Quantization of unstable systems with action unbounded from below-Stochastic quantization by the use of kerneled Langevin equation

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In Parisi-Wu's quantization method, they assumed a stochasitic process according to a fictitious time which was introduced in addition to a real time. Quantum fluctuation was taken into the scheme as white noise term in the Langevin equation which describes the stochastic process. In case of systems with stable ground state, this scheme gives us the same quantum description as usual quantum mechanics in the limit of thermal equilibrium of this fictitious stochastic process.In the stochastic quantization, they are only interested in the limit of thermal equilibrium itself, but not in the detailed process in which a stochastic process develops, as far as an arrow of fictitious time does dot reverse. Thus one can choose time scale arbitrary and take this freedom mathematically into the Langevin equation as an integration kernel. In this research, first we utilize this freedom. If we choose a kernel suitably, in the case where a system has unstable potential globally but has local metastable points, we can develop an algorism which prevents run away solution into unstable region in the Langevin simulation. Furthermore, we investigated the reason why a such mechanism does work in our modified algorism. Secondly, we cleared the mathematical structure of its mechanism analytically, and assure its utility.In actual phenomena, though there exist many interesting unstable or meta-stable systems, we have no reliable method to quantize such systems in a definite way. The stochastic quantization utilizing the kernel is one of possibilities which are looked for and, indeed, it can work well. In this research, using a simple model, we showed mathematically the reason why it can work well, and gave it the theoretical bases. In future, we have to apply this method to more realistic problems.