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Square of White Noise Unitary Evolutions on Boson Fock Space

In: Proceedings of the International Conference on Stochastic Analysis and Applications

Author

Listed:
  • Luigi Accardi

    (Universitá di Roma Tor Vergata, Centro Vito Volterra)

  • Andreas Boukas

    (American College of Greece, Department of Mathematics)

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.

Suggested Citation

  • Luigi Accardi & Andreas Boukas, 2004. "Square of White Noise Unitary Evolutions on Boson Fock Space," Springer Books, in: Sergio Albeverio & Anne Boutet de Monvel & Habib Ouerdiane (ed.), Proceedings of the International Conference on Stochastic Analysis and Applications, pages 267-301, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4020-2468-9_17
    DOI: 10.1007/978-1-4020-2468-9_17
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