  For distinguishable particles it is well known that Brownian motion and a
Feynman-Kac functional can be used to calculate the path integral (for
imaginary times) for a general class of scalar potentials. In order 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 non-interacting system. For motion in 1
dimension, we emphasize that the permutation symmetry of the identical
particles completely determines the domain of Brownian motion and the
appropriate boundary conditions: absorption for fermions, reflection for
bosons. Further analysis of the sample paths for motion in 3 dimensions allows
us to decompose these paths into a superposition of 1-dimensional sample paths.
This reduction expresses the propagator (and consequently the energy and other
thermodynamical quantities) in terms of well-behaved 1-dimensional fermion and
boson diffusion processes and the Feynman-Kac functional.
