Monte Carlo path integration in real time via complex coordinates

A method is described for Monte Carlo path integration that is valid for real time propagation. More specifically, it is shown how matrix elements of the complex-time propagator e−βcH, βc=β/2+it/ℏ, can be evaluated by straightforward Monte Carlo for values t≫βℏ/2. The key feature is that one distorts the path of the integration variables so that the kinetic energy part of the integrand is real. This in turn means that the paths are complex valued, but it is shown that, at least for barrier-type potentials, this causes no difficulties.