The Schrödinger Fock kernel and the no-go theorem for the first order and Renormalized Square of White Noise Lie algebras
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- by Luigi Accardi and Andreas Boukas PDF
- Proc. Amer. Math. Soc. 139 (2011), 2973-2986 Request permission
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.References
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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
- 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
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2973-2986
- MSC (2010): Primary 60B15; Secondary 60H40, 17B45
- DOI: https://doi.org/10.1090/S0002-9939-2010-10716-5
- MathSciNet review: 2801637