Path Integral on Relativistic Spinless Potential Problems

The formulation of the relativistic spinless path integral on the general affine space is presented. For the one dimensional space, the Duru-Kleinert (DK) method and the δ-function perturbation technique are applied to solve the relativistic path integrals of the smooth potential and the Dirichlet boundary condition problems, respectively. Let us first consider a point particle of mass M moving at a relativistic velocity in a (D + 1)-dimensional Minkowski space with a given electromagnetic field. By using t = −iτ = −ix 4 , the path integral representation of the fixed-energy amplitude is conveniently formulated in a (D + 1)-Euclidean spacetime with the Euclidean metric, (g µν) = diag (1, · · · , 1, c 2), (1) and is given by