#### Bulg. J. Phys. vol.35 no.s1 (2008), pp. 175-189

#### Renormalized Powers of Quantum White Noise

Luigi Accardi

go back^{1}, Andreas Boukas^{2}^{1}*Centro Vito Volterra, Università di Roma Tor Vergata, via Columbia 2, 00133 Roma, Italy*^{2}*Department of Mathematics and Natural Sciences, American College of Greece, Aghia Paraskevi, Athens 15342, Greece*Abstract.Giving meaning to the powers of the creation and annihilation densities (quantum white noise) is an old and important problem in quantum field theory. In this paper we present an account of some new ideas that have recently emerged in the attempt to solve this problem. We emphasize the connection between the Lie algebra of the renormalized higher powers of quantum white noise (RHPWN), which can be interpreted as a suitably deformed (due to renormalization) current algebra over the 1-mode full oscillator algebra, and the current algebra over the centerless Virasoro (or Witt)-Zamolodchikov-wLie algebras of conformal field theory. Through a suitable definition of the action on the vacuum vector we describe how to obtain a Fock representation of all these algebras. We prove that the restriction of the vacuum to the abelian subalgebra generated by the field operators gives an infinitely divisible process whose marginal distribution is the beta (or continuous binomial)._{∞}