Summary: Complex Langevin method: When can it be trusted?
Department of Physics, Swansea University, Swansea, United Kingdom
Max-Planck-Institut fušr Physik (Werner-Heisenberg-Institut), Mušnchen, Germany
Institut fušr Theoretische Physik, Universitašt Heidelberg and FEST, Heidelberg, Germany
(Received 18 December 2009; published 22 March 2010)
We analyze to what extent the complex Langevin method, which is in principle capable of solving the
so-called sign problems, can be considered as reliable. We give a formal derivation of the correctness and
then point out various mathematical loopholes. The detailed study of some simple examples leads to
practical suggestions about the application of the method.
DOI: 10.1103/PhysRevD.81.054508 PACS numbers: 11.15.Ha, 11.15.Bt
The complex Langevin method solves in principle the
sign problems arising in simulations of various systems, in
particular, in QCD with finite chemical potential. After
being proposed in the early 1980s by Klauder  and
Parisi , it enjoyed a certain limited popularity [3,4],
but very quickly certain problems were found. The first