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#### Schwinger-Dyson equations and line integrals

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##### Citation

Ssalcedo, L. L., & Seiler, E. (2019). Schwinger-Dyson equations and line integrals.* Journal of Physics A: Mathematical and General Physics,* *52*,
035201. Retrieved from https://publications.mppmu.mpg.de/?action=search&mpi=MPP-2018-231.

Cite as: https://hdl.handle.net/21.11116/0000-0005-D7AF-4

##### 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.