相互作用粒子系统的对偶性与玻色子表示
Duality in interacting particle systems and boson representation
摘要：
在马尔可夫过程的背景下，我们展示了一个新的计划来获得对偶过程和基于玻色子表示的对偶函数。该方案适用于生成元由玻色子产生和湮灭算符表示的情况。对于某些随机过程，已知连续时间马尔可夫过程与离散状态空间和连续状态空间之间的对偶关系。我们澄清，使用生成函数方法和Doi-Peliti方法，出生-死亡过程（或离散随机行走模型）自然地连接到一个连续变量的微分方程，这将被解释为一个对偶马尔可夫过程。推导过程中的关键点是用玻色相干态代替了传统的投影态。作为例子，我们将该方案应用到一个简单的出生凝聚过程和布朗动量过程。布朗动量过程的生成元由SU（1，1）代数的元素表示，并利用SU（1，1）的玻色子实现证明了同样的方案是可行的。
英文摘要：
In the context of Markov processes, we show a new scheme to derive dual processes and a duality function based on a boson representation. This scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators. For some stochastic processes, duality relations have been known, which connect continuous time Markov processes with discrete state space and those with continuous state space. We clarify that using a generating function approach and the Doi-Peliti method, a birth-death process (or discrete random walk model) is naturally connected to a differential equation with continuous variables, which would be interpreted as a dual Markov process. The key point in the derivation is to use bosonic coherent states as a bra state, instead of a conventional projection state. As examples, we apply the scheme to a simple birth-coagulation process and a Brownian momentum process. The generator of the Brownian momentum process is written by elements of the SU(1,1) algebra, and using a boson realization of SU(1,1) we show that the same scheme is available.
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