Lf. Lemmens et al., MANY-BODY DIFFUSION AND PATH-INTEGRALS FOR IDENTICAL PARTICLES, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 53(5), 1996, pp. 4467-4476
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.