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White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.

Duality is established for new spaces of entire functions in two infinite dimensional variables with certain growth rates determined by Young functions. These entire functions characterize the symbols of generalized Fock space operators. As an application, a proper space is found for a solution to a normal-ordered white noise differential equation having highly singular coefficients.

The renormalized stochastic differentials of the square of white noise are defined in Boson–Fock space representation. The Itô multiplication table of these differentials and the module form of the unitarity conditions are obtained.

We prove some no-go theorems on the existence of a Fock representation of the *-Lie algebra generated by , where , b_s are the Hida white noise densities. 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.

An operator on a Fock space is considered as a non-linear and non-commutative function of annihilation and creation operators at points . The derivatives with respect to a_t and , called respectively the annihilation- and creation-derivatives, are formulated within the framework of quantum white noise theory. We prove the differentiability of an admissible white noise operator and give explicit formulae for the derivatives in terms of integral kernel operators. The qwn-derivative is a non-commutative counterpart of the Gross derivative in stochastic analysis.

We formulate a class of differential equations for white noise operators including the quantum white noise derivatives and obtain a general form of the solutions. As application we characterize intertwining operators appearing in the implementation problem for the canonical commutation relation.