Noncommutative Poisson Random Measure and Its Applications

Abstract: 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.