Brownian path integral from Dirac equation: a probabilistic approach to the Foldy-Wouthuysen transformation

The spectrum of the Dirac Hamiltonian H(D) has a positive and a negative part, the first corresponds to the energy levels of the electron, the second to minus the energy levels of the positron. The Foldy-Wouthuysen transformation is a tool to obtain a decomposition of H(D) in a direct sum H(D) = H(e-) + -H(e+), where H(e-) is the Hamiltonian of the electron and H(e+) is that of the positron. Unfortunately, this decomposition has been obtained in a dosed form only when the external field is purely magnetic, while, in the presence of an electric field, only a perturbative expansion is available. In this paper we give a path integral representation of the semigroup phi(x, t) exp {-(t/hBAR)H(e-)}phi0(x) in an external electromagnetic field. Our formula is the relativistic version, for Dirac particles, of the well known Feynman-Kac-Ito formula for Schrodinger semigroups. The result can also be regarded as a tool to obtain the decomposition H(D) = H(e-) + -H(e+) even in the presence of a non-trivial electric field.