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Weyl, Dirac and Maxwell Quantum Cellular Automata

Analitical Solutions and Phenomenological Predictions of the Quantum Cellular Automata Theory of Free Fields

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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.

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Notes

  1. Other approaches to discrete spacetime based on p-adic numbers were studied in Ref. [26].

  2. We denote as \([A,B]_+\) the anticommutator \(AB+BA\). The commutator \(AB-BA\) will be denoted as \([A,B]_-\).

  3. This step would require a more precise mathematical characterization (which we omit) of the presented assumptions. See Ref. [20] for the details.

  4. In order to prove this step one needs a stronger isotropy condition than the one presented in the text. See Ref. [20] for the details.

  5. We denote as \(A^*\) the complex conjugate of A.

  6. Since a QCA describes an evolution discrete in time, the derivative with respect to time is not defined in this context. However we can imagine \(A_\mathbf {k}^{t}\) to be defined for any real value of t and then derive with respect the continuous variable t. This is the construction underlying Eq. (11).

  7. More precisely, Eq. (8) leads to two identical copies of Eq. (17).

  8. As for order of magnitude, we consider numerical values corresponding to ultra high energy cosmic rays (UHECR) [51].

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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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  1. QUIT Group, Dipartimento di Fisica and INFN sezione di Pavia, via Bassi 6, 27100, Pavia, Italy

    Alessandro Bisio, Giacomo Mauro D’Ariano, Paolo Perinotti & Alessandro Tosini

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  1. Alessandro Bisio
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  2. Giacomo Mauro D’Ariano
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  3. Paolo Perinotti
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  4. Alessandro Tosini
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Correspondence to Paolo Perinotti.

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Work presented at the conference Quantum Theory: from Problems to Advances, held on 9–12 June 2014 at Linnaeus University, Vaxjo University, Sweden. This paper, together with Ref. [1], 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.

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Bisio, A., D’Ariano, G.M., Perinotti, P. et al. Weyl, Dirac and Maxwell Quantum Cellular Automata. Found Phys 45, 1203–1221 (2015). https://doi.org/10.1007/s10701-015-9927-0

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