Discrete approximation of the free Fock space

TL;DR

This paper demonstrates that the free Fock space can be approximated by a discrete free product called the free toy Fock space, enabling discrete analysis of free probability processes.

Contribution

It introduces an explicit discrete approximation of the free Fock space using the free toy Fock space, connecting continuous and discrete free probability frameworks.

Findings

The free Fock space is the continuous free product of copies of C^2.

Operators in the free Fock space are limits of elementary operators on the free toy Fock space.

Discrete approximations of free stochastic processes like semi-circular Brownian motion are recovered.

Abstract

We prove that the free Fock space ${\F}(\R^+;\C)$, which is very commonly used in Free Probability Theory, is the continuous free product of copies of the space $\C^2$. We describe an explicit embedding and approximation of this continuous free product structure by means of a discrete-time approximation: the free toy Fock space, a countable free product of copies of $\C^2$. We show that the basic creation, annihilation and gauge operators of the free Fock space are also limits of elementary operators on the free toy Fock space. When applying these constructions and results to the probabilistic interpretations of these spaces, we recover some discrete approximations of the semi-circular Brownian motion and of the free Poisson process. All these results are also extended to the higher multiplicity case, that is, ${\F}(\R^+;\C^N)$ is the continuous free product of copies of the space $\C^{N+1}$.