Relativistic Wave Equations on the lattice: an operational perspective

TL;DR

This paper introduces an operational framework for discretizing and computing solutions to relativistic wave equations like Klein-Gordon and Dirac on a lattice, using exponential generating functions and Fourier transform properties.

Contribution

It develops a novel operational approach for discretized relativistic wave equations leveraging exponential generating functions and Fourier analysis.

Findings

Discrete wave propagators are expressed as convolution operators.

The framework applies to differential-difference and difference-difference evolution problems.

Operational properties facilitate the analysis of relativistic equations on lattices.

Abstract

This paper presents an operational framework for the computation of the discretized solutions for relativistic equations of Klein-Gordon and Dirac type. The proposed method relies on the construction of an evolution-type operador from the knowledge of the \textit{Exponential Generating Function} (EGF), carrying a degree lowering operator $L_t=L(\partial_t)$. We also use certain operational properties of the discrete Fourier transform over the $n-$dimensional \textit{Brioullin zone} $Q_h=\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^n$ -- a toroidal Fourier transform in disguise -- to describe the discrete counterparts of the continuum wave propagators, $\cosh(t\sqrt{\Delta-m^2})$ and $\dfrac{\sinh(t\sqrt{\Delta-m^2})}{\sqrt{\Delta-m^2}}$ respectively, as discrete convolution operators. In this way, a huge class of discretized time-evolution problems of differential-difference and difference-difference type may be studied in the spirit of hypercomplex variables.