Spin Factor in Path Integral Representation for Dirac Propagator in External Fields

We study the spin factor problem both in $3+1$ and $2+1$ dimensions which are essentially different for spin factor construction. Doing all Grassmann integrations in the corresponding path integral representations for Dirac propagator we get representations with spin factor in arbitrary external field. Thus, the propagator appears to be presented by means of bosonic path integral only. In $3+1$ dimensions we present a simple derivation of spin factor avoiding some unnecessary steps in the original brief letter (Gitman, Shvartsman, Phys. Lett. {\bf B318} (1993) 122) which themselves need some additional justification. In this way the meaning of the surprising possibility of complete integration over Grassmann variables gets clear. In $2+1$ dimensions the derivation of the spin factor is completely original. Then we use the representations with spin factor for calculations of the propagator in some configurations of external fields. Namely, in constant uniform electromagnetic field and in its combination with a plane wave field.Comment: 34 pages, LaTe