The Schrodinger Fock Kernel and the no-goudTheorem for the First Order and Renormalized Square of udWhite Noise Lie Alegbras

摘要

Using the non-positive definiteness of the Fock kernel associatedudwith the Schr¨odinger algebra we prove the impossibility of a joint Fock representationudof the first order and Renormalized Square of White Noise Lieudalgebras with the convolution type renormalization δ2(t−s) = δ(s) δ(t−s) forudthe square of the Dirac delta function. We show how the Schr¨odinger algebraudFock kernel can be reduced to a positive definite kernel through a restriction ofudthe set of exponential vectors. We describe how the reduced Schr¨odinger kerneludcan be viewed as a tensor product of a Renormalized Square of White Noiseud(sl(2)) and a First Order of White Noise (Heisenberg) Fock kernel. We alsoudcompute the characteristic function of a stochastic process naturally associatedudwith the reduced Schr¨odinger kernel.