  We introduce a noncommutative Poisson random measure on a von Neumann
algebra. This is a noncommutative generalization of the classical Poisson
random measure. We call this construction Poissonization. Poissonization is a
functor from the category of von Neumann algebras with normal semifinite
faithful weights to the category of von Neumann algebras with normal faithful
states. Poissonization is a natural adaptation of the second quantization to
the context of von Neumann algebras. The construction is compatible with normal
(weight-preserving) homomorphisms and unital normal completely positive
(weight-preserving) maps. We present two main applications of Poissonization.
First Poissonization provides a new framework to construct algebraic quantum
field theories that are not generalized free field theories. Second
Poissonization permits straight-forward calculations of quantum relative
entropies (and other quantum information quantities) in the case of type III
von Neumann algebras.
