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Representations of the $su(1,1)$ current algebra and probabilistic perspectives

Simone Floreani, Sabine Jansen, Stefan Wagner

TL;DR

The paper develops three novel representations of the $su(1,1)$ current algebra: in an extended Fock space, with Gamma random measures, and with Pascal point processes. Each representation yields a concrete realization of the current generators $k^+(\varphi)$, $k^-(\varphi)$, and $k^0(\varphi)$, leading to Laguerre and Meixner polynomial structures via the vacuum and raising operators, and enabling a full Baker–Campbell–Hausdorff formula and explicit unitary actions on exponential vectors. The Gamma and Pascal constructions connect to continuous-time Markov processes (Dawson–Watanabe superprocesses and birth–death dynamics) and provide probabilistic interpretations for the operator framework. The univariate reductions recover well-known $SU(1,1)$ discrete-series representations and reveal explicit differential and difference operator realizations tied to Laguerre and Meixner polynomials. Altogether, the work situates these algebraic representations within Araki’s scheme and the $SL(2,\mathbb{R})$ current-group literature, augmenting the bridge between operator algebras and probabilistic structures.

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

Representations of the $su(1,1)$ current algebra and probabilistic perspectives

TL;DR

The paper develops three novel representations of the current algebra: in an extended Fock space, with Gamma random measures, and with Pascal point processes. Each representation yields a concrete realization of the current generators , , and , leading to Laguerre and Meixner polynomial structures via the vacuum and raising operators, and enabling a full Baker–Campbell–Hausdorff formula and explicit unitary actions on exponential vectors. The Gamma and Pascal constructions connect to continuous-time Markov processes (Dawson–Watanabe superprocesses and birth–death dynamics) and provide probabilistic interpretations for the operator framework. The univariate reductions recover well-known discrete-series representations and reveal explicit differential and difference operator realizations tied to Laguerre and Meixner polynomials. Altogether, the work situates these algebraic representations within Araki’s scheme and the current-group literature, augmenting the bridge between operator algebras and probabilistic structures.

Abstract

We construct three representations of the 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 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 on exponential vectors. We explain how the representations fit in with a general scheme proposed by Araki and with representations of the current group with Vershik, Gelfand and Graev's multiplicative measure.
Paper Structure (31 sections, 21 theorems, 172 equations)

This paper contains 31 sections, 21 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The group SU(1,1)
  4. Current algebra
  5. Canonical commutation relations
  6. Gaussian white noise, Hermite polynomials
  7. Poisson point process. Charlier polynomials
  8. From Gauss and Poisson to Gamma and Pascal laws
  9. Continuous-time Markov processes
  10. Main results
  11. Representation in extended Fock space
  12. Extended Fock space
  13. Exponential vectors
  14. Representation of the $su(1,1)$ current algebra
  15. Unitary operators, Baker-Campbell-Hausdorff formula
  16. ...and 16 more sections

Key Result

Lemma 3.1

The space $\mathcal{D}$ of finite linear combinations of the vacuum $\Psi$ and symmetrized tensor products $f_1\otimes_\mathrm s\cdots \otimes_\mathrm s f_n$ of functions $f_i\in \mathcal{C}$ is dense in $\mathfrak F$.

Theorems & Definitions (42)

  • Definition 2.1
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4: Baker-Campbell-Hausdorff formula
  • Remark
  • Corollary 3.5: Vacuum expectations
  • Lemma 3.6
  • Remark
  • Theorem 3.7
  • ...and 32 more