THE SCHRODINGER FOCK KERNEL AND THE NO-GOTHEOREM FOR THE FIRST ORDER AND RENORMALIZEDSQUARE OF WHITE NOISE LIE ALGEBRAS

摘要

Using the non-positive definiteness of the Fock kernel associated with the Schrodinger algebra we prove the impossibility of a joint Fock rep-resentation of the first order and Renormalized Square of White Noise Lie algebras with the convolution type renormalization δ~2(t — s) = 5(s) 6(t — s) for the square of the Dirac delta function. We show how the Schrodinger 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 Schrodinger 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 Schrodinger kernel.