Approximation algorithm for functional integrals, and the instanton gas

A new algorithm is proposed for the approximation of vacuum expectation values in the functional integral formalism. The integrand $\exp (-S[\phi \left]\right)$ is approximated by a sum (integral) of Gaussian functionals. Every step of the algorithm consists of two parts: (a) finding the important field configurations by comparing the previous approximation with the exact integrand and finding points where the error is significant, and (b) approximating the integrand by an integral of Gaussians using the Gaussian-matching principle at each important field configuration. The method is illustrated with the one-dimensional double-well model in the $\phi (\pm \infty )=1\mathrm{}$ homotopy class. The first step is the semiclassical approximation and the second step generates kink-antikink configurations as solutions of a differential equation. The algorithm resolves the problems of overlapping instantons and gives both perturbative and nonperturbative contributions.