Path integral in a magnetic field using the Trotter product formula
چکیده
There are many ways to derive the path integral. In his original paper, Feynman gave a physical motivation for a particular form and verified its correctness by propagating the wave function for an infinitesimal time step. In Ref. 2 the opposite perspective is used: start with a Hamiltonian and use the Trotter product formula to arrive at the sum-overpaths. Some of Feynman’s poetry is lost, but one connects more easily to familiar formulations. The Trotter-productformula-based derivation in Ref. 2 is limited to Hamiltonians of the form H5p/2m1V(x), and skirts the more delicate case of a particle in a magnetic field. As is well known, the latter situation requires that when writing the classical action for the broken line path in the path integral, the vector potential, A~x!, be approximated by its average at the ends of each straight segment of the path. One can also evaluate A at the midpoint of that segment, giving rise to the ‘‘midpoint rule.’’ This feature is one of the fundamental properties of the path integral, reflecting the close relation between the path integral and the Wiener integral, and the ‘‘infinite velocities’’ or non-differentiability of the paths in both cases. Yet more delicate evaluations are necessary for the path integral on curved spaces, so it would be useful to understand how to deal with these rough paths from the Trotter derivation perspective. In this article we use a slight variation of the derivation in Ref. 2 to show how we can arrive at the correct path integral, including a magnetic field. Some years ago two of us ~B. G. and L. S. S.! published such a derivation, but it was phrased in the language of stochastic processes. The present work uses a framework more familiar to physicists and is more streamlined. The earlier paper includes path integrals on curved spaces, which are not treated here. Because the purpose of this article is to extend familiar path integral techniques, we do not elaborate on how the formulas developed here make their way into the usual path integral representations.