Some Fock spaces with depth two action

TL;DR

This paper studies operators on a specialized Fock space acting on the first two tensor components, unifies previous constructions, and extends key results to a more general setting with matrix-valued generating functions.

Contribution

It unifies and generalizes earlier work on Fock space operators acting on the first two components, extending results to matrix-valued generating functions and broader contexts.

Findings

Derived quadratic relations for free cumulant generating functions

Obtained resolvent forms for Wick polynomial generating functions

Classified cases with tracial vacuum states

Abstract

The subject of this paper are operators represented on a Fock space which act only on the two leading components of the tensor. We unify the constructions from arXiv:math/0702158, arXiv:0709.4334, arXiv:0812.0895, and arXiv:1003.2998, and extend a number of results from these papers to our more general setting. The results include the quadratic relation satisfied by (the kernel of) the free cumulant generating function, the resolvent form of the generating function for the Wick polynomials, and classification results for the case when the vacuum state on the operator algebra is tracial. We handle the generating functions in infinitely many variables by considering their matrix-valued versions.