The Dirac Vacuum in Discrete Spacetime

Chaitanya Gupta; Anthony J. Short

doi:10.22331/q-2025-09-03-1845

The Dirac Vacuum in Discrete Spacetime

H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, U.K

Published:	2025-09-03, volume 9, page 1845
Editor:	Daniel Malz
Eprint:	arXiv:2412.03466v3
Doi:	https://doi.org/10.22331/q-2025-09-03-1845
Citation:	Quantum 9, 1845 (2025).

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Abstract

We consider introducing the Dirac sea in a quantum cellular automata model of fermions in discrete spacetime which approximates the Dirac equation in the continuum limit. However, if we attempt to fill up the `negative' energy states, we run into a problem. A new boundary is created between positive and negative energy states, at which pair creation seems energetically favourable. This happens because of the modular nature of energy in discrete time models. We then suggest a possible remedy by amending the model, in order to pull states away from the new boundary.

Featured image: In discrete time models, energy is usually modular – it lies on a circle. This results into two boundaries between ‘positive’ and ‘negative’ energy particles (when energy is 0 and when energy is $\pm/\pi/\delta t$). In the Dirac walk, all energy states on the circle are possible. We introduce a new walk where possible energy states are pulled away from $\pm/\pi/\delta t$.

Popular summary

We usually think of space and time as fundamentally continuous. However, models such as Quantum Cellular Automata (QCA) and Quantum Walks (QW) can give a description of some particle physics with the assumption that space and time are both discrete, like a chessboard. When we zoom out of the chessboard, we retrieve the dynamics we would expect in the continuum theories. However, we argue that these models may come with a serious problem when we consider the vacuum state. Dirac postulated that the vacuum is not exactly empty but is made up of a ‘sea of negative energy’ particles known as the Dirac sea. We argue that this would become unstable when introducing interactions to commonly used discrete models, with the production of high energy pairs of particles and antiparticles, and propose an alternative model to remedy this.

► BibTeX data

@article{Gupta2025diracvacuumin,
  doi = {10.22331/q-2025-09-03-1845},
  url = {https://doi.org/10.22331/q-2025-09-03-1845},
  title = {The {D}irac {V}acuum in {D}iscrete {S}pacetime},
  author = {Gupta, Chaitanya and Short, Anthony J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1845},
  month = sep,
  year = {2025}
}

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Cited by

[1] Chaitanya Gupta and Anthony J. Short, "Fermion doubling in Dirac quantum walks", Physical Review A 114 1, 012208 (2026).

[2] Hans-Thomas Elze, "The Dirac Equation, Mass and Arithmetic by Permutations of Automaton States", Entropy 27 4, 395 (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 21:10:27) and SAO/NASA ADS (last updated successfully 2026-08-09 21:10:29). The list may be incomplete as not all publishers provide suitable and complete citation data.

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