Many-body diffusion and path integrals for identical particles

Abstract

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.

Momentum distribution of confined bosons: Temperature dependence

“…Using the methods for path integration developed in [5][6][7], we have been able to calculate the momentum distribution function of a gas of parabolically confined bosons with quadratic harmonic interparticle interactions. This calculation is presented in Sec.…”

Characterization of shell filling of interacting polarons in a quantum dot through their optical absorption

“…Thus a many-body extension of the Jensen-Feynman inequality was found, which could be used for interacting identical particles (Ref. [8], p. 4476). A more detailed analysis of this variational principle for both local and retarded interactions can be found in Ref.…”

“…In Refs. [6,7], the ground state and the optical response of a fixed number of identical interacting polarons are analyzed using the variational path-integral method for identical particles [8,9,10]. As far as we investigate a system with a fixed number of identical particles, it should be described using the canonical ensemble [10].…”

Note on the Path-Integral Variational Approach in Many-Body Theory

“…If the potential V (x (t)) is invariant with respect to the permutations of the carthesian components of the particle coordinates 22,23 , the many-body propagator was obtained by four independent processes per pair of particles, defined on a state space (D 3 n in the notations of Ref. 22 ) with well-defined boundary conditions (see Eqs (4.16-4.17) in Ref.…”

“…This implies (see subsection 3.1 for more details) that a many-body extension of the Jensen-Feynman inequality was found, which can be used to evaluate the partition function for interacting identical particles (Ref. 22 , p. 4476, Ref. 23 ).…”