On the approximation of Feynman-Kac path integrals

The symbol D [x (τ)] indicates that the integration is performed over the set of all differentiable curves, x : [0, βh̄]→ R, with x (0) = a and x (βh̄) = b. The integer d reflects the dimensionality, with d = 3N for a system of N -particles in 3-dimensional space. The functional Φ can be derived from the classical action by introducing a relationship between temperature and imaginary time (it = βh̄) [1]. In this paper, we will restrict our attention to the quantum many-body system, for which Φ takes the following form: