| Research Abstract |
The stochastic quantization scheme, proposed by Parisi and Wu, has been so designed as to reproduce ordinary quantum theory in the equilibrium limit of a hypothetical stochastic process w.r.t.the so-called fictitious time. In this project, the possibility of quantizing bottomless systems, classical actions of which are unbounded from below, by means of nonequilibrium stochastic processes has been sought. Some related topics have also been investigated.
Here is a summary of the project. 1.We have set up a kerneled Langevin equation for the simplest (0-dim.) bottomless system and solved it to obtain its exact solution. The method has been extended to systems with many degrees of freedom and the exact solution of the corresponding Fokker-Planck equation was found. 2.These exact solutions clarify that the hypothetical stochastic processes described by the bottomless actions are of diffusion type and therefore have no equilibrium. At the same time, however, it is shown that they can well reprodece the desired weight function e^<-s>, expected in the path integral quantization, in the appropriate finite region corresponding to a finite fictitious time. 3.The so-called excursion phenomena, found in numerical simulations of the kerneled Langevin equation, were analyzed theoretically (Kanenaga). 4.Nonequilibrium stochastic processes have been applied to critical phenomena. It has been shown on the basis of some specific examples that the transient process, far from equilibrium, may provide us with such interesting physical quantities as critical exponents (Okano). 5.We have also endeavored to analyze the role of the Zwanziger stochastic gauge fixing on the basis of a solvable gauge model and to derive the transition probability amplitudes for some exactly solvable systems, within the framework of stochastic quantization.
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