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

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

Random variables and positive definite Kernels associates with the Schrodinger algebra

We show that the Feinsilver‐Kocik‐Schott (FKS) kernel for the Schrödinger algebra is not positive definite. We show how the FKS Schrödinger kernel can be reduced to a positive definite one through a restriction of the defining parameters of the exponential vectors. We define the Fock space associated with the reduced FKS Schrödinger kernel. We compute the characteristic functions of quantum random variables naturally associated with the FKS Schrödinger kernel and expressed in terms of the renormalized higher powers of white noise (or RHPWN) Lie algebra generators

Unitarity conditions for the renormalized square of white noise

The formal unitarity conditions for stochastic equations driven by the renormalized square of white noise are shown to hold rigorously in the framework of sesquilinear forms on the Fock space

Control of quantum stochastic differential equations

We review the basic features of the quantum stochastic calculus. Iteration schemes for the computation of the matrix elements of solutions of unitary quantum stochastic evolutions and associated quantum flows are provided along with a basic error analysis of the convergence of the iteration schemes. The application of quantum stochastic calculus to the solution of the quantum version of the quadratic cost control problem is described

On the Fock representation of the renormalized powers of quantum white noise

We describe the "no-go" theorems recently obtained by Accardi-Boukas-Franz in [\cite{1}] for the Boson case, and by Accardi-Boukas in [\cite{2}] for the $q$-deformed case, on the issue of the existence of a common Fock space representation of the renormalized powers of quantum white noise (RPWN)

The Centrally extended Heisenberg algebra and its connection with Schrodinger, Galilei and renormalized higher powers of quantum white noise Lie algebra

In previous papers we have shown that the one mode Heisenberg algebra Heis(1) admits a unique non-trivial central extensions CeHeis(1) which can be realized as a sub-Lie-algebra of the Schrödinger algebra, in fact the Galilei Lie algebra. This gives a natural family of unitary representations of CeHeis(1) and allows an explicit determination of the associated group by exponentiation. In contrast with Heis(1), the group law for CeHeis(1) is given by nonlinear (quadratic) functions of the coordinates. The vacuum characteristic and moment generating functions of the classical random variables canonically associated to CeHeis(1) are computed. The second quantization of CeHeis(1) is also considered

Unitarity conditions for stochastic differential equations driven by nonlinear quantum noise

We prove the stochastic independence of the basic integrators of the renormalized square of white noise (SWN). We use this result to deduce the unitarity conditions for stochastic differential equations driven by the SWN

Renormalized powers of quantum white noise

We prove some no-go theorems on the existence of a Fock representation of the *-Lie algebra. In particular we prove the nonexistence of such a representation for any *-Lie algebra containing . This drastic difference with the quadratic case proves the necessity of investigating different renormalization rules for the case of higher powers of white noise

Quantum Probability, Renormalization and Infinite-Dimensional *-Lie Algebras

The present paper reviews some intriguing connections which link together a new renormalization technique, the theory of *-representations of infinite dimensional *-Lie algebras, quantum probability, white noise and stochastic calculus and the theory of classical and quantum infinitely divisible processes