Representations of the su(1,1) current algebra and probabilistic perspectives
arxiv(2024)
Abstract
We construct three representations of the su(1,1) current algebra: in
extended Fock space, with Gamma random measures, and with negative binomial
(Pascal) point processes. For the second and third representations, the
lowering and neutral operators are generators of measure-valued branching
processes (Dawson-Watanabe superprocesses) and spatial birth-death processes.
The vacuum is the constant function 1 and iterated application of raising
operators yields Laguerre and Meixner polynomials. In addition, we prove a
Baker-Campbell-Hausdorff formula and give an explicit formula for the action of
unitaries exp( k^+(ξ) - k^-(ξ))exp(2 i k^0(θ)) on
exponential vectors. We explain how the representations fit in with a general
scheme proposed by Araki and with representations of the SL(2,ℝ)
current group with Vershik, Gelfand and Graev's multiplicative measure.
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