Restricted phase-space approximation in real-time stochastic quantization

TL;DR

This paper introduces a restricted phase-space approximation in real-time stochastic quantization to improve numerical stability and accuracy in simulating quantum systems, addressing unphysical fixed-points.

Contribution

The paper proposes a simple truncation scheme called the restricted phase-space approximation to prevent unphysical fixed-points in real-time stochastic quantization simulations.

Findings

The method achieves stable numerical results with good accuracy.

It effectively avoids unphysical fixed-points in the Fokker-Planck dynamics.

The approach enables direct computation of vacuum expectation values without the $i\epsilon$ prescription.

Abstract

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.