Schwinger-Dyson equations and line integrals

TL;DR

This paper rigorously analyzes the Schwinger-Dyson equations and the Complex Langevin method, revealing that solutions can be expressed as integrals along specific paths, explaining convergence issues in nonergodic cases.

Contribution

It proves a theorem showing that solutions satisfying the SDE are linear combinations of path integrals, clarifying the behavior of the CL method in complex variables.

Findings

Solutions are linear combinations of path integrals connecting zeros and noncontractible loops.

The theorem explains why CL can converge to incorrect limits.

Provides a rigorous foundation for a long-standing conjecture.

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.