Discrete dirac equation on a finite half-integer lattice

Abstract

We present a relatively simple representation of the Dirac equation on a finite half-integer spatial lattice but an integer momentum lattice with periodic boundary conditions. We present arguments that the half-integer lattice is not just a simple relabeling of the integer lattice but closely related to the fundamental lattice spacing and the form of the action. Furthermore, the concept of discretevs. differential time variable is discussed from the standpoint of the space-time interval. We concluded that it is inconsistent to treat the time variable differently from space variables. In order to develop the consistency of our approach, we have first developed discrete Fourier transforms on the half-integer lattice, and apply them to the calculation of the dispersion relation for the 2D Dirac equation. Next, we show thatdirect separation of the 4D Dirac equation on the cubic lattice leads to equivalent dispersion relations. This second technique, however, allows us to construct explicitly spinors on the cubic lattice very similar to the continuum example. Finally, we note an interesting property of the lattice Dirac action which preserves unitarity. The finite half-integer spatial lattice treatment of the Dirac equation does not lead to species doubling. This unique feature of the lattice, however, disappears in the continuum limit and seems to prevent applications of this work to the usual lattice gauge theories.