Path integral treatment of pair creation by the step potential

In this paper, we use a formalism including supersymmetry to calculate the Green function for the spinning relativistic particle in the presence of the step potential. This is an example of one of simple but fundamental inhomogeneous fields. We obtain an explicit expression of the Polyakov spin factor and the presence of the step gives rise to a correction of a delta potential in the spin–field coupling. The calculation is done using the perturbation technique. The corresponding positive- and negative-energy states are obtained. This Green function is used to evaluate a pair-creation probability in this potential.