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Free Quantum Field Theory from Quantum Cellular Automata

Derivation of Weyl, Dirac and Maxwell Quantum Cellular Automata

Abstract

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).

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Fig. 1

Notes

  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].

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Acknowledgments

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

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Correspondence to Giacomo Mauro D’Ariano.

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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]) .

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Bisio, A., D’Ariano, G.M., Perinotti, P. et al. Free Quantum Field Theory from Quantum Cellular Automata. Found Phys 45, 1137–1152 (2015). https://doi.org/10.1007/s10701-015-9934-1

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Keywords

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