From CCR to Lévy processes: An excursion in quantum probability

Abstract

This is an expositoryarticle telling a short story made from the leaves of quantum probability with the following ingredients: (i)A special projective, unitary, irreducible and factorizable representation of the euclidean group of a Hilbert space known as the Weyl representation.(ii)The infinitesimal version of the Weyl representation includes the Heisenberg canonical commutation relations (CCR) of quantum theory. It also yields the three fundamental operator fields known as the creation, conservation and annihilation fields.(iii)The three fundamental fields, with the inclusion of time, lead to quantum stochastic integration and a calculus with an Itô’s formula for products of differentials.(iv)Appropriate linear combinations of the fundamental operator processes yield all the Lévy processes of classical probability theory along with the bonus of Itô’s formula for products of their differentials.