Stochastic Quantization of Autonomous Phi**4

TL;DR

This paper applies stochastic quantization to the scalar -model, introducing a momentum-dependent renormalization of the Onsager coefficient, and demonstrates numerical relaxation to the symmetry-breaking vacuum at zero temperature.

Contribution

It presents a novel non-perturbative renormalization approach within stochastic quantization for the -model, including a numerical solution of the Langevin equation for the renormalized mode.

Findings

System relaxes to the symmetry-breaking vacuum at zero temperature

Momentum-dependent renormalization of Onsager coefficient is essential

Numerical solution confirms theoretical predictions

Abstract

The non-perturbative autonomous renormalization of the scalar $\Phi^4$-model is applied in the framework of stochastic quantization. I show that this requires a selective, momentum-dependent renormalization of the Onsager coefficient $\lambda$, a direct consequence of the characteristic wavefunction renormalization applied. As a result, I obtain a Langevin equation for the renormalized constant mode of the field, which is solved numerically. It is demonstrated for temperature zero that, starting from specified initial conditions, the system relaxes to its equilibrium state, the symmetry-breaking vacuum of the ``static'' $\Phi^4$-theory.