Duality in interacting particle systems and boson representation

TL;DR

This paper introduces a novel scheme using boson representation and coherent states to derive dual processes in Markov systems, connecting discrete and continuous models through algebraic and generating function methods.

Contribution

It presents a new boson-based approach to derive dual processes and functions, applicable to generators expressed with boson operators, unifying discrete and continuous stochastic models.

Findings

Derived dual processes using bosonic coherent states.

Connected birth-death processes to differential equations.

Applied scheme to processes with SU(1,1) algebra.

Abstract

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.