Summing over Feynman histories by functional contour integration

Abstract

The authors show how complex paths can be consistently introduced into sums over Feynman histories by using the notion of functional contour integration. For a k-dimensional system specified by a potential with suitable analyticity properties, each coordinate axis is replaced by a copy of the complex plane, and at each instant of time a contour is chosen in each plane. This map from the time axis into the set of complex contours defines a functional contour. The family of contours labelled by time generates a (k+1)-dimensional submanifold of the (2k+1)-dimensional space defined by the cartesian product of the time axis and the coordinate planes. The complex Feynman paths lie on this submanifold. The convergence problems encountered in previous proposals for complex path integrals are avoided by the requirement that each contour is asymptotically pinched to the real coordinate axis. An application of this idea to systems described by absorptive potentials yields a simple derivation of the correct WKB result in terms of a complex path that extremalises the action. The method can also be applied to spherically symmetric potentials by using a partial wave expansion and restricting the contours appropriately.