Coherent-state path-integral calculation of the Wigner function

DOI10.1088/0305-4470/33/29/306zbMATH Open1011.81022arXivquant-ph/0006021OpenAlexW2043703092MaRDI QIDQ4520614

Publication date: 13 December 2000

Published in: Journal of Physics A: Mathematical and General (Search for Journal in Brave)

Abstract: We consider a set of operators hat{x}=(hat{x}_1,..., hat{x}_N) with diagonal representatives P(n) in the space of generalized coherent states |n>; hat{x}=int dn 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 {n_1...L}. The distribution of the path centroid bar{P}=(1/L) sum_{i=1}^{L}P(n_i) tends to the Wigner function W(x), the joint distribution for the operators: W(x)=lim_{L->infinity} <delta(x-bar{P})>_{L}. This result is proved in the case where the Hamiltonian commutes with hat{x}. 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 pi. 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.

Full work available at URL: https://arxiv.org/abs/quant-ph/0006021

zbMATH Keywords

path integral generalized coherent states Berry phases joint distribution for the operators Monte Carlo sign problem

Mathematics Subject Classification ID

Path integrals in quantum mechanics (81S40) Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10) Coherent states (81R30) Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics (81S30) Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory (81Q70)