Path integral for spin: A new approach

The path integral representation for the propagator of a Dirac particle in an external electromagnetic field is derived using the functional derivative formalism with the help of Weyl symbol representation for the Grassmann vector part of the variables. The proposed method simplifies the proof of the path integral representation starting from the equation for the Green function significantly and automatically leads to a precise and unambiguous set of boundary conditions for the anticommuting variables and puts strong restrictions on the choice of the gauge conditions. The same problem is reconsidered using the Polyakov and Batalin-Fradkin-Vilkovisky methods together with the Weyl symbol method and it is shown to yield the same PIR. It is shown that in the last case, the non-trivial first class constraints algebra for a Dirac particle plays an important role in the derivation, and this algebra is the limiting case of the superconformal algebra for a Ramond open string when the width goes to zero. That the approach proposed here can be applied to any point-like particle is illustrated in the propagator for the nonrelativistic Pauli spinning particle in an external electromagnetic field. ((orig.))