On the Hamiltonian Approach and Path Integration for a Point Particle Minimally Coupled to Electromagnetism

We {\em derive} the exact configuration space path integral, together with the way how to evaluate it, from the Hamiltonian approach for any quantum mechanical system in flat spacetime whose Hamiltonian has at most two momentum operators. Starting from a given, covariant or non-covariant, Hamiltonian, we go from the time-discretized path integral to the continuum path integral by introducing Fourier modes. We prove that the limit $N \rightarrow \infty$ for the terms in the perturbation expansion (``Feynman graphs'') exists, by demonstrating that the series involved are uniformly convergent. {\em All} terms in the expansion of the exponent in $<x| \exp (- \Delta t \hat{H} / \hbar) |y>$ contribute to the propagator (even at order $\Delta t$!). However, in the time-discretized path integral the only effect of the terms with $\hat{H}^2$ and higher is to cancel terms which naively seem to vanish for $N \rightarrow \infty$, but, in fact, are nonvanishing. The final result is that the naive correspondence between the Hamiltonian and the Lagrangian approach is correct, after all. We explicitly work through the example of a point particle coupled to electromagnetism. We compute the propagator to order $(\Delta t)^2$ both with the Hamiltonian and the path integral approach and find agreement.