STOCHASTIC QUANTIZATION OF INSTANTONS

The method of Parisi and Wu to quantize classical fields is applied to instanton solutions phi(I) of euclidian non-linear theory in one dime nsion. The solution phi(epsilon) of the corresponding Langevin equatio n is built through a singular perturbative expansion in epsilon=h(1/2) in the frame of the center of mass of the instanton, where the differ ence phi(epsilon)-phi(I) carries only fluctuations of the instanton fo rm. The relevance of the method is shown for the stochastic K dV equat ion with uniform noise in space: the exact solution usually obtained b y the inverse scattering method is retrieved easily by the singular ex pansion. A general diagrammatic representation of the solution is then established which makes a thorough use of regrouping properties of st ochastic diagrams derived in scalar field theory. Averaging over the n oise and in the limit of infinite stochastic time, we obtain explicit expressions for the first two orders in epsilon of the perturbed insta nton and of its Green function. Specializing to the Sine-Gordon and ph i(4) models, the first anharmonic correction is obtained analytically. The calculation is carried to second order for the phi(4) model, show ing good convergence. (C) 1996 Academic Press, Inc.