Representations of homogeneous quantum L\'evy fields

TL;DR

This paper develops a mathematical framework for representing homogeneous quantum Lévy fields using covariant GNS representations, enabling dimension-free quantum stochastic integration over space-time.

Contribution

It introduces a new covariant GNS representation for homogeneous quantum Lévy fields in complex Minkowski space, facilitating quantum stochastic calculus on Itô monoids.

Findings

Constructed a strongly covariant GNS representation for the fields.

Provided a method for quantum stochastic integration over space-time.

Linked infinitely divisible states with covariant representations.

Abstract

We study homogeneous quantum L\'{e}vy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum L\'{e}vy fields on It\^{o} monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.