In this thesis we give self-sufficient introduction to the complex Langevin method, which is a promising simulation method for the quantum field theories with finite chemical potential. We begin by revising the the stochastic methods needed for mathematical understanding of Langevin equation.
This revision is done trough the examples of a Brownian motion. After the stochastical preliminaries are deled with, we also give a short introduction to quantum field theories on the lattice and with both finite temperature and chemical potential. We show that introduction of chemical potential can cause a sign problem, which can yield traditional Monte Carlo simulations non usable for this problem.
We notice a similarity of form between the stochastic path integrals and the Feynman path integrals. We use this similarity to define a stochastic quantization method, that lets us to use stochastic evolution equations to find the correlation function in quantum field theory. This method is then expanded for complex actions that arise from the use of the finite chemical potential in the quantum field theory. We discuss the possible problems with this expansion and deduct that there is no complete mathematical understanding of complex Langevin equation.
After all these theoretical considerations we do our own simulations to inspect if this model works. We find out, that for the U(1) one link model and for the SU(3) spin model, the method works as long as we are not using imaginary noise. Unfortunately we also find out that three dimensional XY-model converges to wrong results. We find no reason for this behavior.
