Duality in interacting particle systems and boson representation

In the context of Markov processes, we show a new scheme to derive a dual process and 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 for a bra state, instead of a usual 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 operators in SU(1, 1), and using a boson realization for operators in SU(1, 1) we show that the same scheme is available. PACS numbers: 05.40.-a, 02.50.-r,82.20.-w Duality in interacting particle systems and boson representation 2