The Schrödinger Fock kernel and the no-go theorem for the first order and Renormalized Square of White Noise Lie algebras

Abstract

Abstract. Using the non-positive definiteness of the Fock kernel associated with the Schrödinger algebra we prove the impossibility of a joint Fock representation of the first order and Renormalized Square of White Noise Lie algebras with the convolution type renormalization δ 2 (t − s) = δ(s) δ(t − s) for the square of the Dirac delta function. We show how the Schrödinger algebra Fock kernel can be reduced to a positive definite kernel through a restriction of the set of exponential vectors. We describe how the reduced Schrödinger kernel can be viewed as a tensor product of a Renormalized Square of White Noise (sl (2)) and a First Order of White Noise (Heisenberg) Fock kernel. We also compute the characteristic function of a stochastic process naturally associated with the reduced Schrödinger kernel. The renormalized higher powers of white noiseThe quantum white noise functionals b † t (creation density) and b t (annihilation density) satisfy the Boson commutation relations, where t, s ∈ R and δ is the Dirac delta function, as well as the duality relations . In order to consider the smeared fields defined by the higher powers of b t and b † t , for a test function f and n, k ∈ {0, 1, 2, ...}, the sesquilinear formswhere dt denotes integration with respect to Lebesgue measure μ, with involution