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Foundations of Physics

, Volume 45, Issue 10, pp 1203–1221 | Cite as

Weyl, Dirac and Maxwell Quantum Cellular Automata

Analitical Solutions and Phenomenological Predictions of the Quantum Cellular Automata Theory of Free Fields
  • Alessandro Bisio
  • Giacomo Mauro D’Ariano
  • Paolo PerinottiEmail author
  • Alessandro Tosini
Article

Abstract

Recent advances on quantum foundations achieved the derivation of free quantum field theory from general principles, without referring to mechanical notions and relativistic invariance. From the aforementioned principles a quantum cellular automata (QCA) theory follows, whose relativistic limit of small wave-vector provides the free dynamics of quantum field theory. The QCA theory can be regarded as an extended quantum field theory that describes in a unified way all scales ranging from an hypothetical discrete Planck scale up to the usual Fermi scale. The present paper reviews the automaton theory for the Weyl field, and the composite automata for Dirac and Maxwell fields. We then give a simple analysis of the dynamics in the momentum space in terms of a dispersive differential equation for narrowband wave-packets. We then review the phenomenology of the free-field automaton and consider possible visible effects arising from the discreteness of the framework. We conclude introducing the consequences of the automaton dispersion relation, leading to a deformed Lorentz covariance and to possible effects on the thermodynamics of ideal gases.

Keywords

Quantum field theory Quantum cellular automata Quantum walks 

Notes

Acknowledgments

This work has been supported in part by the Templeton Foundation under the Project ID# 43796 A Quantum-Digital Universe.

References

  1. 1.
    Bisio, A., D’Ariano, G.M., Perinotti, P., Tosini, A.: Foundations of Physics (2015). (in press)Google Scholar
  2. 2.
    von Neumann, J.: Theory of Self-Reproducing Automata. University of Illinois Press, Urbana (1966)Google Scholar
  3. 3.
    ’tHooft, G.: The Cellular Automaton Interpretation of Quantum Mechanics. A View on the Quantum Nature of our Universe, Compulsory or Impossible? arXiv:1405.1548
  4. 4.
    Elze, H.-T.: Action principle for cellular automata and the linearity of quantum mechanics. Phys. Rev. A 89, 012111 (2014)CrossRefADSGoogle Scholar
  5. 5.
    Feynman, R.: Simulating physics with computers. Int. J. Theoret. Phys. 21(6), 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Schumacher, B., Werner, R.: Reversible quantum cellular automata arXiv:quant-ph/0405174 (2004)
  7. 7.
    Arrighi, P., Nesme, V., Werner, R.: Unitarity plus causality implies localizability. J. Comput. Syst. Sci. 77(2), 372–378 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Gross, D., Nesme, V., Vogts, H., Werner, R.: Index Theory of One Dimensional Quantum Walks and Cellular Automata Communications in Mathematical Physics, pp. 1–36. McGraw-Hill, New York (2012)Google Scholar
  9. 9.
    Grossing, G., Zeilinger, A.: Quantum cellular automata. Complex Syst. 2(2), 197–208 (1988)MathSciNetGoogle Scholar
  10. 10.
    Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)CrossRefADSGoogle Scholar
  11. 11.
    Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of the Thirty-Third annual ACM Symposium on Theory of Computing, pp. 37–49. ACM, New York (2001)Google Scholar
  12. 12.
    Reitzner, D., Nagaj, D., Buẑek, V.: Quantum walks, acta physica slovaca. Rev. Tutor. 61(6), 603–725 (2011)Google Scholar
  13. 13.
    Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 59–68. ACM, New York (2003)Google Scholar
  14. 14.
    Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM J. Comput. 37(2), 413–424 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Farhi, E., Goldstone, J., Gutmann, S.: A quantum algorithm for the Hamiltonian NAND tree, arXiv:quant-ph/0702144 (2007)
  17. 17.
    D’Ariano, G.: On the “principle of the quantumness”, the quantumness of relativity, and the computational grand-unification, CP1232 Quantum Theory: Reconsid. Found. 5, 3 (2010)Google Scholar
  18. 18.
    D’Ariano, G.M.: Physics as quantum information processing: quantum fields as quantum automata. Phys. Lett. A 376, 697 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bisio, A., D’Ariano, G.M., Tosini, A.: Quantum field as a quantum cellular automaton: the Dirac free evolution in one dimension. Ann. Phys. 354, 244–264 (2015)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    D’Ariano, G.M., Perinotti, P.: Derivation of the Dirac equation from principles of information processing. Phys. Rev. A 90, 062106 (2014)CrossRefADSGoogle Scholar
  21. 21.
    Bisio, A., D’Ariano, G.M., Perinotti, P.: Quantum cellular automaton theory of light, arXiv:1407.6928 (2014)
  22. 22.
    Arrighi, P., Nesme, V., Forets, M.: The Dirac equation as a quantum walk: higher dimensions, observational convergence. J. Phys. A 47(46), 465302 (2014)MathSciNetCrossRefADSGoogle Scholar
  23. 23.
    Arrighi, P., Facchini, S.: Decoupled quantum walks, models of the Klein–Gordon and wave equations. Europhys. Lett. 104(6), 60004 (2013)CrossRefADSGoogle Scholar
  24. 24.
    Farrelly, T.C., Short, A.J.: Causal fermions in discrete space-time. Phys. Rev. A 89(1), 012302 (2014)CrossRefADSGoogle Scholar
  25. 25.
    Farrelly, T.C., Short, A.J.: Discrete spacetime and relativistic quantum particles, arXiv:1312.2852 (2013)
  26. 26.
    Albeverio, S., Cianci, R., Khrennikov, AYu.: p-Adic valued quantization. P-Adic Numbers Ultrametr. Anal. Appl. 1(2), 91–104 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Ellis, J., Mavromatos, N., Nanopoulos, D.V.: String theory modifies quantum mechanics. Phys. Lett. B 293(1), 37–48 (1992)MathSciNetCrossRefADSGoogle Scholar
  28. 28.
    Lukierski, J., Ruegg, H., Zakrzewski, W.J.: Classical and quantum mechanics of free \(\kappa \)-relativistic systems. Ann. Phys. 243(1), 90–116 (1995)zbMATHMathSciNetCrossRefADSGoogle Scholar
  29. 29.
    ’t Hooft, G.: Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness. Class. Quantum Grav. 13, 1023 (1996)zbMATHMathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Amelino-Camelia, G.: Testable scenario for relativity with minimum length. Phys. Lett. B 510(1), 255–263 (2001)zbMATHCrossRefADSGoogle Scholar
  31. 31.
    Magueijo, J., Smolin, L.: Lorentz invariance with an invariant energy scale. Phys. Rev. Lett. 88, 190403 (2002)CrossRefADSGoogle Scholar
  32. 32.
    De Broglie, L.: Une nouvelle conception de la lumière, vol. 181. Hermamm & Cie, Paris (1934)Google Scholar
  33. 33.
    Jordan, P.: Zur Neutrinotheorie des Lichtes. Zeitschrift für Physik 93(7–8), 464–472 (1935)CrossRefADSGoogle Scholar
  34. 34.
    Kronig, R.D.L.: On a relativistically invariant formulation of the neutrino theory of light. Physica 3(10), 1120–1132 (1936)CrossRefADSGoogle Scholar
  35. 35.
    Perkins, W.: Statistics of a composite photon formed of two fermions. Phys. Rev. D 5, 1375–1384 (1972)CrossRefADSGoogle Scholar
  36. 36.
    Perkins, W.: Quasibosons. Int. J. Theoret. Phys. 41(5), 823 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Bisio, A., D’Ariano, G.M., Tosini, A.: Dirac quantum cellular automaton in one dimension: Zitterbewegung and scattering from potential. Phys. Rev. A 88, 032301 (2013)CrossRefADSGoogle Scholar
  38. 38.
    D’Ariano, G.M., Mosco, N., Perinotti, P., Tosini, A.: Path-integral solution of the one-dimensional Dirac quantum cellular automaton. Phys. Lett. A 378(43), 3165–3168 (2014). doi: 10.1016/j.physleta.2014.09.020 CrossRefADSGoogle Scholar
  39. 39.
    D’Ariano, G., Mosco, N., Perinotti, P., Tosini, A.: Discrete Feynman propagator for the Weyl quantum walk in 2+ 1 dimensions. EPL 109, 40012 (2015)CrossRefGoogle Scholar
  40. 40.
    Succi, S., Benzi, R.: Lattice Boltzmann equation for quantum mechanics. Physica D 69(3), 327–332 (1993)zbMATHMathSciNetCrossRefADSGoogle Scholar
  41. 41.
    Bialynicki-Birula, I.: Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata. Phys. Rev. D 49(12), 6920 (1994)MathSciNetCrossRefADSGoogle Scholar
  42. 42.
    Meyer, D.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85(5), 551–574 (1996)zbMATHCrossRefADSGoogle Scholar
  43. 43.
    Schrödinger, E.: Über die kräftefreie Bewegung in der relativistischen Quantenmechanik. Akademie der wissenschaften in kommission bei W. de Gruyter u,Company(1930)Google Scholar
  44. 44.
    Lurié, D., Cremer, S.: Zitterbewegung of quasiparticles in superconductors. Physica 50(2), 224–240 (1970)CrossRefADSGoogle Scholar
  45. 45.
    Cannata, F., Ferrari, L.: Effects of the nonrelativistic Zitterbewegung on the electron-phonon interaction in two-band systems. Phys. Rev. B 44(16), 8599 (1991)CrossRefADSGoogle Scholar
  46. 46.
    Ferrari, L., Russo, G.: Nonrelativistic zitterbewegung in two-band systems. Phys. Rev. B 42(12), 7454 (1990)CrossRefADSGoogle Scholar
  47. 47.
    Cannata, F., Ferrari, L., Russo, G.: Dirac-like behaviour of a non-relativistic tight binding Hamiltonian in one dimension. Solid State Commun. 74(4), 309–312 (1990)CrossRefADSGoogle Scholar
  48. 48.
    Zhang, X.: Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal. Phys. Rev. Lett. 100, 113903 (2008). doi: 10.1103/PhysRevLett.100.113903 CrossRefADSGoogle Scholar
  49. 49.
    Kurzyński, P.: Relativistic effects in quantum walks: Klein’s paradox and zitterbewegung. Phys. Lett. A 372(40), 6125–6129 (2008)zbMATHMathSciNetCrossRefADSGoogle Scholar
  50. 50.
    Meyer, D.A.: Quantum lattice gases and their invariants. Int. J. Modern Phys. C 8(04), 717–735 (1997)CrossRefADSGoogle Scholar
  51. 51.
    Takeda, M., Hayashida, N., Honda, K., Inoue, N., Kadota, K., Kakimoto, F., Kamata, K., Kawaguchi, S., Kawasaki, Y., Kawasumi, N., et al.: Extension of the cosmic-ray energy spectrum beyond the predicted Greisen–Zatsepin–Kuz’min cutoff. Phys. Rev. Lett. 81(6), 1163–1166 (1998)CrossRefADSGoogle Scholar
  52. 52.
    Amelino-Camelia, G., Ellis, J., Mavromatos, N., Nanopoulos, D.V., Sarkar, S.: Tests of quantum gravity from observations of \(\gamma \)-ray bursts. Nature 393(6687), 763–765 (1998)CrossRefADSGoogle Scholar
  53. 53.
    Abdo, A., Ackermann, M., Ajello, M., Asano, K., Atwood, W., Axelsson, M., Baldini, L., Ballet, J., Barbiellini, G., Baring, M., et al.: A limit on the variation of the speed of light arising from quantum gravity effects. Nature 462(7271), 331–334 (2009)CrossRefADSGoogle Scholar
  54. 54.
    Vasileiou, V., Jacholkowska, A., Piron, F., Bolmont, J., Couturier, C., Granot, J., Stecker, F., Cohen-Tannoudji, J., Longo, F.: Constraints on Lorentz invariance violation from fermi-large area telescope observations of gamma-ray bursts. Phys. Rev. D 87(12), 122001 (2013)CrossRefADSGoogle Scholar
  55. 55.
    Amelino-Camelia, G., Smolin, L.: Prospects for constraining quantum gravity dispersion with near term observations. Phys. Rev. D 80(8), 084017 (2009)CrossRefADSGoogle Scholar
  56. 56.
    Dunne, M.: High intensity laser physics: recent results and developments at the central laser facility, UK, In: Conference on Lasers and Electro-Optics/Pacific Rim (Optical Society of America), pp. 1–2 (2007)Google Scholar
  57. 57.
    Bisio, A., D’Ariano, G.M., Perinotti, P.: Lorentz symmetry for 3d Quantum Cellular Automata arXiv:1503.0101
  58. 58.
    Bibeau-Delisle, A., Bisio, A., D’Ariano, G.M., Perinotti, P., Tosini, A.: Doubly-special relativity from quantum cellular automata. Europhys. Lett. 109, 50003 (2015)CrossRefADSGoogle Scholar
  59. 59.
    Amelino-Camelia, G.: Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale. Int. J. Modern Phys. D 11(01), 35–59 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  60. 60.
    Amelino-Camelia, G., Piran, T.: Planck-scale deformation of Lorentz symmetry as a solution to the ultrahigh energy cosmic ray and the TeV-photon paradoxes. Phys. Rev. D 64(3), 036005 (2001)CrossRefADSGoogle Scholar
  61. 61.
    Amelino-Camelia, A.: Quantum-gravity phenomenology: status and prospects. Modern Phys. Lett. A 17(15n17), 899–922 (2002)MathSciNetCrossRefADSGoogle Scholar
  62. 62.
    Magueijo, J., Smolin, L.: Generalized Lorentz invariance with an invariant energy scale. Phys. Rev. D 67(4), 044017 (2003)MathSciNetCrossRefADSGoogle Scholar
  63. 63.
    Amelino-Camelia, G., Freidel, L., Kowalski-Glikman, J., Smolin, L.: Relative locality: a deepening of the relativity principle. Int. J. Modern Phys. D 20(14), 2867–2873 (2011)zbMATHMathSciNetCrossRefADSGoogle Scholar
  64. 64.
    Amelino-Camelia, G., Astuti, V., Rosati, G.: Relative locality in a quantum spacetime and the pregeometry of k-Minkowski. Eur. Phys. J. C 73(8), 1–11 (2013). doi: 10.1140/epjc/s10052-013-2521-8 CrossRefGoogle Scholar
  65. 65.
    Connes, A., Lott, J.: Particle models and noncommutative geometry. Nucl. Phys. B 18(2), 29–47 (1991)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Lukierski, J., Ruegg, H., Nowicki, A., Tolstoy, V.N.: q-deformation of Poincaré algebra. Phys. Lett. B 264(3), 331–338 (1991)MathSciNetCrossRefADSGoogle Scholar
  67. 67.
    Majid, S., Ruegg, H.: Bicrossproduct structure of \(\kappa \)-Poincare group and non-commutative geometry. Phys. Lett. B 334(3), 348–354 (1994)zbMATHMathSciNetCrossRefADSGoogle Scholar
  68. 68.
    Amelino-Camelia, G., Loret, N., Mandanici, G., Mercati, F.: UV and IR quantum-spacetime effects for the Chandrasekhar model. Int. J. Modern Phys. D 21(06), 1250052 (2012)CrossRefADSGoogle Scholar
  69. 69.
    Camacho, A.: White dwarfs as test objects of Lorentz violations. Class. Quantum Gravit. 23(24), 7355 (2006)zbMATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.QUIT GroupDipartimento di Fisica and INFN sezione di PaviaPaviaItaly

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