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Cited by 195Cellular Automata and ApplicationsNoncommutative and Quantum Gravity TheoriesQuantum and Classical Electrodynamics

Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata

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Bialynicki-Birula, Iwo·1994·Physical Review D
DOI 10.1103/physrevd.49.6920
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Abstract

Very simply unitary cellular automata on a cubic lattice are introduced to model a discretized time evolution of the wave functions for spinning particles. In each evolution step the undated value of the wave function at a given site depends only on the values at the nearest sites. The discretized evolution is also unitary and preserves chiral symmetry. The case of the spin-1/2 particle is studied in detail and it is shown that every local and unitary automaton on a cubic lattice, under some natural assumptions, leads in the continuum limit to the Weyl equation. The sum over histories is evaluated and is shown to reproduce the retarded propagator in the continuum limit. Generalizations to include massive particles (Dirac theory), spin-1 particles (Maxwell theory), and higher-spin particles are also described.

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Crossref · 10.1103/physrevd.49.6920

Cites

44 of 44 references verified0 citation errors · 0 not checkable

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Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The DTQW matrix has eigenvalues e±iω(k), where ω(k) satisfies the dispersion relation.

    “This matrix has eigenvalues e ± iω ( k ) , where ω ( k ) satisfies the dispersion relation [13]”

    From Relativistic quantum walks (2006) · cites this paper as [13]

    In the section on Dirac and Weyl equations on a lattice (p. 4), the paper describes a momentum-space unitary matrix and states that it has eigenvalues λ±. The formulas are not decoded in our copy, so the specific form e^{±iω(k)} and the asserted dispersion relation cannot be verified. (LLM, conf=0.76)

    Experimental — accuracy not yet measured

  • The (2+1)-dimensional Dirac QW models the (2+1)-dimensional Dirac equation.

    “This Dirac QW [4-6] models the + (2 1)-dimensional Dirac equation.”

    From The Dirac equation as a quantum walk: higher dimensions, observational convergence (2014) · cites this paper as [5]

    In “Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata,” p. 4, the paper says its lattice evolution has the Dirac equation as its continuum limit. Page 3 separately mentions a discretized Weyl equation in two spatial dimensions, but the copy does not establish that the cited model is a (2+1)-dimensional Dirac quantum walk. (LLM, conf=0.85)

    Experimental — accuracy not yet measured

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Cites

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Research works printed but not linked: 1

  1. [24]S. D. Drell, M. Weinstein, and S. Yankielowicz, Phys. Rev. D 14 , 487, 1627 (1976).

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Related

Authors
Bialynicki-Birula, I.
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
1994-06-15
Volume
49
Issue
12
Pages
6920-6927
ISSN
0556-2821 (ISSN-L)
DOI
10.1103/physrevd.49.6920
OpenAlex
W2602988713
Cited by
195
References
44
Citations by year
2026: 6 · 2025: 8 · 2024: 3 · 2023: 1 · 2022: 8 · 2021: 8 · 2020: 12 · 2019: 14 · 2018: 7 · 2017: 10 · 2016: 16 · 2015: 10 · 2014: 13 · 2013: 6 · 2012: 6
FWCI
3.90
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Computational Theory and Mathematics › Cellular Automata and Applications
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
Keywords
lattice discretization, Weyl equation, Dirac equation, Maxwell's equations, spin-1/2 particles, chiral symmetry, continuum limit, sum over histories, retarded propagator, massive particles, spin-1 particles, higher spin particles, cubic lattice

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