Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density

The aim of this paper is to show that fermion and boson random point processes naturally appear from representations of CAR and CCR which correspond to gauge invariant generalized free states (also called quasi-free states). We consider particle density operators $\rho(x)$, $x\in\R^d$, in the representation of CAR describing an infinite free Fermi gas of finite density at both zero and finite temperature, and in the representation of CCR describing an infinite free Bose gas at finite temperature. We prove that the spectral measure of the smeared operators $\rho(f)=\int dx f(x)\rho(x)$ (i.e., the measure $\mu$ which allows to realize the $\rho(f)$'s as multiplication operators by $\la\cdot,f\ra$ in $L^2(d\mu)$) is a well-known fermion, resp. boson measure on the space of all locally finite configurations in $\R^d$.