Relativistic Wave Equations on the lattice: an operational perspective

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 $\mathit{\text{Exponential}}\mathit{\text{Generating}}\mathit{\text{Function}}$ (EGF), carrying a degree lowering operator $L_t=L({\mathrm{\partial }}_t)$. We also use certain operational properties of the discrete Fourier transform over the $n-$dimensional $\mathit{\text{Brioullin}}\mathit{\text{zone}}$ $Q_h={(-\frac{\pi }{h},\frac{\pi }{h}]}^n$ – a toroidal Fourier transform in disguise – to describe the discrete counterparts of the continuum wave propagators, $\cosh (t\sqrt{\mathrm{\Delta }-m^2})$ and ${\displaystyle \frac{\sinh (t\sqrt{\mathrm{\Delta }-m^2})}{\sqrt{\mathrm{\Delta }-m^2}}}$ 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.