Sthochastic Quantization of Constrained Systems: General Theory and Nonlinear Sigma Model

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Abstract
The stochastic quantization method is extended to a dynamical system described by regular Lagrangian under additional holonomic constraints. We first show that the stochastic quantization method surely yields the same result as given by the path-integral quantization, by imposing the constraints on the system throughout the whole hypothetical stochastic process with respect to a fictitious time. Next we propose, on the analogy of the theory of optimization, new types of rather moderate constraints, (i) converging constraints and (ii) fluctuating constraints, which are so desined as to coincide with the original ones only at the infinite fictitious-time limit. The present formalism with the new types of constraints prepares a feasible method to carry out numerical analyses of a dynamical system with nonlinear constraints. We can expect that the method works well in the case of the lattice nonlinear σ-model.

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