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We apply Parisi-Wu type Stochastic Quantization Method to a finite temperature lattice field theory of the real-time formula. In the theory, the time axis is extended to a complex contour proposed by Matsumoto et al. and Niemi and Semenoff. The finite temperature property is guaranteed by (anti-) periodicity of the time contour in the imaginary direction and a part of the time contour along the real axis describes the real evolution of the system. Taking correlations on the real-time part, we can directly obtain the relaxation of the system. We apply numerically this method to a scalar field on the lattice. In the stationary limit of the stochastic process expectation values of physical quantities converge. Taking field correlation on the real-time part, relaxation like behavior of the system appears.

Stochastic quantization of a finite temperature lattice field theory in the real time formula