Quantum stochastic calculus for the uniform measure and Boolean convolution

Abstract

We study a subspace of the Fock space, called Boolean Fock space, and its associated non-communicative processes obtained by combinations of annihilators and creators. These processes include the Boolean Brownian and Poisson processes obtained by replacing the classical convolution by its Boolean counterpart, and a family of Bernoulli processes. Using a quantum stochastic calculus constructed by time changes, we complete the existing non-commutative relations between basic probability laws. In particular the uniform distribution has the role played by the exponential law in the classical setting of tensor independence.