Discrete Feynman propagator for the Weyl quantum walk in 2 + 1 dimensions

Abstract

-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...

Path-sum solution of the Weyl quantum walk in 3 + 1 dimensions

“…The scope of this paper is the study of the QW evolution in terms of a path-sum, recalling the Feynman's formulation of Quantum Mechanics. Such an approach has been effective to obtain the exact analytic solution in position space in a number of cases [22][23][24][25], giving a background for possible generalizations of the method on general graphs. In the present work we will review the method in the setting of Cayley graphs and we will present a solution for the Weyl QW.…”

“…The combinatorics for ι((w (1) ⊕ Sw (1) ) · w (2) ) is the same as that of Ref. [25], and is here reviewed for the convenience of the reader. The idea is to find a classification of the binary strings in terms of the values taken by ι((w (1) ⊕ Sw (1) ) · w (2) ).…”

Re-evaluating Space-Time

“…One way of addressing this concern would be to derive special relativity from an alternative set of assumptions which do not include the principle that all reference frames are equivalent. In particular, this seems more plausible given recent work on quantum particle dynamics in discrete spacetime [4][5][6][7][8][9][10][11][12], in which relativistic symmetries emerge naturally in the continuum limit despite the underlying discrete model having a preferred frame (for example a lattice of spatial points and discrete time steps). In particular, it has been shown that the simplest quantum walks on a lattice behave like massless relativistic particles at scales much larger than the lattice scale,…”

“…A possible solution would be to derive special relativity from a different set of assumptions, such that it emerges naturally even in the evolving state picture. Recent work on particles in discrete space-time suggests that this is highly plausible -relativistic evolution laws emerge naturally there at large scales despite the existence of a preferred frame [4][5][6][7][8][9][10][11][12]. The key is the existence of a bounded speed of information propagation, which is an appealing assumption in any picture.…”

Discrete Time Dirac Quantum Walk in 3+1 Dimensions

“…[6] the authors provided a solution for the Hadamard Walk, whereas Konno derived the solution for an arbitrary coined QW [53]. Considering the application of QWs to the description of relativistic particles, also the Dirac QW in 1 + 1-dimensions and the massless Dirac QW in 2 + 1-dimensions have been analytically solved in position space [54,55]. On the other hand, when a QW is defined on the Cayley graph of an Abelian group, the walk dynamics can be studied in its Fourier representation, providing analytical solutions and also approximate asymptotic solutions in the long-time limit.…”