Coherent-state path integral calculation of the Wigner function

We consider a set of operators x̂ = (x̂1, . . . , x̂N ) with diagonal representatives P (n) in the space of generalized coherent states |n〉 : x̂ = ∫ dμ(n)P (n)|n〉〈n|. We regularize the coherent-state path integral as a limit of a sequence of averages 〈 〉L over polygonal paths with L vertices n1...L. The distribution of the path centroid P̄ = 1 L ∑L l=1 P (nl) tends to the Wigner function W (x), the joint distribution for the operators: W (x) = limL→∞〈δN (x− P̄ )〉L. This result is proved in the case where the Hamiltonian commutes with x̂. The Wigner function is non-positive if the dominant paths with path centroid in a certain region have Berry phases close to odd multiples of π. For finite L the path centroid distribution is a Wigner function convolved with a Gaussian of variance inversely proportional to L. The results are illustrated by numerical calculations of the spin Wigner function from SU(2) coherent states. The relevance to the quantum Monte Carlo sign problem is also discussed. PACS numbers: 03.65.Bz, 02.70.Lq, 31.15.Kb Submitted to: J. Phys. A: Math. Gen. Version 28 March 2000. Accepted 5 June 2000.