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 (using the method of Peter Rowlands) so that there is a nilpotent element in the Clifford algebra such that for a plane wave, the Dirac operator applied to the plane wave returns the wave multiplied by the nilpotent element.

This means that the product of the nilpotent element and the plane wave is a solution to the Dirac equation. We use this formulation to produce solutions of the Dirac equation for (1+1) spacetime in light cone coordinates. We compare and raise questions about this solution in relation to the solutions already understood via the Feynman checkerboard model. We show that the transition to light cone coordinates corresponds to a rewriting of the Clifford algebra for the Dirac equation to a Fermionic algebra linked with a Clifford algebra.