  We perform and extend real-time numerical simulation of a low-dimensional
scalar field theory or a quantum mechanical system using stochastic
quantization. After a brief review of the quantization method and the complex
Langevin dynamics, we calculate the propagator and make a comparison with
analytical results. This is a first step toward general applications, and we
focus only on the vacuum properties of the theory; this enables us to handle
the boundary condition with the $i\epsilon$ prescription in frequency space.
While we can control stability of the numerical simulation for any coupling
strength, our results turn out to flow into an unphysical fixed-point, which is
qualitatively understood from the corresponding Fokker-Planck equation. We
propose a simple truncation scheme, "restricted phase-space approximation," to
avoid the unphysical fixed-point. With this method, we obtain stable results at
reasonably good accuracy. Finally we give a short discussion on the closed-time
path formalism and demonstrate the direct computation of the vacuum expectation
value not with the $i\epsilon$ prescription but from an explicit construction
of the Feynman kernel.
