Relativistic Wave Equations on the lattice: an operational perspective

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.