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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10716-5

Published electronically:
January 3, 2011

MathSciNet review:
2801637

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Abstract | References | Similar Articles | Additional Information

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 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 () 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