Complex Langevin Equation and the Many-Fermion Problem

We study the utility of a complex Langevin (CL) equation as an alternative for the Monte Carlo (MC) procedure in the evaluation of expectation values occurring in fermionic many-body problems. We find that a CL approach is natural in cases where non-positive definite probability measures occur, and remains accurate even when the corresponding MC calculation develops a severe ``sign problem''. While the convergence of CL averages cannot be guaranteed in principle, we show how convergent results can be obtained in three examples ranging from simple one-dimensional integrals over quantum mechanical models to a schematic shell model path integral