective action method, Ann. Phys. (N.Y.) 270, 391 (1998).
  16. A. G. Mirzabekian and G. A. Vilkovisky, Vacuum radiation in covariant perturbation theory, Phys. Lett. B 414, 123 (1997).
  17. D. A. R. Dalvit and F. D. Mazzitelli, Heat kernel and scaling of gravitational constants, Phys. Rev. D 52, 2577 (1995).
  18. Diego A. R. Dalvit and Francisco D. Mazzitelli, Running coupling constants, Newtonian potential and nonlocalities in the effective action, Phys. Rev. D 50, 1001 (1994).
  19. D. López Nacir and F. D. Mazzitelli, Running of Newton’s constant and noninteger powers of the d’Alembertian, Phys. Rev. D 75, 024003 (2007).
  20. R. J. Riegert, A nonlocal action for the trace anomaly, Phys. Lett. 134B, 56 (1984).
  21. Emil Mottola and Ruslan Vaulin, Macroscopic effects of the quantum trace anomaly, Phys. Rev. D 74, 064004 (2006).
  22. A. O. Barvinsky and W. Wachowski, Notes on conformal anomaly, nonlocal effective action, and the metamorphosis of the running scale, Phys. Rev. D 108, 045014 (2023).
  23. John F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett. 72, 2996 (1994).
  24. John F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50, 3874 (1994).
  25. John F. Donoghue and Basem Kamal El-Menoufi, Nonlocal quantum effects in cosmology: Quantum memory, nonlocal FLRW equations, and singularity avoidance, Phys. Rev. D 89, 104062 (2014).
  26. Xavier Calmet, Basem Kamal El-Menoufi, Boris Latosh, and Sonali Mohapatra, Gravitational radiation in quantum gravity, Eur. Phys. J. C 78, 780 (2018).
  27. Xavier Calmet, Roberto Casadio, and Folkert Kuipers, Quantum gravitational corrections to a star metric and the black hole limit, Phys. Rev. D 100, 086010 (2019).
  28. Xavier Calmet, Roberto Casadio, Stephen D. H. Hsu, and Folkert Kuipers, Quantum hair from gravity, Phys. Rev. Lett. 128, 111301 (2022).
  29. Andrés Boasso, Sebastián Franchino Viñas, and Francisco D. Mazzitelli, Nonlocal effective action and particle creation in d dimensions, Phys. Rev. D 111, 085023 (2025).
  30. Andrés Boasso and Francisco D Mazzitelli, Vacuum polarization in the horizonless Bardeen metric, arXiv:2508.06465.
  31. Ilya L. Shapiro, Effective action of vacuum: Semiclassical approach, Classical Quantum Gravity 25, 103001 (2008).
  32. Thomas P. Branson, Peter B. Gilkey, and Dmitri V. Vassilevich, The asymptotics of the Laplacian on a manifold with boundary. II, Boll. Unione Mat. Ital., B 11, 39 (1997).
  33. M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect, International Series of Monographs on Physics Vol. 145 (Oxford University Press, New York, 2009), ISBN [Amazon][WorldCat].
  34. G. Plunien, B. Müller, and W. Greiner, The Casimir effect, Phys. Rep. 134, 87 (1986).
  35. G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, The Casimir force between real materials: Experiment and theory, Rev. Mod. Phys. 81, 1827 (2009).
  36. K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, Singapore, 2001).
  37. G. T. Moore, Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys. (N.Y.) 11, 2679 (1970).
  38. S. A. Fulling and P. C. W. Davies, Radiation from a moving mirror in two-dimensional space-time: Conformal anomaly, Proc. R. Soc. A 348, 393 (1976).
  39. P. C. W. Davies and S. A. Fulling, Radiation from moving mirrors and from black holes, Proc. R. Soc. A 356, 237 (1977).
  40. V. V. Dodonov, Current status of the dynamical Casimir effect, Phys. Scr. 82, 038105 (2010).
  41. Diego A. R. Dalvit, Paulo A. Maia Neto, and Francisco Diego Mazzitelli, Fluctuations, dissipation and the dynamical Casimir effect, Lect. Notes Phys. 834, 419 (2011).
  42. P. D. Nation, J. R. Johansson, M. P. Blencowe, and F. Nori, Colloquium: Stimulating uncertainty: Amplifying the quantum vacuum with superconducting circuits, Rev. Mod. Phys. 84, 1 (2012).
  43. V. V. Dodonov, Fifty years of the dynamical Casimir effect, Physics 2, 67 (2020).
  44. C. D. Fosco and B. C. Guntsche, Quantum dissipative effects for a real scalar field coupled to a time-dependent Dirichlet surface in d+1 dimensions, Phys. Rev. D 110, 105021 (2024).
  45. C. D. Fosco and B. C. Guntsche, Quantum dissipative effects for a real scalar field coupled to a dynamical Neumann surface in d+1 dimensions, Phys. Rev. D 112, 105017 (2025).
  46. M. Setare and Aram Saharian, Particle creation in an oscillating spherical cavity, Mod. Phys. Lett. A 16, 1269 (2001).
  47. D. F. Mundarain and P. A. Maia Neto, Quantum radiation in a plane cavity with moving mirrors, Phys. Rev. A 57, 1379 (1998).
  48. Frank W. J. Olver, Asymptotics and Special Functions, 2nd ed. (A K Peters, Wellesley, Massachusetts, 1997), ISBN [Amazon][WorldCat].

Outline

Information

Phys. Rev. D 114, 025012– Published 15 July, 2026
  • Received 19 May 2026
  • Accepted 23 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation