UNITARY REPRESENTATIONS OF THE WITT AND sl(2, ℝ)-ALGEBRAS THROUGH RENORMALIZED POWERS OF THE QUANTUM PASCAL WHITE NOISE

Abstract

By using an appropriate space of distributions, [Formula: see text], we derive the chaos decomposition property of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure ΛNB. The constructed decomposition is used to define a nuclear triple [Formula: see text] of test and generalized functions, where θ is a Young function satisfying some suitable conditions. A general characterization theorems are proven for the Pascal white noise distributions, white noise test functions and white noise operators in terms of analytical functions with growth condition of exponential type. By using appropriate renormalization procedure, we obtain the representation of the square of white noise obtained by Accardi–Franz–Skeide in Ref. 5. Finally, we investigate the main aim of this paper which is to give unitary equivalent representations of the Witt algebra in the basis of Pascal white noise theory.