Square of White Noise Unitary Evolutions on Boson Fock Space

  • Luigi Accardi
  • Andreas Boukas
Conference paper

Abstract

With the help of Mathematica we deduce an explicit formula for bringing to normal order the product of two normally ordered monomials in the generators of the Lie algebra of SL(2, ℝ). We use this formula to prove the Itô multiplication table for the stochastic differentials of the universal enveloping algebra of the renormalized square of white noise defined on Boson Fock space. Using this Itô table we derive unitarity conditions for processes satisfying quantum stochastic differential equations driven by this noise. From these conditions we deduce in particular that a quantum stochastic differential involving only the three basic integrators (see (2.7)–(2.9) below) and dt cannot have a unitary solution. Computer algorithms for checking these conditions, for computing the product of stochastic differentials, and for iterating the differential of the square of white noise analogue of the Poisson-Weyl operator are also provided.

Keywords

Primary 81S25 Secondary 81S05 
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Copyright information

© Springer Science+Business Media Dordrecht 2004

Authors and Affiliations

  • Luigi Accardi
    • 1
  • Andreas Boukas
    • 2
  1. 1.Centro Vito VolterraUniversitá di Roma Tor VergataRomaItaly
  2. 2.Department of MathematicsAmerican College of GreeceAthensGreece

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