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

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 ) \delta ^2(t-s)=\delta (s)\,\delta (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 ( s l ( 2 ) 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.