 
Summary: Can Stochastic Quantization Evade the Sign Problem? The Relativistic Bose Gas
at Finite Chemical Potential
Gert Aarts
Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
(Received 15 October 2008; published 1 April 2009)
A nonperturbative study of field theories with a complex action, such as QCD at finite baryon density, is
difficult due to the sign problem. We show that the relativistic Bose gas at finite chemical potential has a
sign and ``silver blaze'' problem, similar to QCD. We then apply stochastic quantization and complex
Langevin dynamics to study this theory with nonperturbative lattice simulations. Independence of
chemical potential at small and a transition to a condensed phase at large chemical potential are found.
Lattices of size N4
, with N ž 4, 6, 8, 10, are used. We show that the sign problem is severe, however, we
find that it has no negative effect using this approach. This improves the prospects of applying stochastic
quantization to QCD at nonzero density.
DOI: 10.1103/PhysRevLett.102.131601 PACS numbers: 11.15.Ha, 12.38.Mh
Introduction.Field theories with a complex action are
difficult to treat nonperturbatively. Because the weight in
the path integral eĀS ž jeĀSjei' is not real, standard nu
merical approaches based on a probability interpretation
and importance sampling cannot be applied. This has hin
