  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 $\exp( k^+(\xi) - k^-(\xi))\exp(2 \mathrm i k^0(\theta))$ on
exponential vectors. We explain how the representations fit in with a general
scheme proposed by Araki and with representations of the $SL(2,\mathbb{R})$
current group with Vershik, Gelfand and Graev's multiplicative measure.
