- 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

Pseudoreal quantum fields

Maxim N. Chernodub1,2,*, Peter Millington3,†, and Esra Sablevice3,‡

  • *Contact author: maxim.chernodub@univ-tours.fr
  • †Contact author: peter.millington@manchester.ac.uk
  • ‡Contact author: esra.sablevice@manchester.ac.uk
PDF

Phys. Rev. D 112, 065007 – Published 15 September, 2025

DOI: https://doi.org/10.1103/kn9t-f8vx

Abstract

We introduce the concept of pseudoreality for complex numbers. We show that this concept, applied to quantum fields, provides a unifying framework for two distinct approaches to pseudo-Hermitian quantum field theories. The first approach stems from analytically continuing Hermitian theories into the complex plane, while the second is based on constructing them from first principles. The pseudoreality condition for bosonic fields resolves a longstanding problem with the formulation of gauge theories involving pseudo-Hermitian currents, sheds new light on the resolution of the so-called Hermiticity puzzle, and may allow a consistent minimal coupling of pseudo-Hermitian quantum field theories to gravity. We focus on the iϕ3 cubic scalar theory, obtaining the relevant pseudoreality conditions up to quadratic order in the coupling; a theory of two complex scalar fields with non-Hermitian mass mixing; and the latter’s coupling to a U(1) gauge field. The general principle of pseudoreality, however, is expected to contribute to the ongoing development of the first-principles construction of pseudo-Hermitian quantum field theories, including their formulation in curved spacetimes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (28)

  1. C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  2. C. M. Bender, S. Boettcher, and P. Meisinger, PT symmetric quantum mechanics, J. Math. Phys. (N.Y.) 40, 2201 (1999).
  3. C. M. Bender and P. D. Mannheim, PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues, Phys. Lett. A 374, 1616 (2010).
  4. A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. (N.Y.) 43, 205 (2002).
  5. A. Mostafazadeh, Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys. (N.Y.) 43, 2814 (2002).
  6. A. Mostafazadeh, Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries, J. Math. Phys. (N.Y.) 43, 3944 (2002).
  7. C. M. Bender, D. C. Brody, and H. F. Jones, Scalar quantum field theory with cubic interaction, Phys. Rev. Lett. 93, 251601 (2004).
  8. C. M. Bender, D. C. Brody, and H. F. Jones, Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction, Phys. Rev. D 70, 025001 (2004); 71, 049901(E) (2005).
  9. C. M. Bender, V. Branchina, and E. Messina, Ordinary versus PT-symmetric ϕ3 quantum field theory, Phys. Rev. D 85, 085001 (2012).
  10. C. M. Bender, V. Branchina, and E. Messina, Critical behavior of the PT-symmetric iϕ3 quantum field theory, Phys. Rev. D 87, 085029 (2013).
  11. A. M. Shalaby, Non-perturbative calculations for the effective potential of the PT symmetric and non-Hermitian (−gφ4) field theoretical model, Eur. Phys. J. C 50, 999 (2007).
  12. A. Shalaby and S. S. Al-Thoyaib, Non-perturbative tests for the asymptotic freedom in the PT-symmetric (−ϕ4)3+1 theory, Phys. Rev. D 82, 085013 (2010).
  13. W. Y. Ai, C. M. Bender, and S. Sarkar, PT-symmetric −gφ4 theory, Phys. Rev. D 106, 125016 (2022).
  14. P. D. Mannheim, Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories, Phys. Rev. D 99, 045006 (2019).
  15. J. Alexandre, P. Millington, and D. Seynaeve, Symmetries and conservation laws in non-Hermitian field theories, Phys. Rev. D 96, 065027 (2017).
  16. E. Sablevice and P. Millington, Poincaré symmetries and representations in pseudo-Hermitian quantum field theory, Phys. Rev. D 109, 6 (2024).
  17. J. Alexandre, J. Ellis, and P. Millington, Discrete spacetime symmetries and particle mixing in non-Hermitian scalar quantum field theories, Phys. Rev. D 102, 125030 (2020).
  18. J. Alexandre, J. Ellis, and P. Millington, Discrete spacetime symmetries, second quantization, and inner products in a non-Hermitian Dirac fermionic field theory, Phys. Rev. D 106, 065003 (2022).
  19. M. N. Chernodub and P. Millington, IR/UV mixing from local similarity maps of scalar non-Hermitian field theories, Phys. Rev. D 105, 076020 (2022).
  20. C. M. Bender and D. W. Hook, Universal spectral behavior of x2(ix)ε potentials, Phys. Rev. A 86, 022113 (2012).
  21. A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys. 07, 1191 (2010).
  22. Y. D. Li and Q. Wang, PT-symmetric quantum field theory in path integral formalism and arbitrariness problem, arXiv:2305.05809.
  23. Y. D. Li and Q. Wang, Isospectral local Hermitian theory for the PT-symmetric iϕ3 quantum field theory, Phys. Rev. D 111, 025016 (2025).
  24. A. Fring and T. Taira, Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories, Nucl. Phys. B950, 114834 (2020).
  25. J. Alexandre, J. Ellis, P. Millington, and D. Seynaeve, Gauge invariance and the Englert-Brout-Higgs mechanism in non-Hermitian field theories, Phys. Rev. D 99, 075024 (2019).
  26. P. Millington, Symmetry properties of non-Hermitian PT-symmetric quantum field theories, J. Phys.: Conf. Ser. 1586, 012001 (2020).
  27. J. Alexandre, J. Ellis, P. Millington, and D. Seynaeve, Spontaneously breaking non-Abelian gauge symmetry in non-Hermitian field theories, Phys. Rev. D 101, 035008 (2020).
  28. E. Witten, A note on complex spacetime metrics, in Frank Wilczek: 50 Years of Theoretical Physics, edited by A. Niemi, K. K. Phua, and A. Shapere (World Scientific, Singapore, 2022), pp. 245–280.

Outline

Information

Phys. Rev. D 112, 065007– Published 15 September, 2025
  • Received 29 January 2025
  • Accepted 13 July 2025
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