Restricted phase-space approximation in real-time stochastic quantization

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 iϵ 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 iϵ prescription but from an explicit construction of the Feynman kernel.