• Representations of homogeneous quantum Lévy fields

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      https://www.ias.ac.in/article/fulltext/pmsc/116/04/0477-0488

    • Keywords

       

      Representations; quantum Lévy process; independent increments; symmetry; infinite divisibility; monoid; generating functional; cocycle; conditionally positive definite

    • Abstract

       

      We study homogeneous quantum Lévy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum Lévy fields on Itô monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set.

    • Author Affiliations

       

      V P Belavkin1 L Gregory1

      1. School of Mathematical Sciences, University of Nottingham, Nottingham - NG7 2RD, UK
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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