Complex Space-Time and the Structure of Relativistic Quantum Theory

Year : 2026 | Volume : 03 | Issue : 01 | Page : 22 27
    By

    Abhishek Ranjan Singh*,

  1. Assistant Professor, Department of Physics, Jamuni Lal College, Hajipur, Bihar, India

Abstract

Relativistic quantum mechanics was developed to reconcile the principles of quantum mechanics with Einstein’s theory of relativity. Despite its success in describing high-energy particles, the theory continues to face unresolved conceptual and mathematical difficulties, particularly in relation to the nature of time, causality, and relativistic consistency. In recent years, the idea of extending space-time into the complex domain has emerged as a useful and potentially meaningful approach to these problems. In this work, a theoretical synthesis is presented to examine the role of complex space-time in relativistic quantum mechanics. The study discusses how complex time arises naturally in relativistic wave equations, quantum field theory, and path integral formulations. Established methods such as Wick rotation, complex coordinate transformations, and non-Hermitian but parity–time (PT) symmetric Hamiltonians are reviewed to illustrate how complex space-time contributes to mathematical stability and clearer physical interpretation (Dirac, 1928; Wick, 1954; Bender & Boettcher, 1998). The relevance of complex space-time is further explored in the context of relativistic particle dynamics and quantum cosmology, where it helps address, issues related to vacuum structure, time evolution, and causality (Feynman & Hibbs, 1965; Hartle & Hawking, 1983). The paper suggests that complex space-time should be viewed not only as a computational technique, but as a framework that may reflect deeper features of relativistic quantum phenomena.

Keywords: Complex space-time, PT symmetry, quantum field theory, relativistic quantum mechanics, wick rotation

[This article belongs to Recent Trends in Mathematics ]

How to cite this article:
Abhishek Ranjan Singh*. Complex Space-Time and the Structure of Relativistic Quantum Theory. Recent Trends in Mathematics. 2026; 03(01):22-27.
How to cite this URL:
Abhishek Ranjan Singh*. Complex Space-Time and the Structure of Relativistic Quantum Theory. Recent Trends in Mathematics. 2026; 03(01):22-27. Available from: https://journals.stmjournals.com/rtm/article=2026/view=239225


References

  1. Bender, C. M., & Boettcher, S. (1998). Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters, 80(24), 5243–5246. https://doi.org/10.1103/PhysRevLett.80.5243
  2. Dirac, P. A. M. (1928). The quantum theory of the electron. Proceedings of the Royal Society A, 117(778), 610–624. https://doi.org/10.1098/rspa.1928.0023
  3. Feynman, R. P., & Hibbs, A. R. (1965). Quantum mechanics and path integrals. McGraw–Hill.
  4. Hartle, J. B., & Hawking, S. W. (1983). Wave function of the universe. Physical Review D, 28(12), 2960–2975. https://doi.org/10.1103/PhysRevD.28.2960
  5. Klein, O. (1926). Quantentheorie und fünfdimensionale Relativitätstheorie. Zeitschrift für Physik, 37, 895–906. https://doi.org/10.1007/BF01397481
  6. Wick, G. C. (1954). Properties of Bethe–Salpeter wave functions. Physical Review, 96(4), 1124–1134. https://doi.org/10.1103/PhysRev.96.1124
  7. Ahmavaara Y. The structure of space and the formalism of relativistic quantum theory. I. Journal of Mathematical Physics. 1965 Jan 1;6(1):87-93.
  8. Banai M. Quantum relativity theory and quantum space-time. International journal of theoretical physics. 1984 Nov;23(11):1043-63.
  9. Feynman RP. Space-time approach to non-relativistic quantum mechanics. Reviews of modern physics. 1948 Apr 1;20(2):367.
  10. Ahmavaara Y. The Structure of Space and the Formalism of Relativistic Quantum Theory. II. Journal of Mathematical Physics. 1965 Feb 1;6(2):220-7.
Regular Issue Subscription Review Article
Volume 03
Issue 01
Received 03/02/2026
Accepted 25/02/2026
Published 10/03/2026
Publication Time 35 Days
66

Login


My IP

PlumX Metrics