Charged Particle in a Time-dependent Electric Field: A White Noise Functional Approach

The propagator for a charged particle in a time-dependent electric field is calculated following Hida and Streit's framework [1] where the propagator is the T-transform of Feynman functional. However, we have to regard the potential V = -qE(t)x[s'x following C. C. Bernido and M. V. Carpio-Bernido's [2] prescription of time-dependent potentials. The result agrees with the limiting form of Eqn. (16) of N. Morgenstern Horing, H. L. Cui, and G. Fiorenza [3], when the magnetic field is switched off, and Eqn. (17) of [3] when the electric field is constant in time.