Quantum action principle and path integrals for long-range interactions

Abstract

Schwinger’s action (dynamical) principle for transition amplitudes, in quantum mechanics and field theory, is derived for long-range interactions by using the unescapable fact that a scattering out-(in-) state when translated forward (backward) in time for large positive (negative) times may stilldepend on the quantum-mechanical coupling parameter. For such interactions, the dynamical principle is different from the conventional one used extensively historically in the literature. As a corollary to our main result we show that the integrand in the Feynman path integral formulation is not simply given by exp [i[Action]]. Applications of the derived results are then given to the specific cases for the Coulomb interaction and to quantum electrodynamics.