MANY-BODY DIFFUSION AND PATH-INTEGRALS FOR IDENTICAL PARTICLES

For distinguishable particles it is well known that Brownian motion an d a Feynman-Kac functional can be used to calculate the path integral (for imaginary times) for a general class of scalar potentials. In ord er to treat identical particles, we exploit the fact that this method separates the problem of the potential, dealt with by the Feynman-Kac functional, from the process which gives sample paths of a noninteract ing system. For motion in one dimension, we emphasize that the permuta tion symmetry of the identical particles completely determines the dom ain of Brownian motion and the appropriate boundary conditions: absorp tion for fermions, reflection for bosons. Further analysis of the samp le paths for motion in three dimensions allows us to decompose these p aths into a superposition of one-dimensional sample paths. This reduct ion expresses the propagator (and consequently the energy and other th ermodynamical quantities) in terms of well-behaved one-dimensional fer mion and boson diffusion processes and the Feynman-Kac functional.