From CCR to Levy Processes: An Excursion in Quantum Probability

TL;DR

This paper explores the connection between canonical commutation relations, quantum stochastic calculus, and Levy processes within the framework of quantum probability, highlighting the mathematical structures linking quantum and classical stochastic processes.

Contribution

It provides an expository account of how Levy processes emerge from quantum stochastic calculus derived from CCR and Weyl representations.

Findings

Derivation of Levy processes from quantum stochastic calculus

Development of Ito's formula for quantum and classical processes

Linking quantum probability structures to classical Levy processes

Abstract

This is an expositary article 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. \item 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. (ii) The three fundamental fields, with the inclusion of time, lead to quantum stochastic integration and a calculus with an Ito's formula for products of differentials. (iii) Appropriate linear combinations of the fundamental operator processes yield all the L{\'e}vy processes of classical probability theory along with the bonus of Ito's formula for products of their differentials.