Skip to main content

Free Quantum Field Theory from Quantum Cellular Automata

Derivation of Weyl, Dirac and Maxwell Quantum Cellular Automata


After leading to a new axiomatic derivation of quantum theory (see D’Ariano et al. in Found Phys, 2015), the new informational paradigm is entering the domain of quantum field theory, suggesting a quantum automata framework that can be regarded as an extension of quantum field theory to including an hypothetical Planck scale, and with the usual quantum field theory recovered in the relativistic limit of small wave-vectors. Being derived from simple principles (linearity, unitarity, locality, homogeneity, isotropy, and minimality of dimension), the automata theory is quantum ab-initio, and does not assume Lorentz covariance and mechanical notions. Being discrete it can describe localized states and measurements (unmanageable by quantum field theory), solving all the issues plaguing field theory originated from the continuum. These features make the theory an ideal framework for quantum gravity, with relativistic covariance and space-time emergent solely from the interactions, and not assumed a priori. The paper presents a synthetic derivation of the automata theory, showing how the principles lead to a description in terms of a quantum automaton over a Cayley graph of a group. Restricting to Abelian groups we show how the automata recover the Weyl, Dirac and Maxwell dynamics in the relativistic limit. We conclude with some new routes about the more general scenario of non-Abelian Cayley graphs. The phenomenology arising from the automata theory in the ultra-relativistic domain and the analysis of corresponding distorted Lorentz covariance is reviewed in Bisio et al. (Found Phys 2015, in this same issue).

This is a preview of subscription content, access via your institution.

Fig. 1


  1. The spacetime manifold M is here introduced in a fixed reference frame. The notion of change of reference frame based on the invariance of the QCA dynamics has been the subject of the works [8, 10].

  2. This was firstly noticed by Meyer in Ref. [42] for space dimension \(d=1\).

  3. In a more general scenario scalar field automata can be defined as discussed in Sect. 8.

  4. Since \(W_\mathbf {k}\) has dimension 2, \(W_\mathbf {k}\) and \(W^*_\mathbf {k}\) are similar through \( W_\mathbf {k}^* = \sigma _y W_\mathbf {k}\sigma _y\).

  5. This derivation can be applied, with suitable changes, to the case in which \({f}_\mathbf {k}(\mathbf {q})\) is no longer a constant function, see Ref. [9].


  1. Acevedo, O.L., Roland, J., Cerf, N.J.: Exploring scalar quantum walks on cayley graphs. Quantum Info. Comput. 8(1), 68–81 (2008)

    MATH  Google Scholar 

  2. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)

    Article  ADS  Google Scholar 

  3. 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)

  4. Ariano, G.M.D.: On the missing axiom of quantum mechanicss. AIP Conf. Proc. 810(1), 114–130 (2006). doi:10.1063/1.2158715

    Article  ADS  Google Scholar 

  5. Arrighi, P., Nesme, V., Forets, M.: The Dirac equation as a quantum walk: higher dimensions, observational convergence. J. Phys. A 47(46), 465302 (2014)

    MathSciNet  Article  ADS  Google Scholar 

  6. Bekenstein, J.D.: Black holes and entropy. Phys. Rev. D 7(8), 2333 (1973)

    MathSciNet  Article  ADS  Google Scholar 

  7. Bialynicki-Birula, I.: Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata. Phys. Rev. D 49(12), 6920 (1994)

    MathSciNet  Article  ADS  Google Scholar 

  8. Bibeau-Delisle, A., Bisio, A., D’Ariano, G.M., Perinotti, P., Tosini, A.: Doubly-special relativity from quantum cellular automata. arXiv:1310.6760 (2013)

  9. Bisio, A., D’Ariano, G.M., Perinotti, P.: Quantum cellular automaton theory of light. arXiv:1407.6928 (2014)

  10. Bisio, A., D’Ariano, G.M., Perinotti, P.: Lorentz symmetry for 3d quantum cellular automata. arXiv preprint arXiv:1503.01017 (2015)

  11. Bisio, A., D’Ariano, G.M., Perinotti, P., Tosini, A.: Weyl, Dirac and Maxwell quantum cellular automata: analitical solutions and phenomenological predictions of the quantum cellular automata theory of free fields. Found. Phys. (2015). doi:10.1007/s10701-015-9927-0

  12. 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)

    Article  ADS  Google Scholar 

  13. 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)

    MathSciNet  Article  ADS  Google Scholar 

  14. Bousso, R.: Light sheets and Bekenstein’s entropy bound. Phys. Rev. Lett. 90, 121302 (2003)

    MathSciNet  Article  ADS  Google Scholar 

  15. Bravyi, S.B., Kitaev, A.Y.: Fermionic quantum computation. Ann. Phys. 298(1), 210–226 (2002)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  16. Chiribella, G., D’Ariano, G., Perinotti, P.: Informational derivation of quantum theory. Phys. Rev. A 84, 012311 (2011)

    Article  ADS  Google Scholar 

  17. Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification. Phys. Rev. A 81, 062348 (2010)

    Article  ADS  Google Scholar 

  18. 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)

    Chapter  Google Scholar 

  19. D’Ariano, G.: On the “principle of the quantumness”, the quantumness of relativity, and the computational grand-unification. In: CP1232 Quantum Theory: Reconsideration of Foundations 5, p. 3 (2010)

  20. D’Ariano, G.: Physics as information processing. Advances in quantum theory. AIP Conf. Proc. 1327, 7 (2011)

    Article  ADS  Google Scholar 

  21. D’Ariano, G.: The Dirac quantum automaton: a preview. arXiv:1211.2479 (2012)

  22. D’Ariano, G.M.: A computational grand-unified theory. (2010)

  23. D’Ariano, G.M.: Probabilistic theories: what is special about quantum mechanics? In: Philosophy of Quantum Information and Entanglement p. 85 (2010)

  24. D’Ariano, G.M.: A quantum-digital universe. Adv. Sci. Lett. 17, 130 (2012)

    Article  Google Scholar 

  25. D’Ariano, G.M.: A quantum digital universe. Il Nuovo Saggiatore 28, 13 (2012)

    Google Scholar 

  26. D’Ariano, G.M.: The quantum field as a quantum computer. Phys. Lett. A 376(5), 697–702 (2012)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  27. D’Ariano, G.M., Manessi, F., Perinotti, P., Tosini, A.: Fermionic computation is non-local tomographic and violates monogamy of entanglement. EPL (Europhys. Lett.) 107(2), 20009 (2014)

    Article  Google Scholar 

  28. D’Ariano, G.M., Manessi, F., Perinotti, P., Tosini, A.: The Feynman problem and fermionic entanglement: fermionic theory versus qubit theory. Int. J. Mod. Phys. A 29(17), 1430025 (2014)

    MathSciNet  Article  Google Scholar 

  29. D’Ariano, G.M., Perinotti, P.: Derivation of the Dirac equation from principles of information processing. Phys. Rev. A 90, 90062 (2014)

    Google Scholar 

  30. D’Ariano, G.M., Perinotti, P.: The dirac quantum automaton: a short review. Phys. Scr. 163, 014014 (2014)

    Article  Google Scholar 

  31. D’Ariano, G.M., Perinotti, P.: Quantum theory is an information theory: the operational framework and the axioms. Found. Phys. (2015). doi:10.1007/s10701-015-9935-0

  32. Erba, M.: Non-abelian quantum walks and renormalization. Master Thesis, (2014)

  33. Farrelly, T.C., Short, A.J.: Discrete spacetime and relativistic quantum particles. arXiv:1312.2852 (2013)

  34. Feynman, R.: Simulating physics with computers. Int. J. Theor. Phys. 21(6), 467–488 (1982)

    MathSciNet  Article  Google Scholar 

  35. Fuchs, C.A.: Quantum mechanics as quantum information (and only a little more). quant-ph/0205039 (2002)

  36. Gromov, M.: Infinite groups as geometric objects. Proc. Int. Congr. Math. 1, 2 (1984)

    Google Scholar 

  37. Grossing, G., Zeilinger, A.: Quantum cellular automata. Complex Syst. 2(2), 197–208 (1988)

    MathSciNet  Google Scholar 

  38. Hardy, L.: Quantum theory from five reasonable axioms. quant-ph/0101012 (2001)

  39. Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43(3), 199–220 (1975)

    MathSciNet  Article  ADS  Google Scholar 

  40. de La Harpe, P.: Topics in geometric group theory. University of Chicago Press, Chicago (2000)

    MATH  Google Scholar 

  41. Masanes, L., Müller, M.P.: A derivation of quantum theory from physical requirements. New J. Phys. 13(6), 063001 (2011)

    Article  ADS  Google Scholar 

  42. Meyer, D.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85(5), 551–574 (1996)

    MATH  Article  ADS  Google Scholar 

  43. Wheeler, J.A.: The computer and the universe. Int. J. Theor. Phys. 21(6–7), 557–572 (1982)

    Article  Google Scholar 

  44. Yepez, J.: Relativistic path integral as a lattice-based quantum algorithm. Quantum Inf. Process. 4(6), 471–509 (2006)

    MATH  MathSciNet  Article  Google Scholar 

Download references


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

Author information

Authors and Affiliations


Corresponding author

Correspondence to Giacomo Mauro D’Ariano.

Additional information

Work presented at the conference Quantum Theory: from Problems to Advances, held on 9–12 June 2014 at at Linnaeus University, Vaxjo University, Sweden. This paper, together with Ref. [11], contains future perspective and an original presentation of our most important recent results in the line of research on quantum cellular automata and quantum field theory (see e. g. Ref. [30]) .

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bisio, A., D’Ariano, G.M., Perinotti, P. et al. Free Quantum Field Theory from Quantum Cellular Automata. Found Phys 45, 1137–1152 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Informational principles
  • Quantum field theory
  • Quantum cellular automata
  • Planck scale