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

在paris - wu的量化方法中，他们根据一个虚拟时间假设了一个随机过程，这个虚拟时间是在真实时间之外引入的。该方案将量子涨落作为描述随机过程的朗之万方程中的白噪声项。对于具有稳定基态的系统，该格式给出了与一般量子力学相同的热平衡极限下的量子描述。在随机量子化中，他们只对热平衡本身的极限感兴趣，而对随机过程发展的详细过程不感兴趣，只要一个虚拟的时间箭头没有反转。因此，人们可以任意选择时间尺度，并将这种自由以数学方式作为积分核带入朗之万方程。在这项研究中，我们首先利用这种自由。如果选择合适的核，在系统具有全局不稳定势但有局部亚稳点的情况下，我们可以开发一种算法来防止朗格万模拟中的失控解进入不稳定区域。此外，我们还研究了这种机制在我们改进的算法中起作用的原因。其次，对其机构的数学结构进行了解析，保证了其实用性。在实际现象中，虽然存在许多有趣的不稳定或亚稳定系统，但我们没有可靠的方法来确定这些系统的量化。利用核函数进行随机量化是一种可行的方法，而且效果很好。在本研究中，我们使用一个简单的模型，从数学上说明了它之所以能很好地工作的原因，并给出了理论依据。今后，我们必须把这种方法应用到更实际的问题中去。

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.