Lattice simulations of real-time quantum fields

We investigate lattice simulations of scalar and non-Abelian gauge fields in Minkowski space-time. For $SU(2)$ gauge-theory expectation values of link variables in $3+1$ dimensions are constructed by a stochastic process in an additional (5th) ``Langevin-time.'' A sufficiently small Langevin step size and the use of a tilted real-time contour leads to converging results in general. All fixed point solutions are shown to fulfil the infinite hierarchy of Dyson-Schwinger identities, however, they are not unique without further constraints. For the non-Abelian gauge theory the thermal equilibrium fixed point is only approached at intermediate Langevin-times. It becomes more stable if the complex time path is deformed towards Euclidean space-time. We analyze this behavior further using the real-time evolution of a quantum anharmonic oscillator, which is alternatively solved by diagonalizing its Hamiltonian. Without further optimization stochastic quantization can give accurate descriptions if the real-time extent of the lattice is small on the scale of the inverse temperature.