Advertisement

Foundations of Physics

, Volume 47, Issue 8, pp 1065–1076 | Cite as

Quantum Walks, Weyl Equation and the Lorentz Group

  • Alessandro Bisio
  • Giacomo Mauro D’ArianoEmail author
  • Paolo Perinotti
Article

Abstract

Quantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the small wave-vector limit of quantum automata and walks starting from very general principles. The intrinsic discreteness of this framework is reconciled with the continuous Lorentz symmetry by reformulating the notion of inertial reference frame in terms of the constants of motion of the quantum walk dynamics. In particular, among the symmetries of the quantum walk which recovers the Weyl equation—the so called Weyl walk—one finds a non linear realisation of the Poincaré group, which recovers the usual linear representation in the small wave-vector limit. In this paper we characterise the full symmetry group of the Weyl walk which is shown to be a non linear realization of a group which is the semidirect product of the Poincaré group and the group of dilations.

Keywords

Quantum walk Doubly special relativity Quantum cellular automata Quantum field theory Lorentz transformations 

Notes

Acknowledgements

This publication was made possible through the support of a grant from the John Templeton Foundation under the Project ID# 60609 Causal Quantum Structures. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

References

  1. 1.
    Alexandrov, A.D.: A contribution to chronogeometry. Can. J. Math. 19, 1119–1128 (1967)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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 (2001)Google Scholar
  3. 3.
    Amelino-Camelia, G.: Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale. Int. J. Mod. Phys. D 11(01), 35–59 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)ADSCrossRefGoogle Scholar
  5. 5.
    Amelino-Camelia, G., Freidel, L., Kowalski-Glikman, J., Smolin, L.: Principle of relative locality. Phys. Rev. D 84, 084010 (2011). doi: 10.1103/PhysRevD.84.084010 ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Arnault, P., Debbasch, F.: Quantum walks and discrete gauge theories. Phys. Rev. A 93(5), 052301 (2016)ADSCrossRefGoogle Scholar
  7. 7.
    Arnault, P., Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks and non-abelian discrete gauge theory. Phys. Rev. A 94, 012335 (2016). doi: 10.1103/PhysRevA.94.012335. http://link.aps.org/doi/10.1103/PhysRevA.94.012335
  8. 8.
    Arrighi, P., Facchini, S., Forets, M.: Discrete Lorentz covariance for quantum walks and quantum cellular automata. New J. Phys. 16(9), 093007 (2014). http://stacks.iop.org/1367-2630/16/i=9/a=093007
  9. 9.
    Arrighi, P., Nesme, V., Forets, M.: The Dirac equation as a quantum walk: higher dimensions, observational convergence. J. Phys. A 47(46), 465302 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bialynicki-Birula, I.: Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata. Phys. Rev. D 49(12), 6920 (1994)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bisio, A., D’Ariano, G.M., Perinotti, P.: Quantum cellular automaton theory of light. Ann. Phys. 368, 177–190 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bisio, A., D’Ariano, G.M., Perinotti, P.: Quantum walks, deformed relativity and Hopf algebra symmetries. Philos. Trans. A Math. Phys. Eng. Sci. 374(2068). doi: 10.1098/rsta.2015.0232 (2016)
  13. 13.
    Bisio, A., D’Ariano, G.M., Perinotti, P.: Special relativity in a discrete quantum universe. Phys. Rev. A 94, 041210 (2016). doi: 10.1103/PhysRevA.94.042120 MathSciNetGoogle Scholar
  14. 14.
    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)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Chiribella, G., D’Ariano, G., Perinotti, P.: Informational derivation of quantum theory. Phys. Rev. A 84(012311), 012311–012350 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    Dakic, B., Brukner, C.: Quantum theory and beyond: is entanglement special? In: Halvorson, H. (ed.) Deep Beauty: Understanding the Quantum World through Mathematical Innovation, pp. 365–392. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  17. 17.
    D’Ariano, G.M.: Physics as quantum information processing: quantum fields as quantum automata. Phys. Lett. A 376(697) (2011)Google Scholar
  18. 18.
    D’Ariano, G.M., Perinotti, P.: Derivation of the Dirac equation from principles of information processing. Phys. Rev. A 90, 062106 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    D’Ariano, G.M., Khrennikov, A.: Preface of the special issue quantum foundations: information approach. Philos. Trans. R. Soc. Lond. A 374(2068) (2016). doi:  10.1098/rsta.2015.0244. http://rsta.royalsocietypublishing.org/content/374/2068/20150244
  20. 20.
    Feynman, R.: Simulating physics with computers. Int. J. Theor. Phys. 21(6), 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Fuchs, C.A.: Quantum mechanics as quantum information (and only a little more). quant-ph/0205039 (2002)Google Scholar
  22. 22.
    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 (2012)Google Scholar
  23. 23.
    Hardy, L.: Quantum theory from five reasonable axioms. quant-ph/0101012 (2001)Google Scholar
  24. 24.
    Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44(4), 307–327 (2003). doi: 10.1080/00107151031000110776 ADSCrossRefGoogle Scholar
  25. 25.
    Khrennikov, A., Weihs, G.: Preface of the special issue quantum foundations: theory and experiment. Found. Phys. 42(6), 721–724 (2012). doi: 10.1007/s10701-012-9644-x ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Khrennikov, A., Raedt, H.D., Plotnitsky, A., Polyakov, S.: Preface of the special issue probing the limits of quantum mechanics: theory and experiment, volume 1. Found. Phys. 45(7), 707–710 (2015). doi: 10.1007/s10701-015-9911-8 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kowalski-Glikman, J., Nowak, S.: Doubly special relativity theories as different bases of \(\kappa \)-Poincaré algebra. Phys. Lett. B 539(1), 126–132 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kowalski-Glikman, J., Nowak, S.: Non-commutative space-time of doubly special relativity theories. Int. J. Mod. Phys. D 12(02), 299–315 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lloyd, S.: Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos. Vintage Books, New York (2006)Google Scholar
  30. 30.
    Lukierski, J., Ruegg, H., Nowicki, A., Tolstoy, V.N.: \(q\)-deformation of Poincaré algebra. Phys. Lett. B 264(3), 331–338 (1991)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Magueijo, J., Smolin, L.: Lorentz invariance with an invariant energy scale. Phys. Rev. Lett. 88, 190403 (2002)ADSCrossRefGoogle Scholar
  32. 32.
    Magueijo, J., Smolin, L.: Generalized Lorentz invariance with an invariant energy scale. Phys. Rev. D 67(4), 044017 (2003)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Majid, S., Ruegg, H.: Bicrossproduct structure of \(\kappa \)-Poincare group and non-commutative geometry. Phys. Lett. B 334(3), 348–354 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Meyer, D.: From quantum cellular automata to quantum lattice gases. J. Stat. Phy. 85(5), 551–574 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Meyer, D.A.: From gauge transformations to topology computation in quantum lattice gas automata. J. Phys. A 34(35), 6981 (2001). http://stacks.iop.org/0305-4470/34/i=35/a=323
  36. 36.
    Schumacher, B., Werner, R.: Reversible quantum cellular automata. quant-ph/0405174 (2004)Google Scholar
  37. 37.
    Snyder, H.: Quantized space-time. Phys. Rev. 71, 38–41 (1947)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Susskind, L.: Lattice fermions. Phys. Rev. D 16, 3031–3039 (1977). doi: 10.1103/PhysRevD.16.3031 ADSCrossRefGoogle Scholar
  39. 39.
    Yepez, J.: Relativistic path integral as a lattice-based quantum algorithm. Quantum Inf. Process. 4(6), 471–509 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zeeman, E.C.: Causality implies the Lorentz group. J. Math. Phys. 5(4), 490–493 (1964)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.QUIT Group, Dipartimento di Fisica and INFN sezione di PaviaPaviaItaly

Personalised recommendations