Treatment of Constraints in Stochastic Quantization Method and Covariantized Langevin Equation

Details Treatment of Constraints in Stochastic Quantization Method and Covariantized Langevin Equation Treatment of Constraints in Stochastic Quantization Method and Covariantized Langevin Equation

1992-12-22

arxiv.org/abs/hep-th/9212128v1

Nucl.Phys. B395 (1993) 371-387

Physics

High Energy Physics

High Energy Physics - Theory

20 pages (Plain TeX), 4 figures(not included, sorry!), CHIBA-EP-57

Scientific paper

10.1016/0550-3213(93)90221-A

We study the treatment of the constraints in stochastic quantization method. We improve the treatment of the stochastic consistency condition proposed by Namiki et al. by suitably taking account of the Ito calculus. Then we obtain an improved Langevin equation and the Fokker-Planck equation which naturally leads to the correct path integral quantization of the constrained system as the stochastic equilibrium state. This treatment is applied to $O(N)$ non-linear $\sigma$ model and it is shown that singular terms appearing in the improved Langevin equation cancel out the $\delta^n(0)$ divergences in one loop order. We also ascertain that the above Langevin equation, rewritten in terms of independent variablesis, actually equivalent to the one in the general-coordinate-transformation-covariant and vielbein-rotation-invariant formalism.

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