- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Spacetime quantum mechanics for bosonic and fermionic systems

N. L. Diaz1,2,3,* and R. Rossignoli3,4

  • *Contact author: nldiaz.unlp@gmail.com
PDF

Phys. Rev. Research 8, 013132 – Published 6 February, 2026

DOI: https://doi.org/10.1103/zmky-mmwb

Abstract

We provide a Hilbert space approach to quantum mechanics where space and time are treated on an equal footing. Our approach replaces the standard dependence on an external classical time parameter with a spacetime-symmetric algebraic structure, thereby unifying the axioms that traditionally distinguish the treatment of spacelike and timelike separations. Standard quantum evolution can be recovered from timelike correlators, defined by means of a quantum action operator, a quantum version of the action of classical mechanics. The corresponding map also provides an alternative perspective on the path integral formulation that, in the case of fermions, does not require the use of Grassmann variables. In addition, the formalism can be interpreted in terms of generalized quantum states, codifying both the conventional information of a quantum system at a given time and its evolution. We show that these states are solutions to a quantum principle of stationary action grounded in timelike correlations and pseudo-entropies.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (79)

  1. J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).
  2. A. Einstein, On the electrodynamics of moving bodies, Ann. Phys. (Berlin) 17, 50 (1905).
  3. N. L. Diaz, J. M. Matera, and R. Rossignoli, Spacetime quantum and classical mechanics with dynamical foliation, Phys. Rev. D 109, 105008 (2024).
  4. N. Diaz, J. Matera, and R. Rossignoli, Path integrals from spacetime quantum actions, Ann. Phys. (NY) 479, 170052 (2025).
  5. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de Sitter space/conformal field theory correspondence, Phys. Rev. Lett. 96, 181602 (2006).
  6. M. Van Raamsdonk, Building up space–time with quantum entanglement, Int. J. Mod. Phys. D 19, 2429 (2010).
  7. G. Evenbly and G. Vidal, Tensor network states and geometry, J. Stat. Phys. 145, 891 (2011).
  8. C. Cao, S. M. Carroll, and S. Michalakis, Space from Hilbert space: Recovering geometry from bulk entanglement, Phys. Rev. D 95, 024031 (2017).
  9. R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. 20, 367 (1948).
  10. N. L. Diaz, J. M. Matera, and R. Rossignoli, Spacetime quantum actions, Phys. Rev. D 103, 065011 (2021).
  11. D. N. Page and W. K. Wootters, Evolution without evolution: Dynamics described by stationary observables, Phys. Rev. D 27, 2885 (1983).
  12. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum time, Phys. Rev. D 92, 045033 (2015).
  13. A. Boette, R. Rossignoli, N. Gigena, and M. Cerezo, System-time entanglement in a discrete-time model, Phys. Rev. A 93, 062127 (2016).
  14. N. L. Diaz and R. Rossignoli, History state formalism for Dirac’s theory, Phys. Rev. D 99, 045008 (2019).
  15. N. L. Diaz, J. M. Matera, and R. Rossignoli, History state formalism for scalar particles, Phys. Rev. D 100, 125020 (2019).
  16. P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, Trinity of relational quantum dynamics, Phys. Rev. D 104, 066001 (2021).
  17. T. Favalli and A. Smerzi, A model of quantum spacetime, AVS Quantum Sci. 4, 044403 (2022).
  18. N. L. Diaz, P. Braccia, M. Larocca, J. Matera, R. Rossignoli, and M. Cerezo, Parallel-in-time quantum simulation via Page and Wootters quantum time, Phys. Rev. Res. 7, 033294 (2025).
  19. V. Giovannetti, S. Lloyd, and L. Maccone, Geometric event-based quantum mechanics, New J. Phys. 25, 023027 (2023).
  20. M. Suleymanov, I. L. Paiva, and E. Cohen, Nonrelativistic spatiotemporal quantum reference frames, Phys. Rev. A 109, 032205 (2024).
  21. D. Cafasso, N. Pranzini, J. Y. Malo, V. Giovannetti, and M. Chiofalo, Quantum time and the time-dilation induced interaction transfer mechanism, Phys. Rev. D 110, 106014 (2024).
  22. A. Milekhin, Z. Adamska, and J. Preskill, Observable and computable entanglement in time, arXiv:2502.12240.
  23. Wu-zhong Guo, Spacetime density matrix: Formalism and properties, J. High Energy Phys. 12 (2026) 128.
  24. K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, Pseudoentropy in dS/CFT and timelike entanglement entropy, Phys. Rev. Lett. 130, 031601 (2023).
  25. K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, Timelike entanglement entropy, J. High Energy Phys. 05 (2023) 052.
  26. K. Narayan, De Sitter space, extremal surfaces, and time entanglement, Phys. Rev. D 107, 126004 (2023).
  27. C.-S. Chu and H. Parihar, Time-like entanglement entropy in AdS/BCFT, J. High Energ. Phys. 06 (2023) 173.
  28. J. F. Fitzsimons, J. A. Jones, and V. Vedral, Quantum correlations which imply causation, Sci. Rep. 5, 18281 (2015).
  29. D. Horsman, C. Heunen, M. F. Pusey, J. Barrett, and R. W. Spekkens, Can a quantum state over time resemble a quantum state at a single time? Proc. R. Soc. A 473, 20170395 (2017).
  30. J. Cotler, C. M. Jian, X. Qi, and F. Wilczek, Superdensity operators for spacetime quantum mechanics, J. High Energy Phys. 09 (2018) 093.
  31. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010).
  32. H. Buhrman, R. Cleve, J. Watrous, and R. De Wolf, Quantum fingerprinting, Phys. Rev. Lett. 87, 167902 (2001).
  33. D. Bultrini, M. H. Gordon, P. Czarnik, A. Arrasmith, M. Cerezo, P. J. Coles, and L. Cincio, Unifying and benchmarking state-of-the-art quantum error mitigation techniques, Quantum 7, 1034 (2023).
  34. Z. Holmes, N. J. Coble, A. T. Sornborger, and Y. Subaşı, Nonlinear transformations in quantum computation, Phys. Rev. Res. 5, 013105 (2023).
  35. N. Diaz, D. García-Martín, S. Kazi, M. Larocca, and M. Cerezo, Showcasing a barren plateau theory beyond the dynamical lie algebra, arXiv:2310.11505.
  36. P. Jordan and E. Wigner, Über das paulische äquivalenzverbot, Z. Phys. 47, 631 (1928).
  37. E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. (NY) 16, 407 (1961).
  38. R. Jozsa and A. Miyake, Matchgates and classical simulation of quantum circuits, Proc. R. Soc. A 464, 3089 (2008).
  39. It is clear that if we think of eiεP as a two-qubit operation, it is a matchgate with the convention of [35, 38, 79]: eiεP=exp{iπ4(σ2⊗σ1−σ1⊗σ2)} with σ2⊗σ1,σ1⊗σ2 both a product of two Majoranas [35]. In this sense, eiεP is a variation of the fswap gate [79].
  40. M. Gaudin, Une démonstration simplifiée du théorème de Wick en mécanique statistique, Nucl. Phys. 15, 89 (1960).
  41. T. Evans and D. A. Steer, Wick’s theorem at finite temperature, Nucl. Phys. B 474, 481 (1996).
  42. A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge University Press, Cambridge, 2010).
  43. One can show that while the quantum action is related to Feynman PIs, the operator S̃ is instead related to Schwinger-Keldysh PIs [61]. We employ the second to discuss generalized states since here we are interested in expectation values at a single initial time. See also Sec. 4e.
  44. Y. Aharonov, D. Z. Albert, and L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60, 1351 (1988).
  45. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Colloquium: Understanding quantum weak values: Basics and applications, Rev. Mod. Phys. 86, 307 (2014).
  46. P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer Science & Business Media, New York, 2004).
  47. This type of entropy was introduced in the context of dS-CFT correspondence [24] (for standard QM).
  48. R. Balian, Information in statistical physics, Stud. Hist. Philos. Sci. Part B: Stud. Hist. Philos. Mod. Phys. 36, 323 (2005).
  49. R. Balian, D. Haar, and J. Gregg, From Microphysics to Macrophysics: Methods and Applications of Statistical Physics, Theoretical and Mathematical Physics (Springer, Berlin, Heidelberg, 2006), Vol. 1.
  50. M. A. Nielsen, C. M. Dawson, J. L. Dodd, A. Gilchrist, D. Mortimer, T. J. Osborne, M. J. Bremner, A. W. Harrow, and A. Hines, Quantum dynamics as a physical resource, Phys. Rev. A 67, 052301 (2003).
  51. N. Gigena and R. Rossignoli, Entanglement in fermion systems, Phys. Rev. A 92, 042326 (2015).
  52. H. Kleinert, Path integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (World Scientific Publishing Company, Singapore, 2006).
  53. G. Puddu, P. Bortignon, and R. Broglia, The RPA-SPA approximation to level densities, Ann. Phys. (NY) 206, 409 (1991).
  54. R. Rossignoli, N. Canosa, and P. Ring, Thermal and quantal fluctuations for fixed particle number in finite superfluid systems, Phys. Rev. Lett. 80, 1853 (1998).
  55. Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka, and Z. Wei, New holographic generalization of entanglement entropy, Phys. Rev. D 103, 026005 (2021).
  56. M. E. Peskin, An Introduction to Quantum Field Theory (CRC Press, Boca Raton, FL, 2018).
  57. F. Schwabl, Advanced Quantum Mechanics (Springer, Berlin, Heidelberg, 2008).
  58. B. Thaller, The Dirac Equation (Springer-Verlag, Berlin, Heidelberg, 1992).
  59. R. P. Feynman, The theory of positrons, in Quantum Electrodynamics (CRC Press, Boca Raton, FL, 2018), pp. 167–177.
  60. L. C. Céleri, V. Kiosses, and D. R. Terno, Spin and localization of relativistic fermions and uncertainty relations, Phys. Rev. A 94, 062115 (2016).
  61. J. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2, 407 (1961).
  62. A. J. Parzygnat, T. Takayanagi, Y. Taki, and Z. Wei, SVD entanglement entropy, J. High Energy Phys. 12 (2023) 123.
  63. J. Fullwood and A. J. Parzygnat, Operator representation of spatiotemporal quantum correlations, arXiv:2405.17555.
  64. W. H. Zurek, Environment-assisted invariance, entanglement, and probabilities in quantum physics, Phys. Rev. Lett. 90, 120404 (2003).
  65. W. H. Zurek, Wave-packet collapse and the core quantum postulates: Discreteness of quantum jumps from unitarity, repeatability, and actionable information, Phys. Rev. A 87, 052111 (2013).
  66. N. Gigena and R. Rossignoli, One-body information loss in fermion systems, Phys. Rev. A 94, 042315 (2016).
  67. R. Shankar, Principles of Quantum Mechanics (Springer Science & Business Media, New York, 2012).
  68. P. A. M. Dirac, Generalized Hamiltonian dynamics, Can. J. Math. 2, 129 (1950).
  69. C. Kiefer, Quantum gravity, Int. Ser. Monogr. Phys. 155, 1 (2004).
  70. R. B. Griffiths, Consistent histories and the interpretation of quantum mechanics, J. Stat. Phys. 36, 219 (1984).
  71. R. Omnös, Logical reformulation of quantum mechanics. I. Foundations, J. Stat. Phys. 53, 893 (1988).
  72. M. Gell-Mann and J. B. Hartle, Quantum mechanics in the light of quantum cosmology, in Murray Gell-Mann: Selected Papers (World Scientific, Singapore, 2010), pp. 303–325.
  73. C. J. Isham, Quantum logic and the histories approach to quantum theory, J. Math. Phys. 35, 2157 (1994).
  74. C. J. Isham, N. Linden, and S. Schreckenberg, The classification of decoherence functionals: An analog of gleason’s theorem, J. Math. Phys. 35, 6360 (1994).
  75. A. M. Gleason, Measures on the closed subspaces of a Hilbert space, in The Logico-Algebraic Approach to Quantum Mechanics: Historical Evolution (Springer, New York, 1975), Vol. 1, pp. 123–133.
  76. N. L. Diaz, et al. (unpublished).
  77. K. Savvidou, The action operator for continuous-time histories, J. Math. Phys. 40, 5657 (1999).
  78. K. Chen and L.-A. Wu, A matrix realignment method for recognizing entanglement, Quantum Inf. Comput. 3, 193 (2003).
  79. M. Hebenstreit, R. Jozsa, B. Kraus, S. Strelchuk, and M. Yoganathan, All pure fermionic non-Gaussian states are magic states for matchgate computations, Phys. Rev. Lett. 123, 080503 (2019).

Outline

Information

Phys. Rev. Research 8, 013132– Published 6 February, 2026
  • Received 18 August 2025
  • Accepted 31 December 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation