Fock representation of the renormalized higher powers of white noise and the Virasoro--Zamolodchikov--$w_{\infty} *$--Lie algebra

TL;DR

This paper establishes a Fock representation for the truncated renormalized higher powers of white noise and the Virasoro--Zamolodchikov--$w_{ty}$ algebra, revealing their connection to binomial and beta processes.

Contribution

It introduces a positive definite truncation of the RHPWN Fock kernels and constructs a Fock space representation linking these algebras to stochastic processes.

Findings

Truncated RHPWN Fock spaces are positive definite for orders ≥ 2.

The Fock spaces host continuous binomial and beta processes.

A new Fock representation of the algebras is established.

Abstract

The identification of the $*$--Lie algebra of the renormalized higher powers of white noise (RHPWN) and the analytic continuation of the second quantized Virasoro--Zamolodchikov--$w_{\infty} *$--Lie algebra of conformal field theory and high-energy physics, was recently established in \cite{id} based on results obtained in [1] and [2]. In the present paper we show how the RHPWN Fock kernels must be truncated in order to be positive definite and we obtain a Fock representation of the two algebras. We show that the truncated renormalized higher powers of white noise (TRHPWN) Fock spaces of order $\geq 2$ host the continuous binomial and beta processes.