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Quantum Walks, Weyl Equation and the Lorentz Group

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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.

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Notes

  1. For example, if \(\Gamma \) is the one dimensional lattice which we identify with the set of integers \(\mathbb {Z}\), we may require \(|x-x'| \ge n \Rightarrow \langle x | \langle i |A | x' \rangle | i' \rangle = 0\) for some \(n\ge 1\). More synthetically we can say that the unitary matrix A is block-sparse.

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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.

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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

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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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Correspondence to Giacomo Mauro D’Ariano.

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Bisio, A., D’Ariano, G.M. & Perinotti, P. Quantum Walks, Weyl Equation and the Lorentz Group. Found Phys 47, 1065–1076 (2017). https://doi.org/10.1007/s10701-017-0086-3

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