Nuclear Physics B275 [FS17] (1986) 375-397 North-Holland, Amsterdam THE COMPLEX LANGEVIN EQUATION AND MONTE CARLO SIMULATIONS OF ACTIONS WITH STATIC CHARGES* J. AMBJORN The Niels Bohr Institute, UniversiO, of Copenhagen, Blegdamsvej 17, DK-2 I O0 Copenhagen O, Denmark M. FLENSBURG and C. PETERSON Department of Theoretical Pl~vsics, University of Lund, S~h~egatan 14 A, S 22362 Lund, Sweden Received 14 April 1986 We explore the possibilities of using the complex Langevin equation to Monte Carlo generate configurations of gauge theories with static charges present in the action. Successful results are obtained for systems that possess nontrivial saddle-point (classical) solutions. For that reason the algorithm works impressively for two-dimensional U(1) in the entire fl-range whereas for three- and four-dimensional U(1) the applicability is limited to the weak coupling regime (large fl). The method fails completely for SU(2) and SU(3). 1. Introduction It is often convenient to be able to make Monte Carlo simulations of systems described by complex (effective) actions. Most notable is of course the desire to simulate physical processes without having to rotate from Minkowski space to euclidean space [1]. But even in euclidean field theory one frequently encounters situations, where the effective actions are complex. The integration over fermionic degrees of freedom, the inclusion of topological terms or chemical potentials or certain external fields can generate complex effective actions when rotated from Minkowski space-time to euclidean space [2-6]. In these cases most standard Monte Carlo updating techniques, like Metropolis and the heat bath, cannot be used in a straightforward manner. Even the measure- ment of the simplest "internal" observables can become exceedingly difficult since one has to include the imaginary part of the action in the measurement. If the action can be written as S(O) = SR(O ) q- Si(O), (1.1) * Work supported in part by the Swedish Natural Science Council under contract NFR F 701%109. 0550-3213/86/$03.50'.~:Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)