Schwinger-Dyson equations and line integrals

Abstract

The Complex Langevin (CL) method sometimes shows convergence to the wrong limit, even though the Schwinger-Dyson Equations (SDE) are fulfilled. We analyze this problem in a more general context for the case of one complex variable. We prove a theorem that shows that under rather general conditions not only the equilibrium measure of CL but any linear functional satisfying the SDE on a space of test functions is given by a linear combination of integrals along paths connecting the zeroes of the underlying measure and noncontractible closed paths. This proves rigorously a conjecture stated long ago by one us (L.~L.~S.) and explains a fact observed in nonergodic cases of CL. one us (L.~L.~S.) and explains a fact observed in nonergodic cases of CL.