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


Authors: Luigi Accardi and Andreas Boukas
Journal: Proc. Amer. Math. Soc. 139 (2011), 2973-2986
MSC (2010): Primary 60B15; Secondary 60H40, 17B45
Published electronically: January 3, 2011
MathSciNet review: 2801637
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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 $ \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 ($ 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.


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Additional Information

Luigi Accardi
Affiliation: Centro Vito Volterra, Università di Roma Tor Vergata, via Columbia 2, 00133 Roma, Italy
Email: accardi@volterra.mat.uniroma2.it

Andreas Boukas
Affiliation: Department of Mathematics, The American College of Greece, Aghia Paraskevi, Athens 15342, Greece
Email: andreasboukas@acg.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10716-5
Keywords: Schrödinger algebra, positive definite kernel, Fock representation, renormalized higher powers of white noise
Received by editor(s): April 5, 2010
Received by editor(s) in revised form: July 12, 2010, and July 22, 2010
Published electronically: January 3, 2011
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2011 American Mathematical Society