Quantum cellular automata have been recently considered as a fundamental approach to quantum field theory, resorting to a precise automaton, linear in the field, for the Dirac equation in one dimension. In such linear case a quantum automaton is isomorphic to a quantum walk, and a convenient formulation can be given in terms of transition matrices, leading to a new kind of discrete path integral that we solve analytically in terms of Jacobi polynomials versus the arbitrary mass parameter.

-Recently quantum walks have been considered as a possible fundamental description of the dynamics of relativistic quantum fields. Within this scenario we derive the analytical solution of the Weyl walk in 2 + 1 dimensions. We present a discrete path-integral formulation of the Feynman propagator based on the binary encoding of paths on the lattice. The derivation exploits a special feature of the Weyl walk, that occurs also in other dimensions, that is closure under multiplication of the set of the walk transition matrices. This result opens the perspective of a similar solution in the 3 + 1 case.A simple description of particles propagation on a discrete spacetime was proposed by Feynman in the so called checkerboard problem [1] that consists in finding a simple rule to represent the quantum dynamics of a Dirac particle in 1 + 1 dimensions as a discrete path-integral.The definition of a discrete path-integral is closely related to the underlying notion of "discrete spacetime" and on the dynamical model used to describe the discrete time evolution of the quantum systems. As a consequence, in the absence of an established theory of quantum spacetime, the formulation of a discrete Feynman propagator can be considered within different possible scenarios.Following the original idea of Feynman, and the subsequent progress of Refs. [2,3], in Ref.[4] Kaufmann and Noyes analysed the checkerboard problem, providing a solution of the finite-difference version of Dirac's equation. In Refs. [5, 6] a path-integral formulation for the discrete space-time is presented within the causal set approach of Bombelli and Sorkin [7], with trajectories within the causal set summed over to obtain a particle propagator. More recently, following the pioneering papers [8][9][10], the quantum walks (QWs) have been considered as a discrete model of dynamics for relativistic particles [11][12][13][14][15][16][17][18][19][20].A QW is the quantum version of a (classical) random walk that describes a particle moving in discrete time steps and with certain probabilities from one lattice positionto the neighboring sites. The first QW appeared in [21] where the measurement of the z-component of a spin-1/2 quantum system, also denoted internal degree of freedom or coin system, decides whether the particle moves right or left. Then the measurement was replaced by a unitary operator on the coin system [22] with the QW representing a discrete unitary evolution of a particle state with internal degree of freedom given by the coin. In the more general case the coin at a site x of the lattice can be represented by a finite dimensional Hilbert space H x = C s , with the total Hilbert space of the system given by the direct sum of all sites Hilbert spaces.QWs provide the one-step free evolution of one-particle quantum states, however, replacing the quantum state with a quantum field on the lattice, a QW describes the discrete evolution of non interacting particles with a given statistics-a "second quantization" of the QW. This can be ultimately re...

Abstract:We study the solutions of an interacting Fermionic cellular automaton which is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particle solutions of the automaton recently in the literature, which exploits the symmetries of the evolution operator. In the two-particle sector, the evolution operator is given by the sequence of two steps, the first one corresponding to a unitary interaction activated by two-particle excitation at the same site, and the second one to two independent one-dimensional Dirac quantum walks. The interaction step can be regarded as the discrete-time version of the interacting term of some Hamiltonian integrable system, such as the Hubbard or the Thirring model. The present automaton exhibits scattering solutions with nontrivial momentum transfer, jumping between different regions of the Brillouin zone that can be interpreted as Fermion-doubled particles, in stark contrast with the customary momentum-exchange of the one-dimensional Hamiltonian systems. A further difference compared to the Hamiltonian model is that there exist bound states for every value of the total momentum and of the coupling constant. Even in the special case of vanishing coupling, the walk manifests bound states, for finitely many isolated values of the total momentum. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.

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