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Cited by 60Quantum Mechanics and ApplicationsQuantum Computing Algorithms and ArchitectureNoncommutative and Quantum Gravity Theories

Free-Dirac-particle evolution as a quantum random walk

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Bracken, A. J., Ellinas, D., Smyrnakis, I.·2007·Physical Review A
DOI 10.1103/physreva.75.022322
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Abstract

It is known that any positive-energy state of a free Dirac particle that is initially highly localized evolves in time by spreading at speeds close to the speed of light. As recently indicated by Strauch, this general phenomenon, and the resulting ``two-horned'' distributions of position probability along any axis through the point of initial localization, can be interpreted in terms of a quantum random walk, in which the roles of ``coin'' and ``walker'' are naturally associated with the spin and translational degrees of freedom in a discretized version of Dirac's equation. We investigate the relationship between these two evolutions analytically and show how the evolved probability density on the $x$ axis for the Dirac particle at any time $t$ can be obtained from the asymptotic form of the probability distribution for the position of a ``quantum walker.'' The case of a highly localized initial state is discussed as an example.

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  • Our analysis agrees with the works [42-46] that studied the continuum limit of QWs in comparison with the Dirac or the Klein-Gordon equations.

    “We notice that our analysis agrees with the works [42-46] that studied the continuum limit, i.e. when the lattice spacings and the time steps are sent to 0, of QWs in comparison with the Dirac or the Klein-Gordon equations 3 .”

    From Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension (2015) · cites this paper as [44]

  • The bimodal probability distribution in position space is an indication of the one-dimensional analog of an initially localized Dirac particle, with positive energy, evolving in time which spreads in all directions in position space at speeds close to the speed of light.

    “This bimodal probability distribution in position space is an indication of the one-dimensional analog of an initially localized Dirac particle, with positive energy, evolving in time which spreads in all directions in position space at speeds close to the speed of light53.”

    From Quantum walks and Dirac cellular automata on a programmable trapped-ion quantum computer (2020) · cites this paper as [53]

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Research works printed but not linked: 6

  1. [4][4] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, Proc. 33rd Annual ACM Symposium on Theory of Computing (STOC'01) , 37-49, (2001). -
  2. [7][7] D. Aharonov, A. Ambainis, J. Kempe and U. Vazirani, Proc. 33rd Annual ACM Symposium on Theory of Computing (STOC'01) , 50-59, (2001). -
  3. [14][14] A.J. Bracken, J.A. Flohr and G.F. Melloy, Proc. Roy. Soc. (London) A 461 , 3633 (2005). -
  4. [19][19] D. Ellinas, in Quantum Probability and Infinite Dimensional Analysis: From Foundations to Applications , Eds. M. Schurmann and U. Franz, (World Scientific, Singapore, 2005), pp. 174-200. -
  5. [20][20] A. J. Bracken, D. Ellinas and I. Tsohantjis, J. Phys. A 37, L91 (2004). -
  6. [22][22] D. Ellinas and I. Smyrnakis, ArXiv quant-ph/0510098, Physica A , to appear (2006). - · arXiv

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Authors
Bracken, A. J., Ellinas, D., & Smyrnakis, I.
Affiliations
  • The University of Queensland (AU)
  • Technical University of Crete (GR)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2007-02-21
Volume
75
Issue
2
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.75.022322
OpenAlex
W2020603078
Cited by
60
References
39
Citations by year
2026: 1 · 2025: 3 · 2024: 1 · 2023: 1 · 2022: 3 · 2021: 3 · 2020: 9 · 2019: 5 · 2018: 3 · 2017: 2 · 2016: 3 · 2015: 2 · 2014: 4 · 2013: 5 · 2012: 3
FWCI
2.04
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
Keywords
quantum walks, relativistic quantum mechanics

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