New approach to the path integral representation for the Dirac particle propagator

The path integral representation for the propagator of a spinning particle in an external electromagnetic field is derived using the functional derivative formalism with the help of a Weyl symbol representation. The proposed method essentially simplifies the proof of the path integral representation starting from the equation for the Green function and automatically leads to a precise and unambiguous form of the boundary conditions for the Grassmann variables and puts a strong restriction on the choice of the gauge condition. The path integral representation as in the canonical case has been obtained from the general quantization method of Batalin, Fradkin, and Vilkovisky employing a Weyl symbol representation; being the nontrivial first class constraint algebra for a Dirac particle plays an important role in this derivation. This algebra is the limiting case of the superconformal algebra for a Ramond-type open string when the width of one goes to zero.