Extension of Nelson's Stochastic Quantization to Finite Temperature Using Thermo Field Dynamics

TL;DR

This paper extends Nelson's stochastic quantum mechanics to finite temperature using Thermo Field Dynamics, demonstrating that the formalism reproduces the TFD Schrödinger equation and satisfies the uncertainty relation.

Contribution

It introduces a novel formalism combining stochastic mechanics with TFD to incorporate finite temperature effects in quantum systems.

Findings

Reproduces TFD-type Schrödinger equation via stochastic equations.

Drift terms depend on temperature, modeling thermal fluctuations.

Satisfies position-momentum uncertainty at finite temperature.

Abstract

We present an extension of Nelson's stochastic quantum mechanics to finite temperature. Utilizing the formulation of Thermo Field Dynamics (TFD), we can show that Ito's stochastic equations for tilde and non-tilde particle positions reproduce the TFD-type Schr\"odinger equation which is equivalent to the Liouville-von Neumann equation. In our formalism, the drift terms in the Ito's stochastic equation have the temperature dependence and the thermal fluctuation is induced through the correlation of the non-tilde and tilde particles. We show that our formalism satisfy the position-momentum uncertainty relation at finite temperature.