Rigorous real-time Feynman path integral for vector potentials

In this paper, we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the L-2 transition probability amplitude via improper Riemann integrals. Our formulation will hold for a vector potential Hamiltonian for which its potential and vector potential each carry at most a finite number of singularities and discontinuities.