The Schrodinger Fock Kernel and the no-go Theorem for the First Order and Renormalized Square of White Noise Lie Alegbras

Using the non-positive definiteness of the Fock kernel associated with the Schr¨odinger 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 δ2(t−s) = δ(s) δ(t−s) for the square of the Dirac delta function. We show how the Schr¨odinger 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 Schr¨odinger 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 Schr¨odinger kernel.