SPACE, TIME AND FUNDAMENTAL INTERACTIONS

THE DIRAC EQUATION AND A FERMIONIC CLIFFORD ALGEBRA

Kauffman L. H.

This paper examines the structure of the Dirac equation and gives a new treatment of the Dirac equation in \(1+1\) spacetime. We reformulate the Dirac operator \({\cal{D}}\) so that there is a nilpotent element \(U\), with \(U^2 = 0\) in the Clifford algebra such that for a plane wave \(\psi\), \({\cal{D}}\psi = U\psi\). This means that \(U\psi\) is a solution to the Dirac equation since \({\cal{D}}(U\psi) = U^2\psi = 0\times\psi = 0\). We use this method to reformulate a nilpotent version of the Dirac equation for \((1+1)\) spacetime in light cone coordinates. We then give a solution to the Dirac equation by the method just indicated and compare this solution with the solutions already understood in relation to the Feynman checkerboard model. In the course of this reformulation we see that the transition to light cone coordinates corresponds to a rewriting of the Clifford algebra for the Dirac equation to a new Fermionic algebra.