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

Simone Floreani  Universität Bonn, Institut für Angewandte Mathematik, Abt.
Stochastische Systeme mit Wechselwirkung, Endenicher Allee 60, 53115 Bonn,
Germany<EMAIL_ADDRESS>,  Sabine Jansen  Mathematisches Institut,
Ludwig-Maximilians-Universität, 80333 München; Munich Center for Quantum
Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany.
<EMAIL_ADDRESS>and  Stefan Wagner  Mathematisches Institut, Ludwig-
Maximilians-Universität, 80333 München; Munich Center for Quantum Science and
Technology (MCQST), Schellingstr. 4, 80799 München, Germany.
<EMAIL_ADDRESS>
(Date: 12 February 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 $\exp(k^{+}(\xi)-k^{-}(\xi))\exp(2\mathrm{i}k^{0}(\theta))$ 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.

_MSC2020:_ 81R10; 33C45; 60K35; 60H40

_Keywords_ : current algebras; infinite-dimensional Lie algebras; orthogonal
polynomials; Gamma random measure; negative binomial (Pascal) point processes;
Markov processes

##  1\. Introduction

The generators for many interacting particle systems can be written with
creation or annihilation operators, or raising and lowering operators. This
holds true not only for bosons, fermions, or spin chains in quantum mechanics,
but also for classical systems that evolve according to some continuous-time
Markov process; this helps uncover useful hidden symmetries [50]. Sometimes
the generators of two different Markov semigroup ($P_{t}=\exp(tL)$, $t\geq 0$)
have the _same_ formal expression with raising and lowering operators; the
difference comes from different representations of the raising and lowering
operators and their commutation relations. These observations lead to the
algebraic approach to duality [15, 23, 28, 55].

A concrete example is the following model for energy transport, see Giardinà,
Kurchan, Redig and Vafayi [23] and references therein. It is closely related
to a model by Kipnis, Marchioro and Presutti [31]. The simplest version of the
_Brownian momentum process_ on a finite 1D lattice $\\{1,2,\ldots,\ell\\}$ is
a continuous-time Markov process with state space $\mathbb{R}^{\ell}$ and
formal generator

$L=\sum_{i,j}\Bigl{(}p_{i}\frac{\partial}{\partial
p_{j}}-p_{j}\frac{\partial}{\partial p_{i}}\Bigr{)}^{2}.$

The sum is over nearest-neighbor pairs $\\{i,j\\}$, say with periodic boundary
conditions. (More general versions allow for open systems interacting with
reservoirs at the boundaries and for vector-valued momenta.) The
interpretation is that at each site $i$, sits a particle with momentum
$p_{i}$; two neighboring sites may exchange momentum but the total kinetic
energy $\frac{1}{2}\sum_{i}p_{i}^{2}$ is conserved. The _symmetric inclusion
process_ instead describes particles hopping on the 1D lattice. The state
space is $\mathbb{N}_{0}^{\ell}$ and the formal generator is

$\bigl{(}\tilde{L}f\bigr{)}(\boldsymbol{n})=\sum_{i,j}n_{i}\bigl{(}n_{j}+\frac{1}{2}\bigr{)}\Bigl{(}f(n_{1},\ldots,n_{i}-1,\ldots,n_{j}+1,\ldots,n_{\ell})-f(n_{1},\ldots,n_{\ell})\Bigr{)}.$

Both $L$ and $\tilde{L}$ can be written as

$\sum_{i,j}\Bigl{(}k_{i}^{+}k_{j}^{-}+k_{j}^{+}k_{i}^{-}-2k_{i}^{0}k_{j}^{0}+\frac{1}{8}\Bigr{)}$
(1)

with raising, lowering and neutral operators $k_{i}^{\pm}$, $k_{j}^{0}$ that
satisfy the $su(1,1)$ commutation relations,

$[k_{i}^{-},k_{i}^{+}]=2k_{i}^{0},\quad[k_{i}^{0},k_{i}^{\pm}]=\pm
k_{i}^{\pm}.$

Operators associated with different lattice sites commute. For the Brownian
energy process,

$k_{i}^{+}=\frac{1}{2}p_{i}^{2},\quad
k_{i}^{-}=\frac{1}{2}\frac{\partial^{2}}{\partial p_{i}^{2}},\quad
k_{i}^{0}f=\frac{1}{4}\Bigl{(}\frac{\partial}{\partial
p_{i}}p_{i}f+p_{i}\frac{\partial}{\partial p_{i}}f\Bigr{)}$

For the symmetric inclusion process,

$\displaystyle k_{i}^{+}f(\boldsymbol{n})$
$\displaystyle=n_{i}f(n_{1},\ldots,n_{i-1},n_{i}-1,n_{i+1},\ldots,n_{\ell}),$
$\displaystyle k_{i}^{-}f(\boldsymbol{n})$
$\displaystyle=\Bigl{(}\frac{1}{2}+n_{i}\Bigr{)}f(n_{1},\ldots,n_{i-1},n_{i}+1,n_{i+1},\ldots,n_{\ell}),$
$\displaystyle k_{i}^{0}f(\boldsymbol{n})$
$\displaystyle=\Bigl{(}n_{i}+\frac{1}{4}\Bigr{)}f(\boldsymbol{n}).$

Notice that the lowering operator $k_{i}^{-}$ annihilates the indicator
$\mathbbm{1}_{\\{\boldsymbol{n}=0\\}}$ that there is no particle at all. The
common algebraic formula (1) helps explain dualities between the two processes
[23]. In turn, duality helps investigate transport phenomena [22, 31].

How can we extend the algebraic formalism to systems on $\mathbb{R}^{d}$?
Infinite-dimensional Lie algebras and current algebras provide the proper
algebraic framework; concretely, we look for representations of the $su(1,1)$
current algebra. Those keywords may suggest a daunting background from quantum
field theory, algebra or geometry [20, 29], but the definition we shall adopt
(Definition 2.1), loosely following Araki [5] and Accardi and Boukas [1] is a
simple variant of the familiar canonical commutation relations for bosons. In
a previous article [19], we have already introduced a family of operators that
form a representation of the $su(1,1)$ current algebra, and we have showed how
to deduce intertwining relations for consistent Markov processes. Here we
continue the analysis and place a stronger emphasis on the algebraic
structure.

We give three representations of the $su(1,1)$ current algebra. The first
representation is similar to the representation in Fock space of the canonical
commutation relations from quantum mechanics (Theorem 3.2). The other two
representations are closer to Gaussian models for the Euclidean free fields,
in which Hermite polynomials play a distinguished role [24]. The Hilbert
spaces are $L^{2}$-spaces with respect to some probability measure, the vacuum
is the constant function $1$, and iterated application of raising operators to
the vacuum gives rise to orthogonal polynomials. Gaussian fields are replaced
with Gamma random measures and negative binomial point processes (also called
Pascal point processes). Hermite polynomials are replaced with Laguerre and
Meixner polynomials (Theorem 3.7 and 3.9).

In addition, we prove an infinite-dimensional version of the Baker-Campbell-
Hausdorff formula (Theorem 3.4) and we make explicit the action of a familiy
of unitaries, analogous to the Weyl operators
$\exp(\mathrm{i}(c(f)+c^{\dagger}(f))$ from quantum field theory, on
exponential vectors (Theorem 3.3). The theorem should yield a representation
of the current group for the universal cover of $SU(1,1)$, but we refrain from
making this explicit.

To the best of our knowledge, the three representations are new, however the
ingredients that we use are not and there are closely related results in the
literature. The first representation is similar to Boukas’s representation of
Feinsilver’s finite-difference algebra [12] and Śniady’s representation of
commutation relations for quadratic white noise [52]. All these algebras are
related: Accardi and Skeide [4] linked the representations of the finite-
difference algebra and the renormalized square of white noise, which in turn
is related to to representations of the current algebra of $sl(2,\mathbb{R})$
[3]; finally, as is well known, the groups $SL(2,\mathbb{R})$ and $SU(1,1)$
are isomorphic and so are their Lie algebras.

The extended Fock space used in our first representation also appears in chaos
decompositions in non-Gaussian white noise analysis, see Kondratiev, da Silva,
Streit, and Us [33], Berezansky and Mierzejewski [10], Kondratiev and Lytvynov
[34]. Gamma random measures and negative binomial (Pascal) point processes,
together with raising, lowering, and neutral operators and infinite-
dimensional Meixner and Laguerre polynomials have been intensely analyzed, see
Lytvynov [38] and references therein. Lytvynov also made the connection with
the algebra of the square of the white noise [40]. For an algebraic stance on
Pascal white noise, see Barhoumi, Ouerdiane and Riahi [8].

A slightly different series of articles goes back to Gelfand, Graev and
Vershik [21]. Araki [5] proposed a general scheme for finding factorizable
representations of current groups in Fock spaces—but did not implement it for
$SU(1,1)$ or $SL(2,\mathbb{R})$. Vershik, Gelfand and Graev however did
investigate those groups [60]. For a full account and further references we
refer the reader to Graev and Vershik [58]. We only mention this: A key role
is played by a kind of generalized infinite-dimensional Lebesgue measure,
defined as a specific $\sigma$-finite measure that is absolutely continuous
with respect to the law of a Gamma random measure. Yet another representation
involves instead “quasi-Poisson spaces” [59]. These representations are
different from ours, but in Appendices A and B we explain how our
representations relate to Araki’s scheme and to the infinite-dimensional
Lebesgue measure.

All of the above-mentioned references have a strongly analytic flavor to them.
This makes them hard to access for readers who may not wish to learn about
distribution theory, nuclear spaces, rigged Hilbert spaces, or white noise
analysis. Therefore we have strived to make the setup as simple as possible.
We keep topological assumptions to a minimum and we allow for non-smooth test
functions and reference measures with atoms.

Another distinctive feature of our presentation is the emphasis on
probabilistic interpretations of operators. It turns out that the lowering and
neutral operators in our second and third representations generate continuous-
time Markov processes that are of considerable intrinsic interest: measure-
valued branching processes (Dawson-Watanabe superprocesses) and spatial birth-
death processes. This observation, to the best of our knowledge, is new, not
only for the current algebra but also for the Lie algebra where the associated
processes are diffusions (Laguerre and Bessel) and linear birth-death
processes.

The article is organized as follows. First we recall some basics about the
group $SU(1,1)$, define the kind of representations that we shall study, and
provide some context for our results (Section 2). Then we state our main
results on representations of the $su(1,1)$ current algebra (Section 3). The
simpler representations of the $su(1,1)$ Lie algebra are given in Section 4.
Finally we turn to the proofs of our main results (Section 5). In Appendices A
and B we explain how our approach fits into Araki’s scheme [5] and to the
representation of the $SL(2,\mathbb{R})$ current group with generalized
Lebesgue measure sketched in Tsilevich, Vershik and Yor [57].

##  2\. Preliminaries

Here we gather some relevant background, most importantly, we define the
representations of $su(1,1)$ current algebras that we investigate in this
article (Definition 2.1). In addition, we provide some background that aids
the understanding of our main results: basics on the group $SU(1,1)$;
representations of the canonical commutation relations with Hermite and
Charlier polynomials; how Gamma and and negative binomial (Pascal) laws
naturally appear; and why we may expect relations with continuous-time Markov
processes.

###  2.1. The group $SU(1,1)$

The group $SU(1,1)$ consists of the complex $2\times 2$ matrices

$A=\begin{pmatrix}a&b\\\ \overline{b}&\overline{a}\end{pmatrix}$ (2)

with $|a|^{2}-|b|^{2}=1$. Every such matrix can be written as

$A(\xi,\theta)=\exp\bigl{(}\xi
k^{+}-\overline{\xi}k^{-})\exp(2\mathrm{i}\theta k^{0})$ (3)

where $\xi\in\mathbb{C}$, $\theta\in\mathbb{R}$, and

$k^{+}=\begin{pmatrix}0&\mathrm{i}\\\ 0&0\end{pmatrix},\quad
k^{-}=\begin{pmatrix}0&0\\\ \mathrm{i}&0\end{pmatrix},\quad
k^{0}=\begin{pmatrix}1/2&0\\\ 0&-1/2\end{pmatrix}.$

The matrices satisfy the commutation relations

$[k^{-},k^{+}]=2k^{0},\quad[k^{0},k^{\pm}]=\pm k^{\pm}.$ (4)

The $su(1,1)$ algebra consists of the complex linear combinations of the
matrices $k^{0},k^{+},k^{-}$. The exponential in (3) is easily computed, one
has

$A(\xi,\theta)=\begin{pmatrix}\cosh|\xi|&\mathrm{i}\frac{\xi}{|\xi|}\sinh|\xi|\\\
-\mathrm{i}\frac{\overline{\xi}}{|\xi|}\sinh|\xi|&\cosh|\xi|\end{pmatrix}\begin{pmatrix}\mathrm{e}^{\mathrm{i}\theta}&0\\\
0&\mathrm{e}^{-\mathrm{i}\theta}\end{pmatrix}.$ (5)

Bargmann [7] investigated representations of $SU(1,1)$ in the context of the
Lorentz and Poincaré groups from special relativity. The $su(1,1)$ algebra
also appears in quantum optics for pairs of photons: If $a$ and $a^{\dagger}$
are bosonic annihilation and creation operators, then
$k^{-}=\frac{1}{2}a^{2}$, $k^{+}=\frac{1}{2}(a^{\dagger})^{2}$ and
$k^{0}=\frac{1}{2}a^{\dagger}a+\frac{1}{4}$ satisfy the $su(1,1)$ algebra [14,
16]. The operator $\exp(\frac{1}{2}(\xi
a^{2}-\overline{\xi}(a^{\dagger})^{2}))$ is the _squeeze operator_ [51,
Chapter 2.7].

###  2.2. Current algebra

As is customary with bosonic creation and annihilation operators, we work with
operators indexed by functions. Let $(\mathbb{X},\mathcal{X})$ be a measurable
space and $\alpha$ be a $\sigma$-finite reference measure on $\mathbb{X}$; for
example, $\mathbb{R}^{d}$ with the Borel $\sigma$-algebra and the Lebesgue
measure. We introduce an algebra of test functions

$\mathcal{C}=\bigl{\\{}\varphi:\mathbb{X}\to\mathbb{C}:\,\text{ bounded,
measurable, }\alpha(\\{x:\varphi(x)\neq 0\\})<\infty\bigr{\\}}.$ (6)

The following definition is modeled after a similar definition in Accardi and
Boukas [2] for the algebra of the renormalized square of white noise. Araki
[5] defines the $su(1,1)$ current algebra over $\mathbb{X}$ as the set of
$su(1,1)$-valued, bounded measurable functions, equipped with the pointwise
Lie algebra operations. Our function-indexed operators
$k^{\pm}(\varphi),k^{0}(\theta)$ represent the matrix-valued maps
$x\mapsto\varphi(x)k^{+}$, $x\mapsto\overline{\varphi(x)}k^{-}$,
$x\mapsto\theta(x)k^{0}$ with $k^{\pm},k^{0}$ the $2\times 2$-matrices from
Section 2.1.

######  Definition 2.1.

A family $k^{\pm}(\varphi),k^{0}(\varphi)$, $\varphi\in\mathcal{C}$ of
operators in a Hilbert space $\mathcal{H}$ is a _Fock representation of the
$su(1,1)$ current algebra_ with _vacuum_ $\Psi$ if:

  1. (i)

The operators are defined on a common domain $\mathcal{D}$ that is dense in
$\mathcal{H}$ and they map $\mathcal{D}$ to itself, i.e.,
$k^{\\#}(\varphi)\mathcal{D}\subset\mathcal{D}$.

  2. (ii)

$k^{+}(\varphi)$ and $k^{0}(\varphi)$ are linear in $\varphi$ while
$k^{-}(\varphi)$ is antilinear in $\varphi$.

  3. (iii)

For all $\varphi,\theta\in\mathcal{C}$,

$\begin{gathered}[k^{-}(\varphi),k^{+}(\theta)]=2k^{0}(\overline{\varphi}\theta)\\\
[k^{0}(\varphi),k^{+}(\theta)]=k^{+}(\varphi\theta),\quad[k^{0}(\varphi),k^{-}(\theta)]=-k^{-}(\overline{\varphi}\theta)\end{gathered}$
(7)

and $[k^{\\#}(\varphi),k^{\\#}(\theta)]=0$ for $\\#\in\\{0,+,-\\}$.

  4. (iv)

For all $f,g\in\mathcal{D}$ and $\varphi\in\mathcal{C}$,

$\displaystyle\langle f,k^{0}(\varphi)g\rangle=\langle
k^{0}(\overline{\varphi})f,g\rangle,\quad\langle
f,k^{+}(\varphi)g\rangle=\langle k^{-}(\varphi)f,g\rangle.$ (8)

  5. (v)

$\Psi\in\mathcal{D}$, $||\Psi||=1$ and $k^{-}(\varphi)\Psi=0$ for all
$\varphi\in\mathcal{C}$.

  6. (vi)

$\Psi$ is cyclic: the linear combinations of $\Psi$ and
$k^{+}(\varphi_{1})\cdots k^{+}(\varphi_{n})\Psi$ where $n\in\mathbb{N}$ and
$\varphi_{1},\ldots,\varphi_{n}\in\mathcal{C}$, are dense in $\mathfrak{F}$.

All our representations are _factorizable_ —when test functions have disjoint
support, the associated operators commute. In addition, our three
representations from Section 3 satisfy

$k^{0}(\varphi)\Psi=\Bigl{(}\frac{1}{2}\int\varphi\mathrm{d}\alpha\Bigr{)}\Psi$

for all $\varphi\in\mathcal{C}$. Thus, the measure $\frac{1}{2}\alpha$ plays a
role analogous to the Bargmann index for the discrete series, minimal weight
representations of $SU(1,1)$ (see Section 4).

We remark that the three representations are unitarily equivalent. More
precisely, the isomorphism from [33, Equation (4.9)] associated with the chaos
decomposition of functionals of a Gamma random measure switches between the
representation in extended Fock spaces and the representation outlined in
Section 3.3. Similarly, the isomorphism from [38, Corollary 5.3], which
relates to the chaos decomposition of functionals associated with a Pascal
point process, switches between the representation in extended Fock spaces and
the representation introduced in Section 3.4.

###  2.3. Canonical commutation relations

The raising and lowering operators $k^{+}(f)$, $k^{-}(f)$ are similar to
bosonic creation and annihilation operators $c^{\dagger}(f)$, $c(g)$ with the
canonical commutation relations $[c(f),c^{\dagger}(g)]=\langle f,g\rangle$.
The best known representation, in quantum many-body theory, is in the Fock
space of sequences of symmetric functions $(f_{n})_{n\in\mathbb{N}_{0}}$. For
better comparison with our results, we recall two additional representations.
To keep things concrete in this section $\mathbb{X}$ is $\mathbb{R}^{d}$
equipped with its Borel $\sigma$-algebra and with the Lebesgue measure
$\lambda$. The scalar product in $L^{2}(\mathbb{X},\lambda)$ is $\langle
f,g\rangle=\int\overline{f}\,g\,\mathrm{d}\lambda$.

####  2.3.1. Gaussian white noise, Hermite polynomials

The setting of the first additional representation is similar to that of the
Euclidean quantum free field, we refer to Glimm and Jaffe [24] for background
and proofs. By the Bochner-Minlos theorem, there is a unique probability
measure $\mu$ on $\mathcal{S}^{\prime}_{\mathbb{R}}$, the space of real-valued
tempered distributions on $\mathbb{R}^{d}$, with

$\int\mathrm{e}^{\mathrm{i}\varphi(f)}\mu(\mathrm{d}\varphi)=\exp\Bigl{(}-\frac{1}{2}\int
f(x)^{2}\mathrm{d}x\Bigr{)}.$

for all real-valued Schwartz functions $f$. For all $n\in\mathbb{N}$ and real-
valued Schwartz functions $f_{1},\ldots,f_{n}$, the vector
$(\varphi(f_{1}),\ldots,\varphi(f_{n}))$ is a Gaussian random vector with
covariance matrix $C=(\int f_{i}g_{j}\mathrm{d}\lambda)_{i,j=1,\ldots,n}$. In
particular

  * •

If $f_{1},\ldots,f_{n}$ have disjoint support then $\varphi(f_{1})$,…,
$\varphi(f_{n})$ are independent.

  * •

Each variable $\varphi(f)$ has a normal distribution with mean zero and
variance $\int f^{2}\mathrm{d}\lambda$.

The Hilbert space is
$L^{2}(\mathcal{S}^{\prime}_{\mathbb{R}}(\mathbb{R}^{d}),\mu)$, the space of
complex-valued square-integrable functions. Let $\mathcal{D}$ be the set of
functions $F:\mathcal{S}^{\prime}_{\mathbb{R}}\to\mathbb{C}$ of the form

$F(\varphi)=P_{n}\bigl{(}\varphi(f_{1}),\ldots,\varphi(f_{n})\bigr{)}$

with $n\in\mathbb{N}$, $f_{1},\ldots,f_{n}$ Schwartz functions, and
$P_{n}:\mathbb{C}^{n}\to\mathbb{C}$ a multivariate polynomial. The
annihilation operators are

$c(f)F(\varphi)=\int\overline{f(x)}\frac{\delta}{\delta\varphi(x)}F(\varphi)\mathrm{d}x=\sum_{j=1}^{n}\langle
f,f_{j}\rangle\frac{\partial P_{n}}{\partial
x_{j}}\bigl{(}\varphi(f_{1}),\ldots,\varphi(f_{n})\bigr{)}$

and the creation operators are

$\displaystyle c^{\dagger}(g)F(\varphi)$
$\displaystyle=\varphi(g)F(\varphi)-\int
g(x)\frac{\delta}{\delta\varphi(x)}F(\varphi)\mathrm{d}x$
$\displaystyle=\varphi(g)F(\varphi)-\sum_{j=1}^{n}\langle
f_{j},g\rangle\frac{\partial P_{n}}{\partial
x_{j}}\bigl{(}\varphi(f_{1}),\ldots,\varphi(f_{n})\bigr{)}.$

The vacuum is the constant function $\mathbbm{1}$. Iterated application of
creation operators to the vacuum gives rise to Hermite polynomials [24].

####  2.3.2. Poisson point process. Charlier polynomials

The second additional representation comes from Poisson white noise analysis
(Kondratiev, da Silva, Streit, Us [33] and references therein). It is similar
to the previous representation except that the Gaussian measure $\mu$ is
replaced by the law of a Poisson point process, supported on a simpler space
of point configurations—there is no need for tempered distributions.

Let $\mathbf{N}$ be the set of finite or countable sums of Dirac measures,
equipped with the $\sigma$-algebra generated by the counting variables
$\eta\mapsto\eta(B)$, $B\in\mathcal{X}$. The law $\pi_{\lambda}$ of the
_Poisson point process with intensity measure $\lambda$_ is the uniquely
defined probability measure on $\mathbf{N}$ under which

  * •

Each counting variable $\eta\mapsto\eta(B)$, $B\in\mathcal{X}$, has a Poisson
distribution with parameter $\lambda(B)$

$\pi_{\lambda}\bigl{(}\\{\eta:\eta(B)=n\\}\bigr{)}=\mathrm{e}^{-\lambda(B)}\frac{\lambda(B)^{n}}{n!}.$

  * •

For all disjoint $B_{1},\ldots,B_{n}\in\mathcal{X}$, the associated counting
variables are independent.

Mathematical physicists should think of $\pi_{\lambda}$ as the grand-canonical
Gibbs measure for a classical ideal gas in which the expected number of
particles in a region $B$ is $\lambda(B)$. When $\lambda$ is Lebesgue measure,
the ideal gas is homogeneous with density $1$.

We work in the Hilbert space $L^{2}(\mathbf{N},\pi_{\lambda})$. Abbreviate
$\eta(f)=\int f\mathrm{d}\eta$ and let $\mathcal{D}$ be the space of
polynomials in the variables $\eta(f)=\int f\mathrm{d}\eta$. The annihilation
and creation operators are [33]

$\displaystyle c(f)F(\eta)$
$\displaystyle=\int\overline{f(x)}\bigl{(}F(\eta+\delta_{x})-F(\eta)\bigr{)}\lambda(\mathrm{d}x),$
$\displaystyle c^{\dagger}(f)F(\eta)$ $\displaystyle=\int
f(x)F(\eta-\delta_{x})\eta(\mathrm{d}x)-F(\eta)\int f\mathrm{d}\lambda.$

The vacuum $\Psi$ is again the constant function $1$, and iterated application
of creation operators for indicators $f$ gives rise to _Charlier polynomials_
in the occupation numbers $\eta(B)$, $B\subset\mathbb{X}$. For example, for a
finite-volume set $B\subset\mathbb{R}^{d}$,

$\bigl{(}c^{\dagger}(\mathbbm{1}_{B})\bigr{)}^{n}\mathbf{1}(\eta)=\mathscr{C}_{n}\bigl{(}\eta(B);\lambda(B)\bigr{)}$

where $\mathscr{C}_{n}(x;\beta)=x^{n}+\text{lower order terms}$ is a
univariate Charlier polynomial, orthogonal with respect to the measure
$\nu(\\{x\\})=\mathrm{e}^{-\beta}\beta^{x}/x!$ on $\mathbb{N}_{0}$ [32].

###  2.4. From Gauss and Poisson to Gamma and Pascal laws

Our second and third representations of the $su(1,1)$ current algebra rely on
Gamma and Pascal laws. Here we briefly motivate how these distributions arise,
building on results from quantum stochastic calculus. Let $a^{\dagger}$ and
$a$ be the creation and annihilation operators for the harmonic oscillator in
some Hilbert space $\mathcal{H}$ with vacuum $\psi_{0}$, for example,
$a=\frac{\mathrm{d}}{\mathrm{d}x}$ and
$a^{\dagger}=x-\frac{\mathrm{d}}{\mathrm{d}x}$ in
$L^{2}(\mathbb{R},\exp(-x^{2}/2)\mathrm{d}x)$, see the next subsection. Write
$\langle f(A)\rangle$ for the vacuum expectation values
$\langle\psi_{0},f(A)\psi_{0}\rangle$ of polynomial functions $f$ of an
operator $A$. The following is known:

  1. (i)

The vacuum expectations of $a+a^{\dagger}$ are Gaussian:

$\langle
f(a+a^{\dagger})\rangle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(z)\exp(-z^{2}/2)\mathrm{d}z$

for every polynomial $f:\mathbb{R}\to\mathbb{R}$ (just think of the harmonic
oscillator).

  2. (ii)

$a^{\dagger}+a+ca^{\dagger}a$, where $c\in\mathbb{R}\setminus\\{0\\}$, is
associated with a Poisson law of parameter $\lambda=1/c^{2}$:

$\langle
f(a+a^{\dagger}+ca^{\dagger}a)\rangle=\mathrm{e}^{-\lambda}\sum_{n=0}^{\infty}\frac{\lambda^{n}}{n!}f\bigl{(}c^{-1}(n-\lambda)\bigr{)}.$

See Hudson and Parthasarathy [26] and Meyer [41, Chapter IV.2(5)]. A related
result is given by Ito and Kubo [27, Proposition 5.2].

  3. (iii)

$\frac{1}{2}(a+a^{\dagger})^{2}$ goes with a Gamma distribution:

$\langle
f\Bigl{(}\frac{1}{2}(a+a^{\dagger})^{2}\Bigr{)}\rangle=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}f(x)x^{-1/2}\mathrm{e}^{-x}\mathrm{d}x.$

This follows from (i)—think $x=z^{2}/2$.

  4. (iv)

$a^{2}+(a^{\dagger})^{2}+\beta a^{\dagger}a$ with $\beta>2$ goes with negative
binomial (Pascal) distribution with parameters $\alpha=1/2$ and $p\in(0,1)$
the unique solution of $\sqrt{p}+{\sqrt{p}}^{-1}=\beta$: there exist non-zero
reals $c_{1},c_{2}$ such that

$\langle f\bigl{(}a^{2}+(a^{\dagger})^{2}+\beta
a^{\dagger}a\bigr{)}\rangle=(1-p)^{\alpha}\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}\,p^{n}f\bigl{(}c_{1}n+c_{2}\bigr{)}$

where $(\alpha)_{0}=1$, $(\alpha)_{n}=\alpha(\alpha+1)\cdots(\alpha+n-1)$. See
Accardi, Franz and Skeide [3] and the survey by Accardi and Boukas [1, Section
17]. A similar result holds for $\beta<-2$.

  5. (v)

$a^{2}+(a^{\dagger})^{2}+\beta a^{\dagger}a$ with $\beta\in(-2,2)$ is
associated with a probability measure on $\mathbb{R}$, the orthogonality
measure for the Meixner-Pollaczek polynomials [3].

Remembering that $k^{+}=\frac{1}{2}(a^{\dagger})^{2}$,
$k^{-}=\frac{1}{2}a^{2}$, $k^{0}=\frac{1}{2}a^{\dagger}a+\frac{1}{4}$ satisfy
the commutation relations of the $su(1,1)$ algebra, we see that the operators

$X=k^{+}+k^{-}+2k^{0},\qquad Y=k^{+}+k^{-}+({\sqrt{p}}^{-1}+\sqrt{p})k^{0}$
(9)

with $p\in(0,1)$, are naturally associated with Gamma and negative binomial
(Pascal) laws. Our three representations give priority to different operators:
roughly, we diagonalize $k^{0}$, $X$ or $Y$.

Combinations of the type (9) are also motivated by the theory of orthogonal
polynomials and Jacobi matrices, see Berezansky [9], Lytvynov [39] and
references therein. Consider for example the Laguerre polynomials
$p_{n}(x)=\mathscr{L}_{n}^{(\alpha-1)}(x)$, orthogonal with respect to the
measure $x^{\alpha-1}\exp(-x)\mathrm{d}x$ on $\mathbb{R}_{+}$ and with leading
coefficient $1$. They satisfy the three-term recurrence relation [32, Eq.
(9.12.4)]

$xp_{n}(x)=p_{n+1}(x)+(2n+\alpha)p_{n}(x)+n(n+\alpha-1)p_{n-1}(x).$

Every polynomial $f(x)$ is a linear combination
$f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}g(n)p_{n}(x)$, with at most finitely many
non-zero $g(n)$’s. When multiplied with $x$, it is

$xf(x)=\sum_{n=0}^{\infty}\frac{1}{n!}p_{n}(x)\Bigl{(}ng(n-1)+(2n+\alpha)g(n)+(\alpha+n)g(n+1)\Bigr{)}.$

Thus multiplication by $x$ corresponds to a linear transformation of the
sequence of coefficients, given by an infinite band-diagonal matrix, the
Jacobi matrix. Let us define $J^{+}g(n)=ng(n-1)$,
$J^{0}g(n)=(n+\alpha/2)g(n)$, $J^{-}g(n)=(\alpha+n)g(n+1)$, then
multiplication by $x$ corresponds to the action of $J^{+}+2J^{0}+J^{-}$ in the
space of sequences.

For infinite-dimensional orthogonal polynomials, the linear operators
$J^{\pm}$ and $J^{0}$ are replaced by families of operators $J^{\pm}(\varphi)$
and $J^{0}(\varphi)$, indexed by test functions $\varphi$. The focus in this
context is on the commutative family of operators $J^{+}(\varphi)+\lambda
J^{0}(\varphi)+J^{-}(\varphi)$ (_Jacobi fields_) and not so much on the full
set of commutation relations (see, however, [40]).

###  2.5. Continuous-time Markov processes

Finally we sketch how continuous-time Markov processes naturally enter the
scene, using the Gaussian model for the harmonic oscillator [41, Chapter
III.2(2)]. Let $\mu=\mathcal{N}(0,1)$ be the standard normal distribution on
$\mathbb{R}$. The norm in $L^{2}(\mathbb{R},\mu)$ is

$||f||^{2}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}|f(x)|^{2}\mathrm{e}^{-x^{2}/2}\mathrm{d}x.$

The densely defined operators $a=\frac{\mathrm{d}}{\mathrm{d}x}$,
$a^{\dagger}=x-\frac{\mathrm{d}}{\mathrm{d}x}$ satisfy
$[a,a^{\dagger}]=\mathrm{id}$. Now, the point we wish to highlight is this:
Both the annihilation operator $a$ and $L=-\widehat{n}$, with $\widehat{n}$
the number operator

$\widehat{n}=a^{\dagger}a=-\Bigl{(}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-x\frac{\mathrm{d}}{\mathrm{d}x}\Bigr{)}$

are generators of continuous-time Markov processes. The operator
$L=-\widehat{n}$ generates an Ornstein-Uhlenbeck process for which the
Gaussian measure $\mu$ is reversible. The annihilation operator generates
deterministic motion to the right, $X_{t}=X_{0}+t$. The creation operator
corresponds to time-evolution of densities. Indeed $\exp(ta^{\dagger})$ maps
$f_{0}$ to $f_{t}(x)=f_{0}(x-t)\exp(tx)$. Thus, if $X_{0}$ has probability
density function $\rho_{0}(x)=f_{0}(x)\exp(-x^{2}/2)/\sqrt{2\pi}$, then
$X_{t}=X_{0}+t$ has probability density function
$\rho_{t}(x)=\rho_{0}(x-t)=f_{t}(x)\exp(-x^{2}/2)/\sqrt{2\pi}$.

We will encounter a similar structure for the second and third representations
of the current algebra (and of the Lie algebra, see Section 4): The lowering
and the neutral operator are related to continuous-time Markov processes,
concretely spatial birth-death processes and measure-valued branching
processes.

##  3\. Main results

###  3.1. Representation in extended Fock space

Let $(\mathbb{X},\mathcal{X})$ be a Borel space, for example, a complete
metric separable space with its Borel $\sigma$-algebra. Let $\alpha$ be a
$\sigma$-finite measure on $\mathbb{X}$.

####  3.1.1. Extended Fock space

Define measures $\lambda_{n}$ on $\mathbb{X}^{n}$, equipped with the product
$\sigma$-algebra $\mathcal{X}^{n}$, as follows. Let $\mathfrak{S}_{n}$ be the
set of permutations of $\\{1,\ldots,n\\}$. Every permutation is a product of
cycles with disjoint supports; the number of cycles is denoted $|\sigma|$.
Given $f_{n}:\mathbb{X}^{n}\to\mathbb{C}$ a new function
$f_{n}^{\sigma}:\mathbb{X}^{|\sigma|}\to\mathbb{C}$ is obtained by identifying
variables that belong to the same cycle. For example, for $n=3$ and
$\sigma=(1\ 3)(2)$, we get $f_{3}^{\sigma}(x,y)=f_{3}(x,y,x)$. The measure
$\lambda_{n}$ on $\mathbb{X}^{n}$ is the uniquely defined measure for which

$\int f_{n}\mathrm{d}\lambda_{n}=\sum_{\sigma\in\mathfrak{S}_{n}}\int
f_{n}^{\sigma}\mathrm{d}\alpha^{|\sigma|}$

for all measurable $f_{n}:\mathbb{X}^{n}\to\mathbb{R}_{+}$. For example, we
have $\lambda_{1}=\alpha$ and

$\int f_{2}\mathrm{d}\lambda_{2}=\int
f_{2}(x,y)\alpha(\mathrm{d}x)\alpha(\mathrm{d}y)+\int
f(x,x)\alpha(\mathrm{d}x).$

Let $L^{2}(\lambda_{n})=L^{2}(\mathbb{X}^{n},\mathcal{X}^{n},\lambda_{n})$ be
the space of complex-valued, square-integrable functions $f_{n}$ and
$L_{\mathrm{s}}^{2}(\lambda_{n})$ be the subspace of symmetric functions,
i.e., for every permutation $\sigma\in\mathfrak{S}_{n}$,
$f_{n}(x_{\sigma(1)},\ldots,x_{\sigma(n)})=f_{n}(x_{1},\ldots,x_{n})$ for
$\lambda_{n}$-almost all $(x_{1},\ldots,x_{n})$. The _interacting_ or
_extended Fock space_ $\mathfrak{F}$ [33] consists of the sequences
$(f_{n})_{n\in\mathbb{N}_{0}}$ where $f_{0}\in\mathbb{C}$, $f_{n}\in
L_{\mathrm{s}}^{2}(\lambda_{n})$, and

$|f_{0}|^{2}+\sum_{n=1}^{\infty}\frac{1}{n!}\int|f_{n}|^{2}\mathrm{d}\lambda_{n}<\infty.$

The extended Fock space is equipped with the scalar product

$\langle(f_{n})_{n\in\mathbb{N}_{0}},(g_{n})_{n\in\mathbb{N}_{0}}\rangle=\overline{f_{0}}g_{0}+\sum_{n=1}^{\infty}\frac{1}{n!}\int\overline{f_{n}}g_{n}\mathrm{d}\lambda_{n}.$

The _vacuum_ is the sequence $\Psi=(1,0,0,\ldots)$. The symmetrized tensor
product of functions $f_{1},\ldots,f_{n}:\mathbb{X}\to\mathbb{C}$ is

$\bigl{(}f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}\bigr{)}(x_{1},\ldots,x_{n})=\frac{1}{n!}\sum_{\sigma\in\mathfrak{S}_{n}}f_{1}(x_{\sigma(1)})\cdots
f_{n}(x_{\sigma(n)}).$

We shall often identify a function $F_{n}:\mathbb{X}^{n}\to\mathbb{C}$ with
the sequence $(0,\ldots,F_{n},0,\ldots)$ in $\mathfrak{F}$. With this
identification in mind, the following lemma is useful. Remember the algebra
$\mathcal{C}$ of bounded functions supported on sets of finite measure.

######  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 $\mathfrak{F}$.

The lemma is proven in Section 5.1.

####  3.1.2. Exponential vectors

For $u\in\mathcal{C}$ with $\sup|u|<1$, we define a vector
$\mathcal{E}_{u}\in\mathfrak{F}$ by

$(\mathcal{E}_{u})_{0}=1,\quad(\mathcal{E}_{u})_{n}(x_{1},\ldots,x_{n})=\prod_{i=1}^{n}u(x_{i}).$

In terms of the raising operators introduced below,
$\mathcal{E}_{u}=\exp(k^{+}(u))\Psi$, whence the name _exponential vector_
[41, Chapter IV.1]. (Mathematical physicists may be more familiar with the
notion of _coherent states_ ; for bosons, these are just the normalized
exponential states, but for $SU(1,1)$ the relation is more subtle, see the
remark after Theorem 3.4 below.) We note

$\langle\mathcal{E}_{u},\mathcal{E}_{v}\rangle=\exp\Bigl{(}-\int\log\bigl{(}1-\overline{u(x)}v(x)\bigr{)}\alpha(\mathrm{d}x)\Bigr{)}$
(10)

where $-\log(1-z)=\sum_{j=1}^{\infty}z^{j}/j$ for $|z|<1$. Eq. (10) already
appeared in Boukas [12, 13], however Boukas’s Hilbert space is more abstract
than our interacting Fock space.

The linear combinations of exponential vectors are dense in $\mathfrak{F}$,
moreover if $u_{1},\ldots,u_{n}$ are linearly independent in
$L^{2}(\mathbb{X},\alpha)$, then the associated exponential vectors are
linearly independent in $\mathfrak{F}$. These properties are proven in the
same way as their counterparts in the usual bosonic Fock space, we refer to
Meyer [41, Chapter IV.1.3].

####  3.1.3. Representation of the $su(1,1)$ current algebra

Define _raising, lowering and neutral operators_
$k^{+}(\varphi),k^{-}(\varphi),k^{0}(\varphi):\mathcal{D}\to\mathfrak{F}$,
indexed by $\varphi\in\mathcal{C}$, as follows: for $n\in\mathbb{N}$,

$\displaystyle\bigl{(}k^{+}(\varphi)f\bigr{)}_{n}(x_{1},\ldots,x_{n})$
$\displaystyle=\sum_{i=1}^{n}\varphi(x_{i})f_{n-1}(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{n}),$
$\displaystyle\bigl{(}k^{-}(\varphi)f\bigr{)}_{n}(x_{1},\ldots,x_{n})$
$\displaystyle=(n+1)\int\overline{\varphi(y)}f_{n+1}(y,x_{1},\ldots,x_{n})\alpha(\mathrm{d}y)$
$\displaystyle\qquad\qquad+\sum_{i=1}^{n}\overline{\varphi(x_{i})}f_{n+1}(x_{1},\ldots,x_{n},x_{i}),$
$\displaystyle\bigl{(}k^{0}(\varphi)f\bigr{)}_{n}(x_{1},\ldots,x_{n})$
$\displaystyle=\Bigl{(}\sum_{i=1}^{n}\varphi(x_{i})+\frac{1}{2}\int\varphi\mathrm{d}\alpha\Bigr{)}f_{n}(x_{1},\ldots,x_{n}).$

For $n=0$ we use the same expressions, with $\sum_{i=1}^{0}(\cdots)=0$. In
particular, $(k^{+}(\varphi)f)_{0}=0$,
$(k^{-}(\varphi)f)_{0}=\int\overline{\varphi}f_{1}\mathrm{d}\alpha$, and
$(k^{0}(\varphi)f)_{0}=(\frac{1}{2}\int\varphi\mathrm{d}\alpha)f_{0}$.

Similar operators already appear in Śniady [52] and Lytvynov [39]. Śniady
studies them in the context of the renormalized algebra of the square of white
noise, Lytvynov investigates infinite-dimensional orthogonal polynomials and
relates Jacobi fields to the square of white noise in [40]. Neither of them
explicitly comments on the relation with the $su(1,1)$ current algebra.

######  Theorem 3.2.

The operators $k^{\pm}(\varphi),k^{0}(\varphi):\mathcal{D}\to\mathfrak{F}$
form a Fock representation of the $su(1,1)$ current algebra in $\mathfrak{F}$
with vacuum $\Psi=(1,0,0,\ldots)$.

The operators $k^{\\#}(\varphi)$ may be written as

$k^{+}(\varphi)=a^{\dagger}(\varphi),\quad
k^{-}(\varphi)=a(\varphi)+b(\varphi),\quad
k^{0}(\varphi)=\widehat{n}(\varphi)+\frac{1}{2}\int\varphi\mathrm{d}\alpha$

where

$\displaystyle
a^{\dagger}(\varphi)f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}$
$\displaystyle=(n+1)\varphi\otimes_{\mathrm{s}}f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n},$
$\displaystyle
a(\varphi)f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\langle\varphi,f_{i}\rangle
f_{1}\otimes_{\mathrm{s}}\cdots\widehat{f_{i}}\cdots\otimes_{\mathrm{s}}f_{n},$
$\displaystyle
b(\varphi)f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}$
$\displaystyle=\frac{2}{n}\sum_{1\leq i<j\leq
n}\bigl{(}\overline{\varphi}f_{i}f_{j}\bigr{)}\otimes_{\mathrm{s}}f_{1}\otimes_{\mathrm{s}}\cdots\widehat{f}_{i}\cdots\widehat{f}_{j}\cdots\otimes_{\mathrm{s}}f_{n},$
$\displaystyle\widehat{n}(\varphi)f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}$
$\displaystyle=\sum_{i=1}^{n}f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}(\varphi
f_{i})\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}.$

Here, as usual, $\widehat{f}_{i}$ means omission of $f_{i}$. The scalar
product is $\langle\varphi,f\rangle=\int\overline{\varphi}f\mathrm{d}\alpha$.
In $a^{\dagger}(\varphi)$, $a(\varphi)$ we recognize the bosonic creation and
annihilation operators; $\widehat{n}(\varphi)$ is a smeared bosonic number
operator (think $\int\varphi(x)a_{x}^{\dagger}a_{x}\,\mathrm{d}x$). (Factors
of $\sqrt{n+1}$ or $1/\sqrt{n}$ often included in explicit formulas [46,
Chapter X.7] are missing because the scalar product in our (extended) Fock
space has the factor $1/n!$.) The operators satisfy

$[a(f),a^{\dagger}(g)]=\langle
f,g\rangle,\quad[\widehat{n}(f),a^{\dagger}(g)]=a^{\dagger}(fg),\quad[\widehat{n}(f),a(g)]=-a(f\overline{g}).$
(11)

###  3.2. Unitary operators, Baker-Campbell-Hausdorff formula

For $\xi,\theta\in\mathcal{C}$ with real-valued $\theta$, define unitary
operators, analogous to (3), by

$U(\xi,\theta)=\exp\bigl{(}k^{+}(\xi)-k^{-}(\xi)\bigr{)}\exp\bigl{(}2\mathrm{i}k^{0}(\theta)\bigr{)}.$
(12)

The second exponential on the right is a multiplication operator, with
multiplier
$\exp(\mathrm{i}\int\theta\mathrm{d}\alpha+2\mathrm{i}\sum_{j=1}^{n}\theta(x_{j}))$.
The first exponential is by definition $\exp(\mathrm{i}A)$ with $A$ the self-
adjoint closure of $\frac{1}{i}(k^{+}(\xi)-k^{-}(\xi))$. The latter operator
is essentially self-adjoint on $\mathcal{D}$; the proof with Nelson’s analytic
vector theorem is similar to [19, Lemma 4.1] and Reed and Simon [46, Theorem
X.41(a)] and therefore omitted.

The unitaries $U(\xi,\theta)$ act on exponential vectors in a fairly explicit
way. For $z:\mathbb{X}\to\\{\zeta\in\mathbb{C}:\,|\zeta|<1\\}$, let

$z_{\xi}(x)=\frac{z(x)+\frac{\xi(x)}{|\xi(x)|}\tanh|\xi(x)|}{1+z(x)\frac{\overline{\xi(x)}}{|\xi(x)|}\tanh|\xi(x)|}$
(13)

and

$C_{\xi}(z)=\exp\Biggl{(}-\int\log\Bigl{(}\cosh|\xi(x)|+z(x)\frac{\overline{\xi(x)}}{|\xi(x)|}\sinh|\xi(x)|\Bigr{)}\alpha(\mathrm{d}x)\Biggr{)}.$
(14)

As explained by us in [19, Section 5], the equations are related to the action
of $SU(1,1)$ on the open unit disk by Möbius transforms, and to
representations of $SU(1,1)$ given by Bargmann [7, Section 9].

######  Theorem 3.3.

Let $\xi:\mathbb{X}\to\mathbb{C}$, $\theta:\mathbb{X}\to\mathbb{R}$,
$z:\mathbb{X}\to\mathbb{C}$ be bounded functions in $\mathcal{C}$ with
$\sup|z|<1$. Then

$\displaystyle\exp\bigl{(}k^{+}(\xi)-k^{-}(\xi)\bigr{)}\mathcal{E}_{z}$
$\displaystyle=C_{\xi}(z)\mathcal{E}_{z_{\xi}},$
$\displaystyle\exp\bigl{(}2\mathrm{i}k^{0}(\theta)\bigr{)}\mathcal{E}_{z}$
$\displaystyle=\exp\Bigl{(}\mathrm{i}\int\theta\,\mathrm{d}\alpha\Bigr{)}\mathcal{E}_{\exp(2\mathrm{i}\theta)z}.$

The proof of the first formula is similar to Proposition 5.1 in [19] and
therefore omitted. The second formula is straightforward, we leave the details
to the reader.

The $2\times 2$ matrices in $SU(1,1)$ satisfy the Baker-Campbell-Hausdorff
formula

$\exp\Bigl{(}\xi
k^{+}-\overline{\xi}k^{-}\Bigr{)}=\exp\Bigl{(}\frac{\xi}{|\xi|}(\tanh|\xi|)k^{+}\Bigr{)}\Bigl{(}\frac{1}{\cosh|\xi|}\Bigr{)}^{2k^{0}}\exp\Bigl{(}-\frac{\xi}{|\xi|}(\tanh|\xi|)k^{-}\Bigr{)}.$
(15)

The unitaries $U(\xi,0)$ satisfy a similar equation.

######  Theorem 3.4 (Baker-Campbell-Hausdorff formula).

For all $\xi\in\mathcal{C}$ and all $f\in\mathcal{D}$, we have

$\exp\Bigl{(}k^{+}(\xi)-k^{-}(\xi)\Bigr{)}f=\exp\bigl{(}k^{+}(v)\bigr{)}\exp\bigl{(}k^{0}(w)\bigr{)}\exp\bigl{(}-k^{-}(v)\bigr{)}f,$
(16)

where $v:=\mathbbm{1}_{\\{\xi\neq 0\\}}\frac{\xi}{|\xi|}\tanh|\xi|$ and
$w:=-2\log\cosh|\xi|$.

On the right side $\exp(k^{0}(w))$ is a multiplication operator and $\exp(\pm
k^{\pm}(v))$ are defined by exponential series that converge on the relevant
domains (Lemma 5.2). The theorem is proven in Section 5.2.

######  Remark.

The analogue of the _Perelomov coherent state_ [14, 44] is
$\mathscr{C}_{\xi}=\exp\bigl{(}k^{+}(\xi)-k^{-}(\xi)\bigr{)}\Psi$. Theorem 3.4
yields $\mathscr{C}_{\xi}=\lambda_{\xi}\exp(k^{+}(v))\Psi$ with
$\lambda_{\xi}=\exp(-\int\log(\cosh|\xi|)\,\mathrm{d}\alpha)$ and $v$ as given
in the theorem. Thus, the coherent state $\mathscr{C}_{\xi}$ differs from the
exponential state $\mathcal{E}_{v}$ by a normalization and an additional
reparametrization from $\xi$ to $v$.

Araki [5] emphasizes the role of expectation functionals $g\mapsto E(g)$ for
current groups. In light of this, we mention as a corollary an explicit
expression for the vacuum expectations.

######  Corollary 3.5 (Vacuum expectations).

For all $\xi\in\mathcal{C}$ and all real-valued $\theta\in\mathcal{C}$,

$\langle\Psi,U(\xi,\theta)\Psi\rangle=\exp\Bigl{(}\mathrm{i}\int\theta\,\mathrm{d}\alpha-\int\bigl{(}\log\cosh|\xi|\bigr{)}\,\mathrm{d}\alpha\Bigr{)}.$

###  3.3. Gamma random measure and Laguerre polynomials

Next comes a representation in a probability space of random measures somewhat
analogous to the Gaussian model for the canonical commutation relations from
Section 2.3.1 as the raising and lowering operators involve differential
operators.

Let $\mathbf{M}$ be the space of $s$-finite measures on $\mathbb{X}$ (a
measure is $s$-finite if it is a countable sum of finite measures [36]). The
space $\mathbf{M}$ is equipped with the $\sigma$-algebra generated by the
variables $B\mapsto\mu(B)$ with $B\in\mathcal{X}$. Given the $\sigma$-finite
reference measure $\alpha$, let $\rho_{\alpha}$ be the uniquely defined
probability measure on $\mathbf{M}$ such that:

  * •

For all disjoint $B_{1},\ldots,B_{n}\in\mathcal{X}$, the variables
$\mu\mapsto\mu(B_{i})$, $i=1,\ldots,n$, are independent.

  * •

For every $B\in\mathcal{X}$, the variable $\mu\mapsto\mu(B)$ has a Gamma
distribution with shape parameter $\alpha(B)$ and scale parameter $1$: if
$\alpha(B)<\infty$,

$\rho_{\alpha}\bigl{(}\\{\mu:\mu(B)\leq
y\\}\bigr{)}=\frac{1}{\Gamma(\alpha(B))}\int_{0}^{y}t^{\alpha(B)-1}\mathrm{e}^{-t}\mathrm{d}t\quad(y\geq
0).$

When $\alpha(B)=0$ or $\infty$, we ask that $\mu(B)=0$ or $\infty$,
$\rho_{\alpha}$-almost surely.

The probability measure $\rho_{\alpha}$ is the law of a _Gamma random
measure_. It is the law of a compound Poisson process [36, Chapter 15]: Let
$\Pi=\sum_{i}\delta_{(X_{i},Z_{i})}$ be a Poisson point process on
$\mathbb{X}\times(0,\infty)$ with intensity measure
$\alpha(\mathrm{d}x)z^{-1}\exp(-z)\mathrm{d}z$, then
$\xi=\sum_{i}Z_{i}\delta_{X_{i}}$ has law $\rho_{\alpha}$ [36, Example 15.6].

We work in the Hilbert space $L^{2}(\mathbf{M},\rho_{\alpha})$. Let
$\mathbb{D}$ be the space of polynomials in the variables
$\mu(\varphi)=\int\varphi\mathrm{d}\mu$, where $\varphi$ is in the algebra
$\mathcal{C}$ of test functions from (6). Thus $\mathbb{D}$ consists of the
functions $F$ of the form

$F(\mu)=P_{n}\bigl{(}\mu(f_{1}),\ldots,\mu(f_{n})\bigr{)}$ (17)

with $n\in\mathbb{N}$, $f_{1},\ldots,f_{n}\in\mathcal{C}$, and
$P_{n}:\mathbb{C}^{n}\to\mathbb{C}$ a multivariate polynomial with complex
coefficients. Notice that if $f\in\mathcal{C}$, then $\mu(|f|)<\infty$ for
$\rho_{\alpha}$-almost all $\mu\in\mathcal{M}$, thus $F(\mu)$ is well-defined
except possibly on $\rho_{\alpha}$ null sets on which we may define it to be
zero.

######  Lemma 3.6.

The space $\mathbb{D}$ is dense in $L^{2}(\mathbf{M},\rho_{\alpha})$.

The lemma is a classical infinite-dimensional variant of a result due to Riesz
[47]: For probability measures on $\mathbb{R}$, uniqueness in the moment
problem implies density of polynomials. For the reader’s convenience we give a
self-contained proof in Section 5.3.

For a polynomial $F$ as in (17), $\mu\in\mathbf{M}$ and $x\in\mathbb{X}$, set

$\displaystyle\frac{\delta F}{\delta\mu(x)}(\mu)$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}F(\mu+t\delta_{x})\Big{|}_{t=0}=\sum_{j=1}^{n}f_{j}(x)\partial_{j}P_{n}\bigl{(}\mu(f_{1}),\ldots,\mu(f_{n})\bigr{)},$
$\displaystyle\frac{\delta^{2}F}{\delta\mu(x)^{2}}(\mu)$
$\displaystyle=\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}F(\mu+t\delta_{x})\Big{|}_{t=0}=\sum_{i,j=1}^{n}f_{i}(x)f_{j}(x)\partial_{i}\partial_{j}P_{n}\bigl{(}\mu(f_{1}),\ldots,\mu(f_{n})\bigr{)}$

where $\partial_{j}P_{n}$ is the partial derivative of $P_{n}$ with respect to
the $j$-th variable. Finally, we define lowering, raising and neutral
operators $K^{\\#}(\varphi):\mathbb{D}\to\mathbb{D}$, where
$\varphi\in\mathcal{C}$, by

$\displaystyle K^{-}(\varphi)F(\mu)$
$\displaystyle=\int\overline{\varphi(x)}\frac{\delta^{2}F}{\delta\mu(x)^{2}}(\mu)\mu(\mathrm{d}x)+\int\overline{\varphi(x)}\frac{\delta
F}{\delta\mu(x)}(\mu)\alpha(\mathrm{d}x)$ $\displaystyle K^{+}(\varphi)F(\mu)$
$\displaystyle=\int\varphi(x)\frac{\delta^{2}F}{\delta\mu(x)^{2}}(\mu)\mu(\mathrm{d}x)+\int\varphi(x)\frac{\delta
F}{\delta\mu(x)}(\mu)(\alpha-2\mu)(\mathrm{d}x)$
$\displaystyle\qquad\qquad\qquad+(\mu(\varphi)-\alpha(\varphi))F(\mu)$
$\displaystyle K^{0}(\varphi)F(\mu)$
$\displaystyle=-\int\varphi(x)\frac{\delta^{2}F}{\delta\mu(x)^{2}}(\mu)\mu(\mathrm{d}x)-\int\varphi(x)\frac{\delta
F}{\delta\mu(x)}(\mu)(\alpha-\mu)(\mathrm{d}x)$
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}\alpha(\varphi)F(\mu).$

######  Remark.

For every real-valued test function $\varphi\in\mathcal{C}$,

$K^{+}(\varphi)+K^{-}(\varphi)+2K^{0}(\varphi)=\mu(\varphi).$ (18)

The right side is the multiplication operator with $\mu(\varphi)$. The above
relation is in agreement with the univariate example (iii) from Section 2.4.
It should remind the reader of the definition of Segal’s field operator
$\Phi_{\mathrm{S}}(f)$ as a sum of bosonic creation and annihilation operators
[46, Chapter X.7].

For $\beta>0$, let $\mathscr{L}_{n}^{(\beta-1)}(x)$, $n\in\mathbb{N}_{0}$, be
monic Laguerre polynomials, i.e., the polynomials of degree $n$ with leading
coefficient $1$ that are orthogonal with respect to the measure
$x^{\beta-1}\exp(-x)\mathrm{d}x$ on $\mathbb{R}_{+}$.

######  Theorem 3.7.

The operators $K^{\pm}(\varphi),K^{0}(\varphi):\mathbb{D}\to\mathbb{D}$, with
$\varphi\in\mathcal{C}$, form a Fock representation in
$L^{2}(\mathbf{M},\rho_{\alpha})$ of the $su(1,1)$ current algebra with vacuum
$\Psi=\mathbbm{1}$, the constant function $1$. Moreover,

$K^{+}(\mathbbm{1}_{B_{1}})^{n_{1}}\cdots
K^{+}(\mathbbm{1}_{B_{\ell}})^{n_{\ell}}\mathbbm{1}(\mu)=\prod_{i=1}^{\ell}\mathscr{L}_{n_{i}}^{(\alpha(B_{i})-1)}\bigl{(}\mu(B_{i})\bigr{)}$
(19)

for all $\ell\in\mathbb{N}$, disjoint $B_{1},\ldots,B_{\ell}\in\mathcal{X}$
with $0<\alpha(B_{i})<\infty$, $n_{1},\ldots,n_{\ell}\in\mathbb{N}$ and
$\rho_{\alpha}$-almost all $\mu\in\mathbf{M}$.

The theorem is proven in Section 5.3.

We remark that for non-negative $\varphi\in\mathcal{C}$, the operators have
probabilistic interpretations: The lowering operator $K^{-}(\varphi)$ can be
interpreted as the formal generator of a measure-valued Markov process that is
a variant of superprocesses studied by Stannat [54]. The process does not
leave the distribution $\rho_{\alpha}$ invariant. The raising operator
$K^{+}(\varphi)$ describes the time-evolution of densities: if a process with
generator $K^{-}(\varphi)$ has an initial law $\pi_{0}=F_{0}\rho_{\alpha}$
that is absolutely continuous with respect to $\rho_{\alpha}$ with Radon-
Nikodym derivative $F_{0}$, then the law at time $t$ has Radon-Nikodym
derivative $\exp(tK^{+}(\varphi))F_{0}$.

Moreover, the process with formal generator
$-K^{0}(\varphi)+\frac{1}{2}\alpha(\varphi)$ is known as _Dawson-Watanabe
measure-valued continuous-state branching process_ , see [18, Equation (1.1)]
or [53, Equation (38)]. If $\varphi$ is constant to one and $\alpha$ is a
finite measure, the generator has spectrum $-\mathbb{N}_{0}$. Consequently,
the corresponding Markov semigroup behaves analogously to the Ornstein-
Uhlenbeck semigroup in the Gaussian case [43, Equation (1.67)] or in the
Poisson case [35, Proposition 4].

###  3.4. Pascal point process and Meixner polynomials

We conclude with a representation in a probability space of point
configurations analogous to the Poisson-Charlier representation for the
canonical commutation relations from Section 2.3.2.

Let $\mathbf{N}=\mathbf{N}(\mathbb{X})$ be the space of finite or countable
sums of Dirac measures (including the zero measure) on the Borel space
$(\mathbb{X},\mathcal{X})$. The space $\mathbf{N}$ is equipped with the
$\sigma$-algebra $\mathcal{N}$ generated by the counting variables
$B\mapsto\eta(B)$, $B\in\mathcal{X}$. For $a>0$ the rising factorial
(Pochhammer symbol) is

$(a)_{0}=1,\quad(a)_{n}=a(a+1)\cdots(a+n-1).$

For $p\in(0,1)$ and $\alpha$ a $\sigma$-finite measure $\alpha$ on
$\mathbb{X}$, let $\rho_{p,\alpha}$ be the distribution of a Pascal point
process, i.e., $\rho_{p,\alpha}$ is the uniquely defined measure under which:

  * •

The counting variables $\eta(B_{1}),\ldots,\eta(B_{k})$ are independent, for
all $k\geq 2$ and all pairwise disjoint $B_{1},\ldots,B_{k}\in\mathcal{E}$.

  * •

For each $B\in\mathcal{E}$ with $0<\alpha(B)<\infty$,

$\rho_{p,\alpha}\bigl{(}\\{\eta:\,\eta(B)=n\\}\bigr{)}=(1-p)^{\alpha(B)}\frac{p^{n}}{n!}\bigl{(}\alpha(B)\bigr{)}_{n}.$

If $\alpha(B)=0$ or $\infty$ then $\eta(B)=0$ or $\infty$ respectively,
$\rho_{p,\alpha}$ almost surely.

We work in the Hilbert space $L^{2}(\mathbf{N},\rho_{p,\alpha})$. Functions
are complex-valued and the scalar product is $\langle
f,g\rangle=\int\overline{f}g\,\mathrm{d}\rho_{p,\alpha}$. Let $\mathbb{D}$ be
the set of polynomials in the variables
$\eta(\varphi)=\int\varphi\mathrm{d}\eta$, i.e., $\mathbb{D}$ is the set of
functions of the form

$F(\eta)=P_{n}\bigl{(}\eta(\varphi_{1}),\ldots,\eta(\varphi_{n})\bigr{)}$

with $n\in\mathbb{N}$, $\varphi_{1},\ldots,\varphi_{n}\in\mathcal{C}$, and
$P_{n}:\mathbb{C}^{n}\to\mathbb{C}$ a multivariate polynomial.

######  Lemma 3.8.

The space $\mathbb{D}$ is dense in $L^{2}(\mathbf{N},\rho_{p,\alpha})$.

The proof of the lemma is similar to the proof of Lemma 3.6 and it is omitted.
We introduce difference operators

$\mathrm{D}_{x}^{+}F(\eta)=F(\eta+\delta_{x})-F(\eta),\quad\mathrm{D}_{x}^{-}F(\eta)=F(\eta-\delta_{x})-F(\eta)$

and the constant $c=\sqrt{p}^{-1}-\sqrt{p}$. The backward difference
$\mathrm{D}_{x}F(\eta)$ is only defined when $x$ belongs to the configuration
$\eta$, i.e., $\eta(\\{x\\})\geq 1$. For $F\in\mathbb{D}$, let

$\displaystyle K^{-}(\varphi)F(\eta)$
$\displaystyle=\frac{1}{c}\Bigl{(}\int\overline{\varphi(x)}\,\mathrm{D}_{x}^{+}F(\eta)\,(\alpha+\eta)(\mathrm{d}x)+\int\overline{\varphi(x)}\,\mathrm{D}_{x}^{-}F(\eta)\,\eta(\mathrm{d}x)\Bigr{)},$
(20) $\displaystyle K^{+}(\varphi)F(\eta)$
$\displaystyle=\frac{1}{c}\Bigl{(}p\int\varphi(x)F(\eta+\delta_{x})(\eta+\alpha)(\mathrm{d}x)+\frac{1}{p}\int\varphi(x)F(\eta-\delta_{x})\eta(\mathrm{d}x)\Bigr{)}$
$\displaystyle\qquad\qquad\qquad-\frac{1}{c}\bigl{(}2\eta(\varphi)+\alpha(\varphi)\bigr{)}F(\eta)$
(21) $\displaystyle K^{0}(\varphi)F(\eta)$
$\displaystyle=-\frac{1}{c}\Bigl{(}\sqrt{p}\int\varphi(x)\,\mathrm{D}_{x}^{+}F(\eta)\,(\alpha+\eta)(\mathrm{d}x)+\frac{1}{\sqrt{p}}\int\varphi(x)\,\mathrm{D}_{x}^{-}F(\eta)\,\eta(\mathrm{d}x)\Bigr{)}$
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}\alpha(\varphi)F(\eta).$ (22)

######  Remark.

Let us denote the operator in the first line of (22) by $\widehat{N}(\varphi)$
so that $K^{0}(\varphi)=\frac{1}{2}\alpha(\varphi)+\widehat{N}(\varphi)$. An
explicit computation yields, for real-valued $\varphi\in\mathcal{C}$,

$K^{+}(\varphi)+K^{-}(\varphi)+(\sqrt{p}+{\sqrt{p}}^{-1})\widehat{N}(\varphi)=c\,\eta(\varphi)-\sqrt{p}\,\alpha(\varphi).$

The equation is also similar to relations between quadratic white noise,
Pascal laws and Meixner polynomials, see [1, Section 17], [3, Section 4] and
case (iv) in Section 2.4.

Given $a>0$ and $p\in(0,1)$, let $\mathcal{M}_{n}(x;a,p)$ be the Meixner
polynomials of degree $n$ with leading coefficients $1$, orthogonal with
respect to the negative binomial law $\nu(\\{x\\})=(1-p)^{a}(a)_{x}p^{x}/x!$
on $\mathbb{N}_{0}$ [32].

######  Theorem 3.9.

The operators $K^{+}(\varphi)$, $K^{0}(\varphi)$, $K^{-}(\varphi)$,
$\varphi\in\mathcal{C}$, are a Fock representation in
$L^{2}(\mathbf{N},\mathcal{N},\rho_{p,\alpha})$ of the current algebra of
$su(1,1)$ with vacuum $\mathbbm{1}$, the constant function $1$. Moreover, with
$c={\sqrt{p}}^{-1}-\sqrt{p}$,

$\displaystyle K^{+}(\mathbbm{1}_{B_{1}})^{n_{1}}\cdots
K^{+}(\mathbbm{1}_{B_{\ell}})^{n_{\ell}}\mathbbm{1}(\eta)=\prod_{j=1}^{\ell}c^{n_{j}}\mathcal{M}_{n_{j}}\bigl{(}\eta(B_{j});\alpha(B_{j}),p\bigr{)}$
(23)

for all $\ell\in\mathbb{N}$, disjoint $B_{1},\ldots,B_{\ell}\in\mathcal{X}$
with $0<\alpha(B_{i})<\infty$, $n_{1},\ldots,n_{\ell}\in\mathbb{N}$ and
$\rho_{p,\alpha}$-almost all $\eta\in\mathbf{N}$.

The theorem is proven in Section 5.4.

To conclude, we remark that for non-negative $\varphi$, the operators are
amenable to a probabilistic interpretation. The processes that appear are
variants of the linear birth-death processes on $\mathbb{N}_{0}$ called
discrete Bessel and discrete Laguerre processes by Miclo and Patie [42], see
also Section 4.3 below. The role of Meixner polynomials and the Pascal law
(among others) for linear birth-death processes was already studied by Karlin
and McGregor [30], see also Chapter 3 in Schoutens [49].

The lowering operator $K^{-}(\varphi)$ is the generator of a spatial birth-
and-death process. The neutral operator $K^{0}(\varphi)$ is of the form
$\frac{1}{2}\alpha(\varphi)-L(\varphi)$ and $L(\varphi)$ is the generator of a
spatial birth and death process with reversible measure $\rho_{p,\alpha}$.
Furthermore, when the measure $\alpha$ is finite so that the constant function
$\mathbbm{1}$ is in $\mathcal{C}$, one can show that $L(\mathbbm{1})$ has
discrete spectrum $-\mathbb{N}_{0}$.

##  4\. The univariate case: three representations of $su(1,1)$

Our three representations of the $su(1,1)$ current algebra generalize much
simpler univariate counterparts that we recall here. The benefit is twofold.
First, we connect to standard representation theory for $su(1,1)$ and
univariate orthogonal polynomials. Second, the probabilistic interpretation
brings in well known processes: linear birth-death processes on
$\mathbb{N}_{0}$ and diffusions (squared Bessel and Laguerre) on
$\mathbb{R}_{+}$.

###  4.1. Representation in weighted $\ell^{2}$ space

There are several classes of unitary irreducible representations of $SU(1,1)$.
The representations studied here all belong to what is called the “discrete
class, minimal $m$, $D_{k}^{+}$” [7, §5.i] or “principal discrete series
$D_{j}^{+}$” [37, Section C]. Within that class, the representation is
uniquely characterized by a half-integer
$m_{0}\in\\{\frac{1}{2},1,\frac{3}{2},\ldots\\}$, sometimes called _Bargmann
index_. The neutral operator $k^{0}$ has simple eigenvalues
$m_{0},m_{0}+1,\ldots$ and the Casimir operator
$(k^{0})^{2}-\frac{1}{2}(k^{+}k^{-}+k^{-}k^{+})$, analogous to total spin, is
$m_{0}(1-m_{0})$ times the identity operator.

The following is an example of such a representation. It is essentially taken
from Giardinà, Kurchan, Redig and Vafayi [23]; it differs from the usual
representations in $\ell^{2}(\mathbb{N}_{0})$ by a similarity transformation.
Remember the rising factorial $(\alpha)_{0}=1$,
$(\alpha)_{n}=\alpha(\alpha+1)\cdots(\alpha+n-1)$. Fix $\alpha>0$ and define
weights $w_{\alpha}(n)=(\alpha)_{n}/n!$. We work in the Hilbert space
$\ell^{2}(\mathbb{N}_{0},w_{\alpha})$ and define operators $k^{\pm}$, $k^{0}$
with common dense domain $d$, the space of sequences
$(f(n))_{n\in\mathbb{N}_{0}}$ that have at most finitely many non-zero
entries. The operators are given by

$k^{+}f(n)=nf(n-1),\quad k^{-}f(n)=(\alpha+n)f(n+1),\quad
k^{0}f(n)=\Bigl{(}n+\frac{\alpha}{2}\Bigr{)}f(n).$

Equivalently, with $e_{n}(j)=\delta_{n,j}$,

$k^{+}e_{n}=(n+1)e_{n+1},\quad k^{-}e_{n}=(\alpha+n-1)e_{n-1},\quad
k^{0}e_{n}=\Bigl{(}n+\frac{\alpha}{2}\Bigr{)}e_{n}.$ (24)

and $k^{-}e_{0}=0$. One easily checks that $k^{\pm}$ and $k^{0}$ map $d$ into
itself and on $d$ they satisfy the commutation relations (4). Moreover,
$k^{0}$ is symmetric on $d$ and $\langle f,k^{+}g\rangle=\langle
k^{-}f,g\rangle$ for all $f,g\in d$.

When $\alpha$ is integer, we get a representation of the group $SU(1,1)$ with
Bargmann index $m_{0}=\alpha/2$. The representation maps the $2\times 2$
matrix $A(\xi,\theta)\in SU(1,1)$ from Eq. (5) to the unitary operator
$U(\xi,\theta)=\exp(\xi k^{+}-\overline{\xi}k^{-})\exp(2\mathrm{i}\theta
k^{0})$. When $\alpha$ is not integer, we get instead a representation of the
universal covering group of $SU(1,1)$ [48]. (Notice that
$A(\xi,\theta+2\pi)=A(\xi,\theta)$ no matter what, but
$U(\xi,\theta+2\pi)=U(\xi,\theta)$ only if $\alpha$ is integer.)

###  4.2. Representation with Laguerre polynomials

Let $\beta>-1$. The _Laguerre polynomials_ in hypergeometric form are given by
[32, Chapter 9.12]

$L_{n}^{(\beta)}(x)=\frac{(\beta+1)_{n}}{n!}\sum_{j=0}^{n}\frac{(-n)_{j}}{(\beta+1)_{j}}\frac{x^{j}}{j!}=\frac{(-x)^{n}}{n!}+O(x^{n-1})$

they satisfy the orthogonality relation,

$\frac{1}{\Gamma(\beta+1)}\int_{0}^{\infty}L_{n}^{(\beta)}(x)L_{m}^{(\beta)}(x)x^{\beta}\mathrm{e}^{-x}\mathrm{d}x=\delta_{m,n}\frac{(\beta+1)_{n}}{n!}.$
(25)

For $\alpha>0$, let $\ell^{2}(\mathbb{N}_{0},w_{\alpha})$ be the Hilbert space
with weights $w_{\alpha}(n)=(\alpha)_{n}/n!$ from Section 4.3. Let
$\mu_{\alpha}$ be the probability measure on $\mathbb{R}_{+}$ with density
$\Gamma(\alpha)^{-1}x^{\alpha-1}\exp(-x)$ and
$L^{2}(\mathbb{R}_{+},\mu_{\alpha})$ be the space of Borel-measurable square-
integrable functions (complex-valued). By the orthogonality relation (25), the
linear operator

$U:\ell^{2}(\mathbb{N}_{0},w_{\alpha})\to
L^{2}(\mathbb{R}_{+},\mu_{\alpha}),\quad f\mapsto
Uf=\sum_{n=0}^{\infty}f(n)L_{n}^{(\alpha-1)}$

is an isometry that maps $e_{n}$ to $L_{n}^{(\alpha-1)}$ and in particular
$e_{0}$ to the constant function $\mathbbm{1}$. The linear hull of
$e_{0},\ldots,e_{n}$ is mapped to the space of polynomials of degree at most
$n$. Upon setting $K^{\\#}=U^{-1}k^{\\#}U$ we obtain a representation of the
$su(1,1)$ algebra that satisfies $K^{-}\mathbbm{1}=0$ and

$K^{+}L_{n}=(n+1)L_{n+1},\ K^{-}L_{n}=(\alpha+n-1)L_{n-1},\quad
K^{0}L_{n}=\Bigl{(}n+\frac{\alpha}{2}\Bigr{)}K^{0}L_{n},$ (26)

compare (24). For better readability we have dropped the superscripts on the
Laguerre polynomials. The following proposition identifies $K^{\\#}$ as
differential operators; it is similar to Proposition 4.13 in Groenevelt [25].

######  Proposition 4.1.

We have

$\displaystyle K^{+}$
$\displaystyle=-x\partial_{x}^{2}+(2x-\alpha)\partial_{x}+(\alpha-x),$
$\displaystyle K^{-}$ $\displaystyle=-x\partial_{x}^{2}-\alpha\partial_{x},$
$\displaystyle K^{0}$
$\displaystyle=\frac{\alpha}{2}-\bigl{(}x\partial_{x}^{2}+(\alpha-x)\partial_{x}\bigr{)}.$

Notice that both $L=(\frac{\alpha}{2}-K^{0})$ and $-K^{-}$ are infinitesimal
generators of diffusions on $\mathbb{R}_{+}$. The operator $L$ is the
generator of the _Laguerre semigroup_ [6, Section 2.7.3] which is the Markov
semigroup of the _Cox-Ingersoll-Ross process_ (CIR) in mathematical finance to
model interest rate movements [17]. It has the Gamma distribution
$\mu_{\alpha}$ as a reversible measure. The lowering operator corresponds to a
squared Bessel process.

###### Proof.

The generating function of our Laguerre polynomials is [32, Eq. (9.12.10)]

$G_{t}(x)=\sum_{n=0}^{\infty}t^{n}L_{n}(x)=(1-t)^{-\alpha}\exp\Bigl{(}\frac{xt}{t-1}\Bigr{)}.$

In view of Eq. (26), we have

$\displaystyle\bigl{(}K^{0}-\frac{\alpha}{2}\bigr{)}G_{t}(x)$
$\displaystyle=\sum_{n=0}^{\infty}t^{n}nL_{n}(x)=t\partial_{t}G_{t}(x)$
$\displaystyle=-\Bigl{(}\alpha\frac{t}{t-1}+\frac{xt}{(t-1)^{2}}\Bigr{)}G_{t}(x)$
$\displaystyle=-\Bigl{(}x\frac{t^{2}}{(t-1)^{2}}+(\alpha-x)\frac{t}{t-1}\Bigr{)}G_{t}(x)$
$\displaystyle=-(x\partial_{x}^{2}+(\alpha-x)\partial_{x})G_{t}(x).$

This holds true for all sufficiently small $t$, the differential expression
for $K^{0}$ follows. For the raising operators:

$\displaystyle K^{+}G_{t}(x)$
$\displaystyle=\sum_{n=0}^{\infty}t^{n}(n+1)L_{n+1}(x)=\sum_{n=1}^{\infty}nt^{n-1}L_{n}(x)=\partial_{t}G_{t}(x)$
$\displaystyle=\Bigl{(}-\frac{\alpha}{t-1}-\frac{x}{(t-1)^{2}}\Bigr{)}G_{t}(x)$
$\displaystyle=\Bigl{(}-x\frac{t^{2}}{(t-1)^{2}}+(2x-\alpha)\frac{t}{t-1}+\alpha-x\Bigr{)}G_{t}(x)$
$\displaystyle=\Bigl{(}-x\partial_{x}^{2}+(2x-\alpha)\partial_{x}+(\alpha-x)\Bigr{)}G_{t}(x).$

Finally

$\displaystyle K^{-}G_{t}(x)$
$\displaystyle=\sum_{n=1}^{\infty}t^{n}(\alpha+n-1)L_{n-1}(x)$
$\displaystyle=\alpha tG_{t}(x)+t^{2}\partial_{t}G_{t}(x)$
$\displaystyle=\Bigl{(}\alpha
t-\alpha\frac{t^{2}}{t-1}-x\frac{t^{2}}{(t-1)^{2}}\Bigr{)}G_{t}(x)$
$\displaystyle=\Bigl{(}-\alpha\frac{t}{t-1}-x\frac{t^{2}}{(t-1)^{2}}\Bigr{)}G_{t}(x)$
$\displaystyle=-\bigl{(}x\partial_{x}^{2}+\alpha\partial_{x}\bigr{)}G_{t}(x).\qed$

###  4.3. Representation with Meixner polynomials

For $p\in(0,1)$ and $\alpha>0$, the Meixner polynomials are given by [32,
Chapter 9.10]

$M_{n}(x;\alpha,p)=\sum_{j=0}^{n}\frac{(-n)_{j}(-x)_{j}}{(\alpha)_{j}}\frac{1}{j!}\Bigl{(}1-\frac{1}{p}\Bigr{)}^{j}=\frac{x^{n}}{(\alpha)_{n}}\Bigl{(}1-\frac{1}{p}\Bigr{)}^{n}+O(x^{n-1})$

They satisfy the orthogonality relation

$(1-p)^{\alpha}\sum_{x=0}^{\infty}\frac{(\alpha)_{x}}{x!}p^{x}M_{n}(x;\alpha,p)M_{m}(x;\alpha,p)=\Bigl{(}\frac{(\alpha)_{n}}{n!}p^{n}\Bigr{)}^{-1}\delta_{m,n}.$
(27)

Consider the probability weights

$\rho_{p,\alpha}(x)=(1-p)^{-\alpha}\frac{p^{x}}{x!}(\alpha)_{x}\quad(x\in\mathbb{N}_{0})$

and the Hilbert space $\ell^{2}(\mathbb{N}_{0},\rho_{p,\alpha})$. By the
orthogonality relation (27), the operator

$U:\ell^{2}(\mathbb{N}_{0},w_{\alpha})\to\ell^{2}(\mathbb{N}_{0},\rho_{p,\alpha}),\quad
f\mapsto
Uf=\sum_{n=0}^{\infty}f(n)\frac{(\alpha)_{n}}{n!}\sqrt{p}^{n}M_{n}(\cdot;\alpha,p).$

is an isometry. Let $\mathcal{M}_{n}(x;\alpha,p)$ be the Meixner polynomials
with leading coefficient $1$, then

$Ue_{n}=\frac{1}{n!}\bigl{(}\sqrt{p}-\sqrt{p}^{-1}\bigr{)}^{n}\mathcal{M}_{n}(\cdot;\alpha,p).$

Let us define $K^{\\#}=U^{-1}k^{\\#}U$, then we obtain operators with common
domain the set of polynomials, that satisfy

$K^{+}\mathbbm{1}=0,\quad(K^{+})^{n}\mathbbm{1}=(\sqrt{p}-\sqrt{p}^{-1})^{n}\mathcal{M}_{n}(\cdot;\alpha,p)$
(28)

in analogy with $k^{-}e_{0}=0$ and $(k^{+})^{n}e_{0}=n!e_{n}$. The following
proposition identifies the operators $K^{\\#}$. We define forward and backward
difference operators

$\partial^{+}f(x)=f(x+1)-f(x),\quad\partial^{-}f(x)=f(x-1)-f(x)$

with $f(-1)=0$. The proposition is similar to Proposition 4.7 in Groenevelt
[25].

######  Proposition 4.2.

For every polynomial $f:\mathbb{N}_{0}\to\mathbb{C}$,

$\displaystyle K^{+}f(x)$
$\displaystyle=-\frac{\sqrt{p}}{1-p}\Bigl{(}p(\alpha+x)f(x+1)+\frac{1}{p}xf(x-1)-(\alpha+2x)f(x)\Bigr{)},$
$\displaystyle K^{-}f(x)$
$\displaystyle=-\frac{\sqrt{p}}{1-p}\Bigl{(}(\alpha+x)\partial^{+}f(x)+x\partial^{-}f(x)\Bigr{)},$
$\displaystyle K^{0}f(x)$
$\displaystyle=\frac{\alpha}{2}f(x)-\frac{1}{1-p}\bigl{(}p(\alpha+x)\partial^{+}f(x)+x\partial^{-}f(x)\bigr{)}.$

We remark that up to additive and multiplicative constants, the lowering and
the neutral operators are the generators of linear birth-death chains on
$\mathbb{N}_{0}$. As mentioned after Theorem 3.9, those processes were already
studied in connection with orthogonal polynomials by Karlin and McGregor[30];
Miclo and Patie [42] provided intertwining relations between these linear
birth-death chains and squared Bessel and Laguerre diffusions. Our
representation clarifies the algebraic similarity of the processes: they admit
the same formal expression with the $su(1,1)$ algebra.

###### Proof of Proposition 4.2.

The generating function of the Mexiner polynomials is [32]

$G_{t}(x)=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}\,t^{n}M_{n}(x)=\Bigl{(}1-\frac{t}{p}\Bigr{)}^{x}(1-t)^{-\alpha-x}.$

We note

$\displaystyle\partial_{+}G_{t}(x)$
$\displaystyle=G_{t}(x+1)-G_{t}(x)=\Bigl{(}\frac{1-t/p}{1-t}-1\Bigr{)}G_{t}(x)=\frac{1-1/p}{1-t}\,tG_{t}(x).$

and

$\partial_{-}G_{t}(x)=\Bigl{(}\frac{1-t}{1-t/p}-1\Bigr{)}G_{t}(x)=\frac{1/p-1}{1-t/p}\,tG_{t}(x).$

Therefore, in view of $(K^{0}-\frac{\alpha}{2})M_{n}=nM_{n}$, we get

$\displaystyle\bigl{(}K^{0}-\frac{\alpha}{2}\bigr{)}G_{t}(x)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}t^{n}nM_{n}(x)=t\partial_{t}G_{t}(x)$
$\displaystyle=\Bigl{(}-\frac{x}{p}\bigl{(}1-\frac{t}{p}\bigr{)}^{-1}+(\alpha+x)(1-t)^{-1}\Bigr{)}tG_{t}(x)$
$\displaystyle=-\frac{x}{1-p}\partial_{-}G_{t}(x)-\frac{p(\alpha+x)}{1-p}\partial_{+}G_{t}(x).$

Turning to the lowering operators, we note first, $K^{-}M_{0}=0$ and, for
$n\geq 1$,

$\displaystyle K^{-}\frac{(\alpha)_{n}}{n!}t^{n}M_{n}$
$\displaystyle=\bigl{(}\frac{t}{\sqrt{p}}\bigr{)}^{n}K^{-}Ue_{n}=\bigl{(}\frac{t}{\sqrt{p}}\bigr{)}^{n}Uk^{-}e_{n}$
$\displaystyle=\bigl{(}\frac{t}{\sqrt{p}}\bigr{)}^{n}(\alpha+n-1)Ue_{n}$
$\displaystyle=\bigl{(}\frac{t}{\sqrt{p}}\bigr{)}^{n}(\alpha+n-1)\frac{(\alpha)_{n-1}}{(n-1)!}{\sqrt{p}}^{n-1}M_{n-1}$
$\displaystyle=\frac{1}{\sqrt{p}}\frac{(\alpha)_{n}}{(n-1)!}\,t^{n}M_{n-1}.$

We change the summation index in $G_{t}$ from $n$ to $\ell=n-1$, note
$(\alpha)_{n}=(\alpha)_{\ell+1}=(\alpha+\ell)(\alpha)_{\ell}$ and obtain

$\displaystyle K^{-}G_{t}(x)$
$\displaystyle=\frac{1}{\sqrt{p}}\sum_{\ell=0}^{\infty}\frac{(\alpha)_{\ell}}{\ell!}(\alpha+\ell)t^{\ell+1}M_{\ell}(x)$
$\displaystyle=\frac{1}{\sqrt{p}}\bigl{(}\alpha
t+t^{2}\partial_{t}\bigr{)}G_{t}(x)$
$\displaystyle=\frac{1}{\sqrt{p}}\Bigl{(}\alpha
t-\frac{xt^{2}}{p-t}+\frac{(\alpha+x)t^{2}}{1-t}\Bigr{)}G_{t}(x)$
$\displaystyle=\frac{1}{\sqrt{p}}\Bigl{(}-\frac{x}{1-t/p}+\frac{\alpha+x}{1-t}\Bigr{)}tG_{t}(x)$
$\displaystyle=-\frac{\sqrt{p}}{1-p}\bigl{(}x\partial_{-}G_{t}(x)+(\alpha+x)\partial_{+}G_{t}(x)\bigr{)}$

The above relations hold true for all sufficiently small $t$, the expressions
for $K^{0}$ and $K^{-}$ follow. The expression for the raising operator is
proven by similar explicit computations that we omit. Alternatively, the
reader may check the adjointness $\langle f,K^{+}g\rangle=\langle
K^{-}f,g\rangle$ on a dense subspace of polynomials in
$\ell^{2}(\mathbb{N}_{0},\rho_{p,\alpha})$. ∎

##  5\. Proofs

In this section we prove our main results, namely the three representations of
the $su(1,1)$ current algebra, the Baker-Campbell-Hausdorff formula, and the
explicit action of the unitaries
$\exp(k^{+}(\xi)-k^{-}(\xi))\exp(2\mathrm{i}k^{0}(\theta))$ on exponential
vectors. In addition, we prove an integration by parts formula for the Gamma
random measure (Proposition 5.6) that is of interest in its own right.

###  5.1. Representation in extended Fock space. Proof of Theorem 3.2

We start with a lemma that helps prove the adjointness relations for the
lowering and raising operators in extended Fock space.

######  Lemma 5.1.

For all $n\in\mathbb{N}$ and non-negative or integrable
$f:\mathbb{X}^{n+1}\to\mathbb{C}$,

$\int f\mathrm{d}\lambda_{n+1}=\int\Bigl{(}\int
f(x_{1},\ldots,x_{n},y)\alpha(\mathrm{d}y)+\sum_{i=1}^{n}f(x_{1},\ldots,x_{n},x_{i})\Bigr{)}\lambda_{n}(\mathrm{d}\boldsymbol{x}).$
(29)

###### Proof.

We have to evaluate $\sum_{\sigma\in\mathfrak{S}_{n+1}}\int
f^{\sigma}\mathrm{d}\alpha^{|\sigma|}$. The permutations with
$\sigma(n+1)=n+1$ are in one-to-one correspondence with permutations in
$\mathfrak{S}_{n}$, they contribute the first term on the right in (29). Any
permutation with $\sigma(n+1)=i\leq n$ admits a cycle decomposition with a
single cycle containing $n+1$ and $i$ and remaining cycles containing neither
$n+1$ nor $i$. Deleting $n+1$ from the cycle containing $n+1$ (e.g., $n+1\to
i\to 3\to n+1$ becomes $i\to 3\to i$), we obtain the cycle decomposition of a
permutation $\tau\in\mathfrak{S}_{n}$ that has the same number of cycles.
These permutations contribute the second term on the right in (29). ∎

###### Proof of Lemma 3.1.

For $n\in\mathbb{N}$, let $V_{n}$ be the space of linear combinations of
symmetrized tensor products of indicators of sets $B_{j}\in\mathcal{X}$ with
$\alpha(B_{j})<\infty$. It is enough to show that, for every $n\in\mathbb{N}$,
the space $V_{n}$ is dense in $L^{2}_{\mathrm{s}}(\lambda_{n})$.

Fix $n\in\mathbb{N}$. Cartesian products $B_{1}\times\cdots\times B_{n}$ of
measurable sets with finite mass $\alpha(B_{i})<\infty$ form a generating
$\pi$-system of the product $\sigma$-algebra $\mathcal{X}^{n}$. Therefore, by
the functional monotone class theorem [11, Theorem 2.12.9], the linear
combinations of the constant function and indicators of Cartesian products are
dense, with respect to the supremum norm, in the space of bounded measurable
functions from $\mathbb{X}^{n}$ to $\mathbb{C}$. It follows that the linear
hull of $V_{n}$ and the constant function $1$ is dense, with respect to the
supremum norm, in the space of bounded, measurable symmetric functions.

If $\alpha$ has finite total mass, then $\lambda_{n}(\mathbb{X}^{n})<\infty$
as well and we have $\mathbbm{1}\in V_{n}\subset
L_{\mathrm{s}}^{2}(\lambda_{n})$. Furthermore, uniform convergence implies
$L^{2}(\lambda_{n})$-convergence. Thus, every bounded function in
$L_{\mathrm{s}}^{2}(\lambda_{n})$ can be approximated by a function in
$V_{n}$. As the bounded functions are dense in
$L_{\mathrm{s}}^{2}(\lambda_{n})$, we find that $V_{n}$ is dense.

If $\alpha(\mathbb{X})=\infty$, then the previous argument still applies to
$L^{2}_{\mathrm{s}}(\lambda_{n}\mathbbm{1}_{C^{n}})$, for every set
$C\in\mathcal{X}$ with $\alpha(C)<\infty$ (hence $\lambda_{n}(C^{n})<\infty$).
As $\alpha$ is $\sigma$-finite, every function $f\in
L_{\mathrm{s}}^{2}(\lambda_{n})$ is the limit in $L^{2}$-norm of a sequence
$(f\mathbbm{1}_{C_{j}^{n}})_{j\in\mathbb{N}}$ with $\alpha(C_{j})<\infty$ and
the proof of the lemma is easily concluded. ∎

###### Proof of Theorem 3.2.

By the expressions given above, the operators $k^{\\#}(\varphi)$ map
symmetrized tensor products to combinations of symmetrized tensor products,
hence $\mathcal{D}$ is invariant. The common domain $\mathcal{D}$ is dense by
Lemma 3.1. Clearly $k^{+}(\varphi)$ and $k^{-}(\varphi)$ are linear in
$\varphi$ while $k^{-}(\varphi)$ is antilinear in $\varphi$, moreover the
vacuum is a unit vector annihilated by all lowering operators. The vacuum is
cyclic because repeated application of raising operators gives symmetrized
tensor products, whose linear hull is precisely the dense space $\mathcal{D}$.

For the commutation relations (7), we note

$\displaystyle
b(\varphi)a^{\dagger}(\psi)f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}=$
$\displaystyle 2\sum_{j=1}^{n}(\overline{\varphi}\psi
f_{j})\otimes_{\mathrm{s}}f_{1}\cdots\widehat{f}_{j}\cdots\otimes_{\mathrm{s}}f_{n}$
$\displaystyle+2\sum_{1\leq i<j\leq
n}(\overline{\varphi}f_{i}f_{j})\otimes_{\mathrm{s}}\psi\otimes_{\mathrm{s}}f_{1}\cdots\widehat{f}_{i}\cdots\widehat{f}_{j}\cdots\otimes_{\mathrm{s}}f_{n}$
$\displaystyle
a^{\dagger}(\psi)b(\varphi)f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}=$
$\displaystyle 2\sum_{1\leq i<j\leq
n}\psi\otimes_{\mathrm{s}}\bigl{(}\overline{\varphi}f_{i}f_{j}\bigr{)}\otimes_{\mathrm{s}}f_{1}\otimes_{\mathrm{s}}\cdots\widehat{f}_{i}\cdots\widehat{f}_{j}\cdots\otimes_{\mathrm{s}}f_{n}$

hence

$\bigl{[}b(\psi),a^{\dagger}(\varphi)\bigr{]}f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}=2\widehat{n}(\overline{\psi}\varphi)\,f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}$

and

$\bigl{[}k^{-}(\psi),k^{+}(\varphi)\bigr{]}=\bigl{[}a(\psi),a^{\dagger}(\psi)\bigr{]}+\bigl{[}b(\psi),a^{\dagger}(\psi)\bigr{]}=\langle\psi,\varphi\rangle+2\widehat{n}(\overline{\psi}\varphi)=2k^{0}(\overline{\psi}\varphi).$

The other commutation relations, as well as Eq. (11), are proven by similar
explicit computations that we omit; see also Śniady [52, Theorem 4].

The adjointness relation for the neutral operator is straightforward:
$k^{0}(\varphi)$ is a multiplication operator that multiplies with
$\sum_{i=1}^{n}\varphi(x_{i})+\frac{1}{2}\int\varphi\mathrm{d}\alpha$, its
adjoint multiplies with the complex conjugate. For the raising and lowering
operators, let $\varphi\in\mathcal{C}$ and $(F_{n})$, $(G_{n})$ in
$\mathcal{D}$. We have

$\displaystyle\frac{1}{(n+1)!}\int\overline{G_{n+1}}\,(k^{+}(\varphi)F_{n})\,\mathrm{d}\lambda_{n+1}$
$\displaystyle\qquad=\frac{1}{(n+1)!}\int\overline{G_{n+1}(x_{1},\ldots,x_{n+1})}\Bigl{(}\sum_{i=1}^{n+1}\varphi(x_{i})F_{n}(x_{1},\ldots,\widehat{x}_{i},\ldots,x_{n+1})\Bigr{)}\lambda_{n+1}(\mathrm{d}\boldsymbol{x})$
$\displaystyle\qquad=\frac{1}{n!}\int\overline{G_{n+1}(x_{1},\ldots,x_{n+1})}\varphi(x_{n+1})F_{n}(x_{1},\ldots,x_{n})\lambda_{n+1}(\mathrm{d}\boldsymbol{x}).$

We apply Lemma 5.1 to
$f(x_{1},\ldots,x_{n},y)=\overline{G_{n+1}(x_{1},\ldots,x_{n},y)}\varphi(y)F_{n}(x_{1},\ldots,x_{n})$.
The inner integral on the right of (29) becomes

$\int\varphi(y)\overline{G_{n+1}(\boldsymbol{x},y)}F_{n}(\boldsymbol{x})\alpha(\mathrm{d}y)+\sum_{i=1}^{n}\varphi(x_{i})\overline{G_{n+1}(\boldsymbol{x},x_{i})}F_{n}(\boldsymbol{x})$

in which we recognize $\overline{k^{-}(\overline{\varphi})G_{n+1}}F_{n}$.
Thus, we get

$\frac{1}{(n+1)!}\int\overline{G_{n+1}}\,(k^{+}(\varphi)F_{n})\,\mathrm{d}\lambda_{n+1}=\frac{1}{n!}\int\overline{k^{-}(\overline{\varphi})G_{n+1}}F_{n}\,\mathrm{d}\lambda_{n}.$

Summation over $n\in\mathbb{N}_{0}$ yields the required adjointness relation.
See also Śniady [52, Theorem 3] or Lytvynov [39, Section 2]. ∎

###  5.2. Baker-Campbell-Hausdorff formula. Proof of Theorem 3.4

First we check that the product of exponentials
$\exp(k^{+}(v))\exp(k^{0}(w))\exp(-k^{-}(v))$ in Theorem 3.4 is well defined
on $\mathcal{D}$.

######  Lemma 5.2.

Let $f\in\mathcal{D}$ and $v,w\in\mathcal{C}$. Then:

  1. (a)

$k^{-}(v)^{n}f$ vanishes for all except finitely many $n$, hence
$\exp(-k^{-}(v))f\in\mathcal{D}$.

  2. (b)

$\exp(k^{0}(w))f$ is in $\mathcal{D}$.

  3. (c)

If $\sup|v|<1$, then
$\sum_{n=0}^{\infty}\frac{1}{n!}\left(k^{+}(v)\right)^{n}f$ converges
absolutely in norm.

The condition $\sup|v|<\infty$ is satisfied for $|v|=\tanh|\varphi|$ as
$\varphi\in\mathcal{C}$ is bounded.

###### Proof.

It is enough to treat functions $f=k^{+}(\varphi_{1})\cdots
k^{+}(\varphi_{m})\Psi$ with $\varphi_{1},\ldots,\varphi_{m}\in\mathcal{C}$,
$m\in\mathbb{N}_{0}$. Item (a) of the lemma follows from
$\left(k^{-}(\varphi)\right)^{n}f=0$ for $n>m$. For item (b), notice that

$\exp\bigl{(}k^{0}(w)\bigr{)}k^{+}(\varphi_{1})\cdots
k^{+}(\varphi_{m})\Psi=\exp\Bigl{(}\frac{1}{2}\int
w\>\mathrm{d}\alpha\Bigr{)}k^{+}(\varphi_{1}e^{w})\cdots
k^{+}(\varphi_{m}e^{w})\Psi\in\mathcal{D}.$ (30)

For item (c), let $C>0$, $0<c<1$ and $B\in\mathcal{X}$ be such that
$\alpha(B)<\infty$ and ${\left|\varphi_{j}\right|}\leq C\mathbbm{1}_{B}$ for
$j\in{\left\\{1,\ldots,m\right\\}}$ and ${\left|v\right|}<c\mathbbm{1}_{B}$.
Then

$\bigl{(}k^{+}(v)^{n}f\bigr{)}_{\ell}(x_{1},\ldots,x_{\ell})\\\
=\delta_{n+m,\ell}\,\sum_{\sigma\in\mathfrak{S}_{n+m}}v(x_{1})\cdots
v(x_{n})\varphi_{\sigma(n+1)}(x_{n+1})\cdots\varphi_{\sigma(n+m)}(x_{n+m}).$

Thus,

$\left\lVert
k^{+}(v)^{n}f\right\rVert^{2}\leq\frac{1}{(n+m)!}\Bigl{(}(n+m)!c^{n}C^{m}\Bigr{)}^{2}\lambda_{n+m}(B)$

and

$\sum_{n=0}^{\infty}\frac{1}{n!}\left\lVert(k^{+}(v))^{n}f\right\rVert\leq
C^{m}\sum_{n=0}^{\infty}\frac{c^{n}}{n!}\Bigl{(}(n+m)!(\alpha(B))_{(n+m)}\Bigr{)}^{1/2}<\infty.\qed$

For the proof of Theorem 3.4 we adapt Truax [56], with extra care for
unbounded operators. For $t\geq 0$, set

$\displaystyle v_{t}:=\mathbbm{1}_{\\{\varphi\neq
0\\}}\frac{\varphi}{{\left|\varphi\right|}}\tanh{t{\left|\varphi\right|}},\quad
w_{t}:=-2\log\cosh{t{\left|\varphi\right|}}$

and define

$\displaystyle
S_{t}f:=\exp\bigl{(}k^{+}(v_{t})\bigr{)}\exp\bigl{(}k^{0}(w_{t})\bigr{)}\exp\bigl{(}-k^{-}(v_{t})\bigr{)}f,\qquad
f\in\mathcal{D}.$

The operator $S_{t}$ is well-defined on $\mathcal{D}$ by Lemma 5.2. We show
that $t\mapsto S_{t}f$ is norm-differentiable and solves the same differential
equation as $\exp(t(k^{+}(\varphi)-k^{-}(\varphi))f$.

######  Lemma 5.3.

The map $t\mapsto S_{t}f$ is norm-differentiable on $\mathbb{R}_{+}$, for
every $f\in\mathcal{D}$, with derivative

$\partial_{t}S_{t}f=\Big{(}\mathrm{e}^{k^{+}(v_{t})}k^{+}(\partial_{t}v_{t})\mathrm{e}^{k^{0}(w_{t})}\mathrm{e}^{-k^{-}(v_{t})}+\mathrm{e}^{k^{+}(v_{t})}k^{0}(\partial_{t}w_{t})\mathrm{e}^{k^{0}(w_{t})}\mathrm{e}^{-k^{-}(v_{t})}\\\
-\mathrm{e}^{k^{+}(v_{t})}\mathrm{e}^{k^{0}(w_{t})}k^{-}(\partial_{t}v_{t})\mathrm{e}^{-k^{-}(v_{t})}\Big{)}f.$
(31)

Note that $\partial_{t}v_{t}$, $\partial_{t}w_{t}\in\mathcal{C}$. The proof of
the lemma is based on straightforward but somewhat tedious bounds and it is
omitted.

The following commutation relations help us express the right side of (31)
with $k^{+}(\varphi)$ and $k^{-}(\varphi)$ only. We remark that
$k^{+}(\theta)$, $k^{0}(\theta)$ and $k^{-}(\theta)$ are closable because they
have densely defined adjoints. We use the same letters for the closed
operators.

######  Lemma 5.4.

For all $v,w,\theta\in\mathcal{C}$ with $\sup|v|<\infty$, and all
$f\in\mathcal{D}$, the following holds true:

  1. (a)

$[k^{0}(\theta),\exp(k^{+}(v))]f=\exp(k^{+}(v))k^{+}(\theta v)f$.

  2. (b)

$\exp(k^{+}(v))f$ is in the domain of the closure of $k^{-}(\theta)$ and

$\bigl{[}k^{-}(\theta),\mathrm{e}^{k^{+}(v)}\bigr{]}f=\bigl{(}-k^{+}(\overline{\theta}v^{2})+2k^{0}(\overline{\theta}v)\bigr{)}\mathrm{e}^{k^{+}(v)}f.$

  3. (c)

$\mathrm{e}^{k^{0}(w)}k^{-}(\theta)f=k^{-}(\theta\mathrm{e}^{-\overline{w}})\mathrm{e}^{k^{0}(w)}f$.

###### Proof.

An induction over $n$, based on $[A,B^{n+1}]=[A,B]B^{n}+B[A,B^{n}]$ and the
commutation relations for $k^{0}(\theta)$ and $k^{+}(v)$, yields

$[k^{0}(\theta),k^{+}(v)^{n}]=nk^{+}(\theta v)k^{+}(v)^{n-1}$

from which one deduces

$[k^{0}(\theta),\exp(k^{+}(v))]=\exp(k^{+}(v))k^{+}(\theta v).$ (32)

Next we check by induction over $n$ that

$[k^{-}(\theta),k^{+}(v)^{n}]=n(n-1)k^{+}(\overline{\theta}v^{2})k^{+}(v)^{n-2}+2nk^{+}(v)^{n-1}k^{0}(\overline{\theta}v).$
(33)

For $n=1$ the previous relation reads
$[k^{-}(\theta),k^{+}(v)]=2k^{0}(\overline{\theta}v)$, which holds true. For
the induction step from $n$ to $n+1$, we write

$\displaystyle[k^{-}(\theta),k^{+}(v)^{n+1}]$
$\displaystyle=\bigl{[}k^{-}(\theta),k^{+}(v)\bigr{]}k^{+}(v)^{n}+k^{+}(v)\bigl{[}k^{0}(\theta),k^{+}(v)^{n}\bigr{]}$
$\displaystyle=2\Bigl{[}k^{0}(\overline{\theta}v),k^{+}(v)^{n}\Bigr{]}+2k^{+}(v)^{n}k_{0}(\overline{\theta}v)+k^{+}(v)\Bigl{[}k^{0}(\theta),k^{+}(v)^{n}\Bigr{]}$
$\displaystyle=2nk^{+}(\overline{\theta}v^{2})k^{+}(v)^{n-1}+2k^{+}(v)^{n}k_{0}(\overline{\theta}v)+k^{+}(v)\Bigl{[}k^{0}(\theta),k^{+}(v)^{n}\Bigr{]}$
$\displaystyle=n(n+1)k^{+}((\overline{\theta}v^{2})k^{+}(v)^{n-1}+2(n+1)k^{+}(v)^{n}k^{0}(v\overline{\theta}).$

The inductive proof of (33) is complete. Now let $f\in\mathcal{D}$ and
$N\in\mathbb{N}$. Eq. (33) yields

$k^{-}(\theta)\sum_{n=0}^{N}\frac{1}{n!}k^{+}(v)^{n}f\\\
=\Bigl{(}\sum_{n=0}^{N}\frac{1}{n!}k^{+}(v)^{n}\Bigr{)}k^{-}(\theta)f+\Bigl{(}\sum_{n=0}^{N-2}\frac{1}{n!}k^{+}(v)^{n}\Bigr{)}k^{+}(\overline{\theta}v^{2})f+2\Bigl{(}\sum_{n=0}^{N-1}\frac{1}{n!}k^{+}(v)^{n}\Bigr{)}k^{0}(\overline{\theta}v)f.$

The right side converges in norm as $N\to\infty$ by Lemma 5.2, hence so does
the left side. It follows that $\exp(k^{+}(v))f$ is in the domain of the
closure of $k^{-}(\theta)$ and we obtain

$\bigl{[}k^{-}(\theta),\mathrm{e}^{k^{+}(v)}\bigr{]}f=\mathrm{e}^{k^{+}(v)}\bigl{(}k^{+}(\overline{\theta}v^{2})+2k^{0}(\overline{\theta}v)\bigr{)}f.$

In view of (32), we may also write

$\bigl{[}k^{-}(\theta),\mathrm{e}^{k^{+}(v)}\bigr{]}f=\bigl{(}-k^{+}(\overline{\theta}v^{2})+2k^{0}(\overline{\theta}v)\bigr{)}\mathrm{e}^{k^{+}(v)}f.$

For (c) it is enough to consider functions $f=k^{+}(\varphi_{1})\cdots
k^{+}(\varphi_{m})\Psi$ for $m\in\mathbb{N}$,
$\varphi_{1},\ldots,\varphi_{m}\in\mathcal{C}$. We write down the proof for
$m=2$, the general case is similar. We have

$\displaystyle k^{-}(\theta)f$
$\displaystyle=k^{-}(\theta)k^{+}(\varphi_{1})k^{+}(\varphi_{2})\Psi$
$\displaystyle=\langle\theta,\varphi_{1}\rangle
k^{+}(\varphi_{2})\Psi+\langle\theta,\varphi_{2}\rangle
k^{+}(\varphi_{1})\Psi+2k^{+}(\varphi_{1}\overline{\theta}\varphi_{2})\Psi.$

By (30),

$\mathrm{e}^{k^{0}(w)}k^{-}(\theta)f=\mathrm{e}^{\int
w\mathrm{d}\alpha/2}\Bigl{(}\langle\theta,\varphi_{1}\rangle
k^{+}(\varphi_{2}\mathrm{e}^{w})\Psi+\langle\theta,\varphi_{2}\rangle
k^{+}(\varphi_{1}\mathrm{e}^{w})\Psi+2k^{+}(\varphi_{1}\overline{\theta}\varphi_{2}\mathrm{e}^{w})\Psi\Bigr{)}.$

Similarly,

$k^{-}(\psi)\mathrm{e}^{k^{0}(w)}f\\\ =\mathrm{e}^{\int
w\mathrm{d}\alpha/2}\Bigl{(}\langle\psi,\mathrm{e}^{w}\varphi_{1}\rangle
k^{+}(\varphi_{2}\mathrm{e}^{w})\Psi+\langle\psi,\mathrm{e}^{w}\varphi_{2}\rangle
k^{+}(\varphi_{1}\mathrm{e}^{w})\Psi+2k^{+}(\varphi_{1}\mathrm{e}^{w}\overline{\psi}\varphi_{2}\mathrm{e}^{w})\Psi\Bigr{)}.$

The expressions are equal for $\psi=\exp(-\overline{w})\theta$. ∎

######  Lemma 5.5.

Let $f\in\mathcal{D}$. Then, $S_{t}f$ is in the domain of the closure of
$k^{+}(\varphi)-k^{-}(\varphi)$, denoted by $k^{+}(\varphi)-k^{-}(\varphi)$ as
well. Moreover,

$\partial_{t}S_{t}f=\left(k^{+}(\varphi)-k^{-}(\varphi)\right)S_{t}f.$

###### Proof.

In Lemma 5.4(b) we proved that $\exp(k^{+}(v))f$ is in the domain of the
closure of $k^{-}(\theta)$ by proving that
$k^{-}(\theta)\sum_{n=0}^{N}n!^{-1}k^{+}(v)^{n}f$ converges in norm as
$N\to\infty$. The same argument shows that $\exp(k^{+}(v))f$ also belongs to
the domain of the closure of $k^{+}(\varphi)-k^{-}(\varphi)$. As
$\exp(k^{0}(w_{t}))$ and $\exp(-k^{-}(v_{t}))$ map $\mathcal{D}$ to
$\mathcal{D}$ by Lemma 5.2, it follows that $S_{t}f$ is in th domain of the
closure, for all $f\in\mathcal{D}$. By Lemma 5.3, the derivative
$\partial_{t}S_{t}f$ is a sum of three terms. The first is
$k^{+}(\partial_{t}v_{t})S_{t}f$. The second term is

$\mathrm{e}^{k^{+}(v_{t})}k^{0}(\partial_{t}w_{t})\mathrm{e}^{k^{0}(w_{t})}\mathrm{e}^{-k^{-}(v_{t})}f=\bigl{(}k^{0}(\partial_{t}w_{t})-k^{+}(v_{t}\partial_{t}w_{t})\bigr{)}S_{t}f,$

where we have used Lemma 5.4(a). The third term is

$\displaystyle-\mathrm{e}^{k^{+}(v_{t})}\mathrm{e}^{k^{0}(w_{t})}k^{-}(\partial_{t}v_{t})\mathrm{e}^{-k^{-}(v_{t})}f$
$\displaystyle\qquad=-\mathrm{e}^{k^{+}(v_{t})}k^{-}(\mathrm{e}^{-w_{t}}\partial_{t}v_{t})\mathrm{e}^{k^{0}(w_{t})}\mathrm{e}^{-k^{-}(v_{t})}f$
$\displaystyle\qquad=\Bigl{(}-k^{+}(\mathrm{e}^{-w_{t}}\overline{\partial_{t}v_{t}}v_{t}^{2})+2k^{0}(\mathrm{e}^{-w_{t}}\overline{\partial
v_{t}}vt)-k^{-}(\mathrm{e}^{-w_{t}}\partial_{t}v_{t})\Bigr{)}S_{t}f.$

This time we have used Lemma 5.4(c) and then (b). Altogether, we get

$\partial_{t}S_{t}f=\Bigl{(}k^{+}\left(\partial_{t}v_{t}-(\partial_{t}w_{t})v_{t}-v_{t}^{2}\overline{(\partial_{t}v_{t})}\right)\\\
+k^{0}\left(\partial_{t}w_{t}+2v_{t}\overline{(\partial_{t}v_{t})}\right)-k^{-}\left((\partial_{t}v_{t})e^{-w_{t}}\right)\Bigr{)}S_{t}f.$

The proof is concluded with the differential equations

$\displaystyle\partial_{t}v_{t}-(\partial_{t}w_{t})v_{t}-v_{t}^{2}\overline{(\partial_{t}v_{t})}e^{-w_{t}}$
$\displaystyle=\varphi,$
$\displaystyle\partial_{t}w_{t}+2v_{t}\overline{(\partial_{t}v_{t})}e^{-w_{t}}$
$\displaystyle=0,$ $\displaystyle(\partial_{t}v_{t})e^{-w_{t}}$
$\displaystyle=\varphi.\qed$

Now we have all the ingredients to prove the Baker-Campbell-Hausdorff formula.

###### Proof of Theorem 3.4.

As noted after Eq. (12), the closure $A$ of
$-\mathrm{i}(k^{+}(\varphi)-k^{-}(\varphi))$ is self-adjoint. By Stone’s
theorem, for every $f\in\mathcal{D}$, the function
$t\mapsto\exp(\mathrm{i}tA)f$ is norm-differentiable with derivative
$\partial_{t}\exp(\mathrm{i}tA)f=\mathrm{i}A\exp(\mathrm{i}tA)f$. Hence
$g_{t}:=\exp(t\left(k^{+}(\varphi)-k^{-}(\varphi)\right))f$ is norm-
differentiable with derivative

$\partial_{t}g_{t}=\left(k^{+}(\varphi)-k^{-}(\varphi)\right)g_{t}.$

The map $t\mapsto S_{t}f$ satisfies the same differential equation by Lemma
5.5, moreover $S_{0}f=g_{0}=f$, therefore $g_{t}=S_{t}f$ for all $t\geq 0$. In
particular, $t=1$ corresponds to the Baker-Campbell-Hausdorff formula. ∎

###  5.3. Gamma random measure. Proof of Theorem 3.7

First we prove that the space of polynomials $\mathbb{D}$ is dense in
$L^{2}(\mathbf{M},\rho_{\alpha})$. We actually prove a slightly stronger
statement: the space of polynomials in the masses $\mu(B)$, with
$\alpha(B)<\infty$, is dense.

###### Proof of Lemma 3.6.

Let $\mathscr{U}$ be the set of finite linear combinations of indicators
$\mathbbm{1}_{B}$, with non-negative coefficients, of sets with finite mass
$\alpha(B)<\infty$. For $u\in\mathscr{U}$, let $\mathscr{H}_{0}$ be the set of
exponential maps $F_{u}:\mathbf{N}\to\mathbb{R}_{+}$,
$\mu\mapsto\exp(-\mu(u))$. Further let $\mathscr{H}$ be the closure of
$\mathscr{H}_{0}$, with respect to the supremum norm, of the linear
combinations of exponentials. The set $\mathscr{H}_{0}$ is closed with respect
to pointwise multiplication and the vector space $\mathscr{H}$ contains the
constant function $1$ and is closed in $\mathscr{L}^{\infty}(\mathbf{M})$.
Therefore, by the functional monotone class theorem [11, Theorem 2.12.9],
$\mathscr{H}$ contains all bounded functions that are measurable with respect
to $\sigma(\mathscr{H}_{0})$. In view of $\mu(B)=-\log
F_{\mathbbm{1}_{B}}(\eta)$, the set $\mathscr{H}_{0}$ generates the full
$\sigma$-algebra on $\mathbf{M}$. Thus, $\mathscr{H}$ contains all bounded
measurable functions.

As a consequence, the linear combinations of exponentials
$\mu\mapsto\exp(-\mu(u))$ are dense in $\mathscr{L}^{\infty}(\mathbf{M})$ with
respect to uniform convergence, hence a fortiori in
$L^{2}(\mathbf{M},\rho_{\alpha})$ with respect to $L^{2}$-norm. To conclude,
we note that the exponential in turn can be approximated in $L^{2}$-norm by
polynomials: Simply consider a truncated exponential series and bound the
remainder, using that there exists $\varepsilon=\varepsilon_{u}>0$ such that
$\int\exp(\varepsilon\mu(u))\rho_{\alpha}(\mathrm{d}\mu)<\infty$.

We obtain that polynomials in the variables $\mu(u)$ are dense in
$L^{2}(\mathbf{M},\rho_{\alpha})$. Every polynomial in $\mu(u)$, with $u$ a
linear combination of indicators, is also a a polynomial in masses
$\mu(B_{i})$, $\alpha(B_{i})<\infty$. The lemma and the stronger claim stated
before this proof follow. ∎

Next we prove an identity for Gamma random measures that will help us prove
the adjointness relations (8). A straightforward integration by part shows
that for every polynomial $f:\mathbb{R}\to\mathbb{C}$, given $a>0$,

$\int_{0}^{\infty}xf^{\prime}(x)x^{a-1}\mathrm{e}^{-x}\mathrm{d}x=\int_{0}^{\infty}(x-a)f(x)x^{a-1}\mathrm{e}^{-x}\mathrm{d}x.$

The generalization for Gamma random measures is as follows. We write
$\mathbb{E}_{\rho_{\alpha}}[F(\mu)]=\int F(\mu)\rho_{\alpha}(\mathrm{d}\mu)$.

######  Proposition 5.6.

Let $\xi$ be a Gamma random measure with law $\rho_{\alpha}$. For all
$\varphi\in\mathcal{C}$ and all polynomials $F\in\mathbb{D}$,

$\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta
F}{\delta\mu(x)}(\mu)\,\mu(\mathrm{d}x)\Bigr{]}=\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\bigl{(}\mu(\varphi)-\alpha(\varphi)\bigr{)}F(\mu)\Bigr{]}.$

###### Proof.

We consider first exponential functions $F(\mu)=\exp(\mu(f))$ and exploit that
the Gamma random measure is compound Poisson. Let
$\Pi=\sum_{i}\delta_{(X_{i},Z_{i})}$ be a Poisson point process on
$\mathbb{X}\times\mathbb{R}_{+}$ with intensity measure
$\alpha(\mathrm{d}x)z^{-1}\exp(-z)\mathrm{d}z$ and
$\xi=\sum_{i}Z_{i}\delta_{X_{i}}$. By the Mecke equation for $\Pi$ [36],

$\mathbb{E}\Bigl{[}\sum_{i}G(X_{i},Z_{i};\Pi)\Bigr{]}=\int_{\mathbb{X}}\Bigl{\\{}\int_{0}^{\infty}\mathbb{E}\Bigl{[}G(x,z;\Pi+\delta_{(x,z)})\Bigr{]}z^{-1}\mathrm{e}^{-z}\mathrm{d}z\Bigr{\\}}\alpha(\mathrm{d}x)$

for all measurable
$G:\mathbb{X}\times\mathbb{R}_{+}\times\mathbf{N}(\mathbb{X}\times\mathbb{R}_{+})\to\mathbb{R}_{+}$.
Let $\varphi$ and $f$ be non-negative test functions in $\mathcal{C}$. We
assume that $||f||_{\infty}$ is sufficiently small so that
$\mathbb{E}_{\rho_{\alpha}}[\exp(2\mu(f))]<\infty$. The Mecke equation applied
to

$G\Bigl{(}x,z;\sum_{j}\delta_{(x_{j},z_{j})}\Bigr{)}=z\varphi(x)f(x)\exp\Bigl{(}\sum_{j}z_{j}f(x_{j})\Bigr{)}$

yields

$\mathbb{E}\Bigl{[}\xi(\varphi
f)\mathrm{e}^{\xi(f))}\Bigr{]}=\int_{\mathbb{X}}\Bigl{\\{}\int_{0}^{\infty}\mathbb{E}\Bigl{[}z\varphi(x)f(x)\mathrm{e}^{zf(x)}\mathrm{e}^{\xi(f)}\Bigr{]}z^{-1}\mathrm{e}^{-z}\mathrm{d}z\Bigr{\\}}\alpha(\mathrm{d}x).$
(34)

An integration by parts gives

$\int_{0}^{\infty}f(x)\mathrm{e}^{zf(x)}\mathrm{e}^{-z}\mathrm{d}z=-1+\int_{0}^{\infty}\mathrm{e}^{zf(x)}\mathrm{e}^{-z}\mathrm{d}z.$

We insert this identity on the right side in Eq. (34) (after changing the
order of integration). This gives rise to two terms. The first, coming from
$-1$, stays as is. For the second term, we apply again the Mecke equation. At
the end, we obtain

$\mathbb{E}\Bigl{[}\xi(\varphi
f)\mathrm{e}^{\xi(f))}\Bigr{]}=-\alpha(\varphi)\mathbb{E}\Bigl{[}\mathrm{e}^{\xi(f)}\Bigr{]}+\mathbb{E}\Bigl{[}\xi(\varphi)\mathrm{e}^{\xi(f)}\Bigr{]}.$

The same identity holds true for $f$ replaced by $sf$ with $s\in(-1,1)$. We
expand in powers of $s$, equate coefficients on the left and right side, and
obtain

$\mathbb{E}\Bigl{[}\xi(\varphi
f)n\xi(f)^{n-1}\Bigr{]}=\mathbb{E}\Bigl{[}\bigl{(}\xi(\varphi)-\alpha(\varphi)\bigr{)}\xi(f)^{n}\Bigr{]}.$

The compound Poisson process $\xi$ has law $\rho_{\alpha}$ and every
polynomial is a linear combination of monomials $\mu\mapsto\mu(f)^{n}$. It
follows that

$\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\mu(\varphi
f)P^{\prime}\bigl{(}\mu(f)\bigr{)}\Bigr{]}=\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\bigl{(}\mu(\varphi)-\alpha(\varphi)\bigr{)}P\bigl{(}\mu(f)\bigr{)}\Bigr{]}$

which is precisely the statement that we were looking for. ∎

Let us show right away how Proposition 5.6 implies the adjointness relations.

######  Lemma 5.7.

The operators $K^{\\#}(\varphi)$ defined in Section 3.3 satisfy the
adjointness relations (8).

###### Proof.

Pick $f,\varphi\in\mathcal{C}$ and $P$ a univariate polynomial. Set
$F(\mu)=P(\mu(f))$ and $G(\mu)=P^{\prime}(\mu(f))$. Notice that

$\frac{\delta^{2}F}{\delta\mu(x)^{2}}(\mu)=f(x)^{2}P^{\prime\prime}\bigl{(}\mu(x)\bigr{)}=f(x)\frac{\delta
G}{\delta\mu(x)}(\mu)$

Proposition 5.6 applied to $\psi=\varphi f$ and $G$ thus yields

$\displaystyle\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta^{2}F}{\delta\mu(x)^{2}}(\mu)\mu(\mathrm{d}x)\Bigr{]}$
$\displaystyle=\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\bigl{(}\mu(\varphi
f)-\alpha(\varphi f)\bigr{)}P^{\prime}\bigl{(}\mu(f)\bigr{)}\Bigr{]}$
$\displaystyle=\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta
F}{\delta\mu(x)}(\mu)\,(\mu-\alpha)(\mathrm{d}x)\Bigr{]},$ (35)

The statement extends, by linearity, to all polynomials $F\in\mathbb{D}$, and
it is equivalent to

$\mathbb{E}_{\rho_{\alpha}}\Bigl{[}k^{-}(\varphi)F(\mu)\Bigr{]}=\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta
F}{\delta\mu(x)}(\mu)\mu(\mathrm{d}x)\Bigr{]}=\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\bigl{(}\mu(\varphi)-\alpha(\varphi)\bigr{)}F(\mu)\Bigr{]}.$
(36)

(For the last equality we have used again Proposition 5.6.)

Now let $F$ and $G$ be two polynomials in $\mathbb{D}$ and $\varphi$ be a test
function in $\mathcal{C}$. We apply Eq. (36) to the function $F\overline{G}$,
keep in mind the product rule for the variational derivatives
$\frac{\delta}{\delta\mu(x)}$, and get

$\langle G,k^{-}(\varphi)F\rangle+\langle
k^{-}(\overline{\varphi})G,F\rangle+2\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta
F}{\delta\mu(x)}\frac{\delta\overline{G}}{\delta\mu(x)}(\mu)\mu(\mathrm{d}x)\Bigr{]}\\\
=\langle\bigl{(}\mu(\overline{\varphi})-\alpha(\overline{\varphi})\bigr{)}G,F\rangle.$
(37)

On the other hand Proposition 5.6 implies

$\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\overline{G(\mu)}\frac{\delta
F}{\delta\mu(x)}(\mu)\mu(\mathrm{d}x)\Bigr{]}=-\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta\overline{G}}{\delta\mu(x)}(\mu)F(\mu)\mu(\mathrm{d}x)\Bigr{]}\\\
+\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\bigl{(}\mu(\varphi)-\alpha(\varphi)\bigr{)}\overline{G}F(\mu)\Bigr{]}$

and then, by an argument similar to (35),

$\displaystyle\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta\overline{G}}{\delta\mu(x)}\frac{\delta
F}{\delta\mu(x)}(\mu)\mu(\mathrm{d}x)\Bigr{]}$
$\displaystyle\quad=-\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta^{2}\overline{G}}{\delta\mu(x)}(\mu)F(\mu)\mu(\mathrm{d}x)\Bigr{]}+\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta\overline{G}}{\delta\mu(x)}(\mu)F(\mu)(\mu-\alpha)(\mathrm{d}x)\Bigr{]}$
$\displaystyle\quad=-\langle
k^{-}(\overline{\varphi})G,F\rangle+\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta\overline{G}}{\delta\mu(x)}(\mu)F(\mu)\mu(\mathrm{d}x)\Bigr{]}.$

We combine this equation with (37) and get

$\langle G,k^{-}(\varphi)F\rangle=\langle
k^{-}(\overline{\varphi})G,F\rangle-2\mathbb{E}_{\rho_{\alpha}}\Bigl{[}\int\varphi(x)\frac{\delta\overline{G}}{\delta\mu(x)}(\mu)F(\mu)\mu(\mathrm{d}x)]\\\
+\langle\bigl{(}\mu(\overline{\varphi})-\alpha(\overline{\varphi})\bigr{)}G,F\rangle.$
(38)

The right side is precisely $\langle K^{+}(\overline{\varphi})F,G\rangle$.
Thus we have proven the adjointness relation for the raising and lowering
operators. The symmetry of the neutral operators is proven by similar
computations that we omit. ∎

Now we have all we need to prove Theorem 3.7.

###### Proof of Theorem 3.7.

(i) The set of polynomials $\mathbb{D}$ is dense in
$L^{2}(\mathbf{M},\rho_{\alpha})$ by Lemma 3.6 and one easily checks that
$K^{\pm}(\varphi)$ and $K^{0}(\varphi)$ map polynomials to polynomials.

(ii) Cleary $K^{-}(\varphi)$ is antilinear in $\varphi$ and $K^{0}(\varphi)$
and $K^{+}(\varphi)$ are linear in $\varphi$.

(iii) It is enough to check the commutation relations on monomials
$F(\mu)=\mu(f)^{n}$ (every other monomial is a linear combination of such
simple powers, e.g., $\mu(f)\mu(g)=\frac{1}{4}(\mu(f+g)^{2}-\mu(f-g)^{2})$).
Thus, let $F(\mu)=P(\mu(f))$ with $P(x)=x^{n}$.

We show first that the lowering operators commute. Explicit computations yield

$K^{-}(\varphi)F(\mu)=\mu(\overline{\varphi}f^{2})P^{\prime\prime}\bigl{(}\mu(f)\bigr{)}+\alpha(\overline{\varphi}f)P^{\prime}\bigl{(}\mu(f)\bigr{)}$

and

$\displaystyle K^{-}(\theta)K^{-}(\varphi)F(\mu)=$
$\displaystyle\mu(\overline{\theta}f^{2})\mu(\overline{\varphi}f^{2})P^{(4)}\bigl{(}\mu(f)\bigr{)}$
$\displaystyle+\Bigl{(}2\mu(\overline{\varphi}\overline{\theta}f^{3})+\alpha(\overline{\varphi}f)\mu(\overline{\theta}f^{2})+\alpha(\overline{\theta}f)\mu(\overline{\varphi}f^{2})\Bigr{)}P^{(3)}\bigl{(}\mu(f)\bigr{)}$
$\displaystyle+\Bigl{(}\alpha(\overline{\theta}\overline{\varphi}f^{2})+\alpha(\overline{\theta}f)\alpha(\overline{\varphi}f)\Bigr{)}P^{\prime\prime}\bigl{(}\mu(f)\bigr{)}.$

The right side is symmetric in $\theta$ and $\varphi$, hence
$[K^{-}(\theta),K^{-}(\varphi)]F=0$. Turning to the commutation relation
between the lowering and the neutral operators, we note that
$K^{0}(\varphi)=-K^{-}(\overline{\varphi})+R(\varphi)+\frac{1}{2}\alpha(\varphi)$
with

$R(\varphi)F(\mu)=\int\varphi(x)\frac{\delta
F}{\delta\mu(x)}\mu(\mathrm{d}x)=\mu(\varphi
f)P^{\prime}\bigl{(}\mu(f)\bigr{)}.$

To lighten notation, we drop the variables $\mu(f)$ from the polynomial $P$
and its derivatives in the final expressions. We compute

$\displaystyle R(\varphi)K^{-}(\theta)F(\mu)$
$\displaystyle=\int\varphi(x)\frac{\delta}{\delta\mu(x)}\Bigl{(}\mu(\overline{\theta}f^{2})P^{\prime\prime}\bigl{(}\mu(f)\bigr{)}+\alpha(\overline{\theta}f)P^{\prime}\bigl{(}\mu(f)\bigr{)}\Bigr{)}\mu(\mathrm{d}x)$
$\displaystyle=\mu(\varphi\overline{\theta}f^{2})P^{\prime\prime}+\mu(\overline{\theta}f^{2})\mu(\varphi
f)P^{(3)}+\alpha(\overline{\theta}f)\mu(\varphi f)P^{\prime\prime}$

and

$\displaystyle K^{-}(\theta)R(\varphi)F(\mu)$
$\displaystyle=\int\overline{\theta(x)}\frac{\delta^{2}}{\delta\mu(x)^{2}}\mu(\varphi
f)P^{\prime}\bigl{(}\mu(f)\bigr{)}\mu(\mathrm{d}x)$
$\displaystyle\qquad\qquad+\int\overline{\theta(x)}\frac{\delta}{\delta\mu(x)}\mu(\varphi
f)P^{\prime}\bigl{(}\mu(f)\bigr{)}\alpha(\mathrm{d}x)$
$\displaystyle=2\mu(\overline{\theta}\varphi
f^{2})P^{\prime\prime}+\mu(\varphi f)\mu(\overline{\theta}f^{2})P^{(3)}$
$\displaystyle\qquad\qquad+\alpha(\overline{\theta}\varphi
f)P^{\prime}+\alpha(\overline{\theta}f)\mu(\varphi f)P^{\prime\prime}.$

It follows that

$\displaystyle[K^{0}(\varphi),K^{-}(\theta)]F(\mu)$
$\displaystyle=[R(\varphi),K^{-}(\theta)]F(\mu)$
$\displaystyle=-\mu(\overline{\theta}\varphi
f^{2})P^{\prime\prime}-\alpha(\overline{\theta}\varphi)P^{\prime}$
$\displaystyle=-K^{-}(\theta\overline{\varphi})F(\mu).$

The other commutation relations are proven by similar computations that we
omit.

(iv) The adjointness relations have been proven in Lemma 5.7.

(v) The constant function $1$ has norm $1$ in
$L^{2}(\mathbf{M},\rho_{\alpha})$ because $\rho_{\alpha}$ is a probability
measure. It is annihilated by the lowering operators because the latter are
differential operators.

(vi) We show first that raising operators yield Laguerre polynomials. Let
$\ell\in\mathbb{N}$ and $B_{1},\ldots,B_{\ell}\in\mathcal{X}$ be disjoint sets
with $\alpha(B_{i})<\infty$. Further let $n_{1},\ldots,n_{\ell}\in\mathbb{N}$.
Remember the univariate operators $K^{+}$ from Proposition 4.1. The operator
used a scalar $\alpha>0$ as a parameter, but now $\alpha$ is a measure; we
define $K_{i}^{+}$ as the univariate operator with scalar $\alpha(B_{i})>0$.
Write $\mathbf{1}$ for the constant function $\mathbb{N}_{0}\ni n\mapsto 1$.
An induction over $n_{1}+\cdots+n_{\ell}$ yields

$K^{+}(\mathbbm{1}_{B_{1}})^{n_{1}}\cdots
K^{+}(\mathbbm{1}_{B_{\ell}})^{n_{\ell}}\mathbbm{1}(\eta)=\prod_{i=1}^{\ell}\bigl{(}(-K_{i}^{+})^{n_{i}}\mathbf{1}\bigr{)}\bigl{(}\mu(B_{i})\bigr{)}.$

The minus sign arises because we made slightly different choices when defining
$K^{\\#}$ and $K^{\\#}(\varphi)$: In the univariate case, we prioritized the
definition through unitary equivalence, for the current algebra we changed
signs so that $K^{+}(\mathbbm{1}_{B})^{n}\mathbbm{1}$ is a polynomial with
positive leading coefficient. Next we note

$(-K_{i}^{+})^{n_{i}}(\mu(B_{i}))^{n_{i}}=(-1)^{n_{i}}n_{i}!L_{n_{i}}^{(\alpha(B_{i})-1)}\bigl{(}\mu(B_{i})\bigr{)}=\mathscr{L}_{n_{i}}^{(\alpha(B_{i})-1)}\bigl{(}\mu(B_{i})\bigr{)}.$

Eq. (19) in Theorem 3.7 follows. In particular, the linear hull of
$\mathbbm{1}$ and the iterates
$\prod_{i=1}^{\ell}K^{+}(\mathbbm{1}_{B_{i}})^{n_{i}}\mathbbm{1}$ contains all
polynomials in occupation numbers $\mu(B)$, where $B$ runs through the sets of
finite measure $\alpha(B)<\infty$. As noted at the beginning of this section,
these polynomials are dense in $L^{2}(\mathbf{M},\rho_{\alpha})$, therefore
the constant function $1$ is cyclic. ∎

###  5.4. Pascal point process. Proof of Theorem 3.9

For the commutation relations for $K^{\pm}(\varphi)$, $K^{+}(\theta)$, it is
convenient to introduce another family of operators $k^{\\#}(\varphi)$,
$\varphi\in\mathcal{C}$, defined on the space $\mathbb{D}$ of polynomials by

$\displaystyle k^{-}(\varphi)F(\eta)$
$\displaystyle=\frac{1}{\sqrt{p}}\int\overline{\varphi(x)}F(\eta+\delta_{x})(\alpha+\eta)(\mathrm{d}x),$
$\displaystyle k^{+}(\varphi)F(\eta)$
$\displaystyle=\sqrt{p}\int\varphi(x)F(\eta-\delta_{x})\eta(\mathrm{d}x),$
$\displaystyle k^{0}(\varphi)F(\eta)$
$\displaystyle=F(\eta)\int\varphi(x)\Bigl{(}\eta+\frac{1}{2}\alpha\Bigr{)}(\mathrm{d}x).$

These operators are similar to their counterparts on the extended Fock space.
Notice

$\displaystyle K^{-}(\varphi)$
$\displaystyle=\frac{1}{c}\left(\sqrt{p}\,k^{+}(\overline{\varphi})+\frac{1}{\sqrt{p}}\,k^{-}(\varphi)-2k^{0}(\overline{\varphi})\right),$
$\displaystyle K^{+}(\varphi)$
$\displaystyle=\frac{1}{c}\left(\frac{1}{\sqrt{p}}\,k^{+}(\varphi)+\sqrt{p}\,k^{-}(\overline{\varphi})-2k^{0}(\varphi)\right),$
$\displaystyle K^{0}(\varphi)$
$\displaystyle=\frac{1}{c}\left(-k^{+}(\varphi)-k^{-}(\overline{\varphi})+\bigl{(}{\sqrt{p}}^{-1}+\sqrt{p}\bigr{)}k^{0}(\varphi)\right).$

######  Lemma 5.8.

The operators $k^{\\#}(\varphi)$, $\varphi\in\mathcal{C}$, map $\mathbb{D}$ to
itself and on $\mathbb{D}$ they satisfy the commutation relations (7).

###### Proof.

Consider a monomial $F(\eta)=\eta(f_{1})\cdots\eta(f_{n})$, with
$n\in\mathbb{N}$ and $f_{1},\ldots,f_{n}\in\mathcal{C}$. Then

$\displaystyle k^{+}(\varphi)F(\eta)$
$\displaystyle=\sqrt{p}\int\varphi(x)\prod_{i=1}^{n}\bigl{(}\eta(f_{i})-f_{i}(x)\bigr{)}\eta(\mathrm{d}x)$
$\displaystyle=\sqrt{p}\sum_{I\subset[n]}(-1)^{n-|I|}\Biggl{(}\prod_{i\in
I}\eta(f_{i})\Biggr{)}\eta\Bigl{(}\varphi\prod_{i\in[n]\setminus
I}f_{i}\Bigr{)}$

where $[n]=\\{1,\ldots,n\\}$. It follows that $k^{+}(\varphi)F$ is a
polynomial. The constant function $\mathbbm{1}$ is mapped to
$\sqrt{p}\eta(\varphi)$, which is a polynomial as well. As every polynomial is
a linear combination of $\mathbbm{1}$ and monomials of degree $n\geq 1$, it
follows that $k^{+}(\varphi)\mathbb{D}\subset\mathbb{D}$. The reasoning for
$k^{-}(\varphi)$ and $k^{0}(\varphi)$ is similar and therefore omitted.

For the commutation relations, we note

$\displaystyle k^{-}(\varphi)k^{+}(\psi)F(\eta)$
$\displaystyle=\int\overline{\varphi(x)}\Bigl{(}\int\psi(y)F(\eta+\delta_{x}-\delta_{y})(\eta+\delta_{x})(\mathrm{d}y)\Bigr{)}(\alpha+\eta)(\mathrm{d}x),$
$\displaystyle k^{+}(\psi)k^{-}(\varphi)F(\eta)$
$\displaystyle=\int\psi(y)\Bigl{(}\int\overline{\varphi(x)}f(\eta+\delta_{x}-\delta_{y})(\alpha+\eta-\delta_{y})(\mathrm{d}x)\Bigr{)}\eta(\mathrm{d}y).$

When we subtract the second line from the first, everything cancels out except
the Dirac terms $\delta_{x}(\mathrm{d}y)$ and $-\delta_{y}(\mathrm{d}x)$ in
the inner integrals. Hence

$\displaystyle[k^{-}(\varphi),k^{+}(\psi)]F(\eta)$
$\displaystyle=\Bigl{(}\int\overline{\varphi(x)}\psi(x)(\alpha+\eta)(\mathrm{d}x)+\int\psi(y)\overline{\varphi(y)}\eta(\mathrm{d}y)\Bigr{)}F(\eta)$
$\displaystyle=2k^{0}(\overline{\varphi}\psi)F(\eta).$

The other commutation relations are proven by similar explicit computations,
we leave the details to the reader. ∎

######  Lemma 5.9.

The operators $k^{\\#}(\varphi)$ satisfy the adjointness relations (8) in
$L^{2}(\mathbf{N},\rho_{p,\alpha})$.

###### Proof.

The neutral operators $k^{0}(\varphi)$ is a multiplication operators which
multiplies with $\eta(\varphi)+\frac{1}{2}\alpha(\varphi)$. The adjoint
multiplies with the complex conjugate, i.e., with
$\eta(\overline{\varphi})+\frac{1}{2}\alpha(\overline{\varphi})$, hence
$\langle F,k^{0}(\varphi)G\rangle=\langle
k^{0}(\overline{\varphi})F,G\rangle$.

For $k^{\pm}(\varphi)$, we exploit the relation

$\iint F(x,\eta)\eta(\mathrm{d}x)\rho_{p,\alpha}(\mathrm{d}\eta)=\iint
F(x,\eta+\delta_{x})p(\alpha+\eta)(\mathrm{d}x)\rho_{p,\alpha}(\mathrm{d}\eta).$
(39)

It is valid for all measurable
$F:\mathbf{N}\times\mathbb{X}\to\mathbb{R}_{+}$, see [19, Proposition 3.1].
Let $\varphi\in\mathcal{C}$ be non-negative and $f,g$ be non-negative
polynomials, e.g., $f(\eta)=\eta(f_{1})\cdots\eta(f_{n})$ with
$f_{i}:\mathbb{X}\to\mathbb{R}_{+}$ in $\mathcal{C}$, similarly for $g$. Eq.
(39) applied to $F(x,\eta)=\varphi(x)\overline{f(\eta+\delta_{x})}g(\eta)$
yields $\langle f,k^{+}(\varphi)g\rangle=\langle k^{-}(\varphi)f,g\rangle$,
compare [19, Lemma 3.2]. The relation is extended to all polynomials and all
$\varphi\in\mathcal{C}$ by taking linear combinations. ∎

Now we are equipped for the proof of Theorem 3.9.

###### Proof of Theorem 3.9.

(i) The space $\mathbb{D}$ of polynomials is dense in
$L^{2}(\mathbf{N},\rho_{p,\alpha})$ by Lemma 3.8 and the operators
$K^{\\#}(\varphi)$, as linear combinations of the operators
$k^{\\#}(\varphi)$, map $\mathbb{D}$ to itself by Lemma 5.8.

(ii) Clearly $K^{+}(\varphi)$ and $K^{0}(\varphi)$ are linear in $\varphi$ and
$K^{-}(\varphi)$ is antilinear in $\varphi$.

(iii) The commutation relations are deduced from Lemma 5.8 as follows. The
commutator $[K^{-}(\varphi),K^{+}(\theta)]$, times $c^{2}$, is a sum of three
terms. First,

$\Bigl{[}\sqrt{p}\,k^{+}(\overline{\varphi})+\frac{1}{\sqrt{p}}k^{-}(\varphi),\frac{1}{\sqrt{p}}\,k^{+}(\theta)+\sqrt{p}\,k^{-}(\overline{\theta})\Bigr{]}=-2pk^{0}(\overline{\varphi}\theta)+\frac{2}{p}\,k^{0}(\overline{\varphi}\theta).$

Second,

$\Bigl{[}\sqrt{p}\,k^{+}(\overline{\varphi})+\frac{1}{\sqrt{p}}k^{-}(\varphi),-2k^{0}(\theta)\Bigr{]}=2\sqrt{p}\,k^{+}(\overline{\varphi}\theta)-\frac{2}{\sqrt{p}}k^{-}(\varphi\overline{\theta}).$

Third,

$\Bigl{[}-2k^{0}(\overline{\varphi}),\frac{1}{\sqrt{p}}\,k^{+}(\theta)+\sqrt{p}\,k^{-}(\overline{\theta})\Bigr{]}=-\frac{2}{\sqrt{p}}\,k^{+}(\overline{\varphi}\theta)+2\sqrt{p}\,k^{-}({\varphi}\overline{\theta}).$

The sum of these three expressions is

$-2c\bigl{(}k^{+}(\overline{\varphi}\theta)+k^{-}({\varphi}\overline{\theta})\bigr{)}+2c\bigl{(}\sqrt{p}^{-1}+\sqrt{p})\bigr{)}k^{0}(\overline{\varphi}\theta\bigr{)}=2c^{2}K^{0}(\overline{\varphi}\theta).$

This completes the proof of
$[K^{-}(\varphi),K^{+}(\theta)]=2K^{0}(\overline{\varphi}\theta)$. The proof
of the other commutation relations is similar.

(iv) The adjointness relations for $K^{\\#}(\varphi)$ follow from the
relations for $k^{\\#}(\varphi)$, proven in Lemma 5.9.

(v) The constant function $\Psi=\mathbbm{1}$ has norm $1$ in
$L^{2}(\mathbf{N},\rho_{p,\alpha})$ because $\rho_{p,\alpha}$ is a probability
measure. The function $\Psi=\mathbbm{1}$ is annihilated by all lowering
operators $K^{-}(\varphi)$ because
$\mathrm{D}^{+}_{x}\mathbbm{1}(\eta)=\mathrm{D}_{x}^{-}\mathbbm{1}(\eta)=0$
for all $x,\eta$.

(vi) We show first that raising operators yield Meixner polynomials. Let
$\ell\in\mathbb{N}$ and $B_{1},\ldots,B_{\ell}\in\mathcal{X}$ be disjoint sets
with $\alpha(B_{i})<\infty$. Further let $n_{1},\ldots,n_{\ell}\in\mathbb{N}$.
Remember the univariate operators $K^{+}$ from Proposition 4.2. The operator
used a scalar $\alpha>0$ as a parameter, but now $\alpha$ is a measure; we
define $K_{i}^{+}$ as the univariate operator with scalar $\alpha(B_{i})>0$.
Write $\mathbf{1}$ for the constant function $\mathbb{N}_{0}\ni n\mapsto 1$.
An induction over $n_{1}+\cdots+n_{\ell}$ yields

$K^{+}(\mathbbm{1}_{B_{1}})^{n_{1}}\cdots
K^{+}(\mathbbm{1}_{B_{\ell}})^{n_{\ell}}\mathbbm{1}(\eta)=\prod_{i=1}^{\ell}\bigl{(}(-K_{i}^{+})^{n_{i}}\mathbf{1}\bigr{)}\bigl{(}\eta(B_{i})\bigr{)}.$

The minus sign arises because we made slightly different choices when defining
$K^{\\#}$ and $K^{\\#}(\varphi)$: In the univariate case, we prioritized the
definition through unitary equivalence, for the current algebra we changed
signs so that $K^{+}(\mathbbm{1}_{B})^{n}\mathbbm{1}$ is a polynomial with
positive leading coefficient. By Eq. (28),

$(-K_{i}^{+})^{n_{i}}(\eta(B_{i}))=(-1)^{n_{i}}\bigl{(}\sqrt{p}-{\sqrt{p}}^{-1}\bigr{)}^{n_{i}}\mathcal{M}_{n_{i}}\bigl{(}\eta(B_{i});\alpha(B_{i}),p\bigr{)}.$

The prefactor in front of the Meixner polynomial is precisely $c^{n_{i}}$ and
Eq. (23) in Theorem 3.9 follows. In particular, the linear hull of
$\mathbbm{1}$ and the iterates
$\prod_{i=1}^{\ell}K^{+}(\mathbbm{1}_{B_{i}})^{n_{i}}\mathbbm{1}$ contains all
polynomials in occupation numbers $\eta(B)$, where $B$ runs through the sets
of finite measure. By an argument completely analogous to the proof of Lemma
3.6 given in Section 5.3, this space is dense in
$L^{2}(\mathbf{N},\rho_{p,\alpha})$. Thus $\Psi=\mathbbm{1}$ is cyclic. ∎

##  Appendix A Representation in standard Fock space. Araki’s scheme

Araki’s factorizable representations of a current group $G$ are of the form
[5]

$\mathcal{U}_{g}\mathscr{E}_{f}=\exp(\mathrm{i}c_{g})\exp\Bigl{(}-\frac{1}{2}||\varphi_{g}||^{2}+\langle\varphi_{g},f\rangle\Bigr{)}\mathscr{E}_{Q_{g}(f-\varphi_{g})}.$

The unitaries $\mathcal{U}_{g}$ operate in a Fock space $\mathfrak{H}$ for
which the single-particle Hilbert space has the structure of a fiber integral
$\mathfrak{h}_{1}=\int^{\oplus}\mathfrak{h}\mathrm{d}\alpha$. The exponential
states $\mathscr{E}_{f}$ are dense. The unitaries $Q_{g}$ form a
representation of $G$ in the one-particle Hilbert space $\mathfrak{h}_{1}$,
they are diagonal with respect to the fiber decomposition. The phase
$c_{g}\in\mathbb{R}$ and the map $g\mapsto\varphi_{g}\in\mathfrak{h}_{1}$ have
to satisfy some consistency conditions (cocycle conditions). The difficulty in
Araki’s approach is to find cocycles.

Here we sketch how our representation in extended Fock space on $\mathbb{X}$
may be lifted to a representation in a Fock space on
$\mathbb{X}\times\mathbb{N}$ and how it fits Araki’s scheme.

###  A.1. Isometric embedding into a standard Fock space. Chinese restaurant
process

We start with a known isomorphism of the extended Fock space with a subspace
of a standard Fock space [33, 38]. Let $\mathfrak{H}$ be a standard Fock space
for the one-particle Hilbert space
$L^{2}(\mathbb{X}\times\mathbb{N},\alpha\otimes\sum_{n=1}^{\infty}\frac{1}{n}\delta_{n})$.
We include a factorial in the norm, i.e., the norm of an element
$\boldsymbol{F}=(F_{m})_{m\in\mathbb{N}_{0}}$ of $\mathfrak{H}$ is

$||\boldsymbol{F}||^{2}=|F_{0}|^{2}\\\
+\sum_{m=1}^{\infty}\frac{1}{m!}\int\sum_{n_{1},\ldots,n_{m}}\bigl{|}F_{m}\bigl{(}(x_{1},n_{1}),(x_{2},n_{2}),\ldots(x_{m},n_{m})\bigr{)}\bigr{|}^{2}\frac{1}{n_{1}}\cdots\frac{1}{n_{m}}\,\alpha^{m}(\mathrm{d}\boldsymbol{x}).$

In the picturesque language of the _Chinese restaurant process_ [45, Chapter
3.1], a configuration $(x_{1},n_{1}),\ldots,(x_{m},n_{m})$ represents a
seating plan in a restaurant with $m$ tables; table no. $i$ is located at
place $x_{i}$ and has $n_{i}$ customers. (More prosaically, think of $m$ piles
with $n_{i}$ particles sitting atop each other on a location $x_{i}$.) The
total number of customers (particles) is $n=n_{1}+\cdots+n_{m}$ and we may
list the locations of the customers by repeating the entry $x_{i}$, for each
$i=1,\ldots,m$, $n_{i}$ times:

$(x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{m}^{n_{m}})=(x_{1},\ldots
x_{1},\ldots,x_{m},\ldots,x_{m})\in\mathbb{X}^{n_{1}+\cdots+n_{m}}.$

This induces a mapping from the extended Fock space $\mathfrak{F}$ to the
Hilbert space $\mathfrak{H}$: Given an element
$\boldsymbol{f}=(f_{n})_{n\in\mathbb{N}_{0}}$ of $\mathfrak{F}$, we define a
new sequence $\boldsymbol{F}=(F_{m})_{m\in\mathbb{N}_{0}}$ by

$F_{0}=f_{0},\quad
F_{m}\bigl{(}(x_{1},n_{1}),\ldots,(x_{m},n_{m})\bigr{)}=f_{\sum
n_{i}}(x_{1}^{n_{1}}\cdots x_{m}^{n_{m}}).$

The map $\mathfrak{U}:\mathfrak{F}\to\mathfrak{H}$,
$\boldsymbol{f}\mapsto\boldsymbol{F}$ is norm-preserving. It maps exponential
states to exponential states: The image of $\mathcal{E}_{z}\in\mathfrak{F}$ is
given by $F_{0}=0$ and

$F_{m}\bigl{(}(x_{1},n_{1}),\ldots,(x_{m},n_{m})\bigr{)}=\prod_{i=1}^{m}z(x_{i})^{n_{i}}=\prod_{i=1}^{m}f(x_{i},n_{i})$
(40)

with $f(x,n)=z(x)^{n}$. That is, $\mathfrak{U}\mathcal{E}_{z}=\mathscr{E}_{f}$
where $\mathscr{E}_{f}$ is an exponential state in $\mathfrak{H}$.

###  A.2. Lifting the representation

The operators $k^{\\#}(\varphi)$ from Section 3.1 are lifted to operators
$K^{\\#}(\varphi)$ in $\mathfrak{H}$ with
$\mathfrak{U}k^{\\#}(\varphi)=K^{\\#}(\varphi)\mathfrak{U}$, as follows. The
neutral operator in $\mathfrak{H}$ is a multiplication operator

$K^{0}(\varphi)F_{m}\bigl{(}(x_{1},n_{1}),\ldots,(x_{m},n_{m})\bigr{)}\\\
=\Bigl{(}\frac{1}{2}\alpha(\varphi)+\sum_{i=1}^{m}n_{i}\varphi(x_{i})\Bigr{)}F_{m}\bigl{(}(x_{1},n_{1}),\ldots,(x_{m},n_{m})\bigr{)}.$

The lowering operator is

$K^{-}(\varphi)F_{m}\bigl{(}(x_{1},n_{1}),\ldots,(x_{m},n_{m})\bigr{)}\\\
=\sum_{i=1}^{m}\overline{\varphi(x_{i})}n_{i}F_{m}\bigl{(}(x_{1},n_{1})\ldots(x_{i},n_{i}+1)\ldots(x_{m},n_{m})\bigr{)}\\\
+\int\overline{\varphi(x)}F_{m+1}\bigl{(}(x_{1},n_{1}),\ldots,(x_{m},n_{m}),(y,1)\bigr{)}\alpha(\mathrm{d}y)$

(a new customer may join a table or open a new table). The raising operator is

$K^{+}(\varphi)F_{m}\bigl{(}(x_{1},n_{1}),\ldots,(x_{m},n_{m})\bigr{)}\\\
=\sum_{i=1}^{m}\varphi(x_{i})n_{i}\Bigl{(}\mathbbm{1}_{\\{n_{i}\geq
2\\}}F_{m}\bigl{(}(x_{1},n_{1})\ldots(x_{i},n_{i}-1)\ldots(x_{m},n_{m})\bigr{)}\\\
+\mathbbm{1}_{\\{n_{i}=1\\}}F_{m-1}\bigl{(}(x_{1},n_{1})\ldots\widehat{(x_{i},1)}\ldots(x_{m},n_{m})\bigr{)}\Bigr{)}$

(if a solo diner leaves, the number $m$ of tables decreases). Exponentiating
as in Theorem 3.3, we should obtain a family of unitary operators in
$\mathfrak{H}$ and a representation of the current group for the universal
cover of $SU(1,1)$. The action of

$\mathcal{U}(\xi,\theta)=\exp\bigl{(}K^{+}(\xi)-K^{-}(\xi)\bigr{)}\exp\bigl{(}2\mathrm{i}K^{0}(\theta)\bigr{)}$

on exponential states $\mathscr{E}_{f}$ with $f(x,n)=z(x)^{n}$ (see (40)) is

$\mathcal{U}(\xi,\theta)\mathscr{E}_{f}=\mathcal{C}_{\xi,\theta}(f)\mathscr{E}_{\widehat{f}},\quad\widehat{f}(x,n)=z_{\xi,\theta}(x)^{n}$
(41)

with

$\mathcal{C}_{\xi,\theta}(f)=\exp\Bigl{(}\mathrm{i}\int\theta\mathrm{d}\alpha-\int\log\Bigl{(}\cosh|\xi|+\mathrm{e}^{2\mathrm{i}\theta}z\frac{\overline{\xi}}{|\xi|}\sinh|\xi|\Bigr{)}\mathrm{d}\alpha\Bigr{)}$
(42)

and

$z_{\xi,\theta}(x)=\frac{\mathrm{e}^{2\mathrm{i}\theta(x)}z(x)+\frac{\xi(x)}{|\xi(x)|}\tanh|\xi(x)|}{1+\mathrm{e}^{2\mathrm{i}\theta(x)}z(x)\frac{\overline{\xi(x)}}{|\xi(x)|}\tanh|\xi(x)|}.$
(43)

###  A.3. Matching the lifted representation to Araki’s scheme

We introduce a single-table, non-spatial Hilbert space
$\mathfrak{h}=\ell^{2}(\mathbb{N},\sum_{n=1}^{\infty}\frac{1}{n}\delta_{n})$,
and operators

$\mathrm{k}^{+}f(n)=n\mathbbm{1}_{\\{n\geq
2\\}}f(n-1),\quad\mathrm{k}^{-}f(n)=nf(n+1),\quad\mathrm{k}^{0}f(n)=nf(n).$

These operators satisfy the $su(1,1)$ commutation relations. The vacuum is
$\psi_{0}(n)=\delta_{n,1}$, the Bargmann index is $m_{0}=2$ (because
$\mathrm{k}^{0}\psi_{0}=\frac{2}{2}\psi_{0}$). Set

$Q_{\xi,\theta}=\exp(\xi\mathrm{k}^{+}-\overline{\xi}\mathrm{k}^{-})\exp(2\mathrm{i}\theta\mathrm{k}^{0}).$

We use the same letters for the operators in
$L^{2}(\mathbb{X}\times\mathbb{N},\alpha\otimes\sum_{n=1}^{\infty}\frac{1}{n}\delta_{n})$
that act on the $n$-variable only. Further set

$\varphi_{\xi,\theta}(x,n)=\Bigl{(}-\mathrm{e}^{2\mathrm{i}\theta}\frac{\xi}{|\xi|}\tanh|\xi|\Bigr{)}^{n}.$

######  Proposition A.1.

Let $\xi,\theta\in\mathcal{C}$ be complex- and real-valued functions in
$\mathcal{C}$, respectively, and $f\in
L^{2}(\mathbb{X}\times\mathbb{N},\alpha\otimes\sum_{n=1}^{\infty}\frac{1}{n}\delta_{n})$.
The unitary $\mathcal{U}(\xi,\theta)$ acts on the exponential states
$\mathscr{E}_{f}\in\mathfrak{H}$ as

$\mathcal{U}(\xi,\theta)\mathscr{E}_{f}=\exp\Bigl{(}\mathrm{i}\int\theta\mathrm{d}\alpha-\frac{1}{2}||\varphi_{\xi,\theta}||^{2}+\langle\varphi_{\xi,\theta},f\rangle\Bigr{)}\mathscr{E}_{Q_{\xi,\theta}(f-\varphi_{\xi,\theta})}.$
(44)

###### Proof sketch.

For $\xi\equiv 0$, the vector $\varphi_{\xi,\theta}$ vanishes and
$\exp(2\mathrm{i}\theta\mathrm{k}^{0})$ is multiplication with
$\exp(2\mathrm{i}\theta(x))$. Thus, the right side of Eq. (44) is
$\exp(\mathrm{i}\alpha(\theta))\mathscr{E}_{\exp(2\mathrm{i}\theta)f}$, which
is indeed the same as $\exp(2\mathrm{i}K^{0}(\theta))\mathscr{E}_{f}$. For
$\theta\equiv 0$, one may replace $\xi$ by $t\xi$ on both sides in (44),
differentiate with respect to $t$, and check that the differentiated equality
holds true. The procedure is similar to the proof of Proposition 5.1 in our
article [19], we omit the details. The full statement is deduced by
concatenating
$\mathcal{U}(\xi,\theta)=\mathcal{U}(\xi,0)\mathcal{U}(0,\theta)$. ∎

We complement the proof sketch by a consistency check with Eqs. (41)–(43). For
$f(x,n)=z(x)^{n}$ with $\sup|z|<1$, we compute

$\langle\varphi_{\xi,\theta},f\rangle=\sum_{n=1}^{\infty}\frac{1}{n}\int\overline{\varphi_{\xi,\theta}(x,n)}f(x,n)\alpha(\mathrm{d}x)=-\int\log\Bigl{(}1+z\frac{\overline{\xi}}{|\xi|}\tanh|\xi|\Bigr{)}\mathrm{d}\alpha.$

With the equality $1-(\tanh s)^{2}=1/(\cosh s)^{2}$, we get

$-\frac{1}{2}||\varphi_{\xi,\theta}||^{2}=\frac{1}{2}\int\log\bigl{(}1-(\tanh|\xi|)^{2}\bigr{)}\mathrm{d}\alpha=-\int\log(\cosh|\xi|)\mathrm{d}\alpha.$

It follows that the exponential on the right side in (44) is equal to the
factor $\mathcal{C}_{\xi,\theta}$ from (41) and (42). It remains to check that
$\mathscr{E}_{Q_{\xi,\theta}(f-\varphi_{\xi,\theta})}$ corresponds to
$\prod_{j}z_{\xi,\theta}(x_{j})^{n_{j}}$ with $z_{\xi,\theta}(x)$. For
simplicity we only consider $\theta\equiv 0$ and real-valued $\xi$. Checking
the desired equality boils down to a computation in
$\ell^{2}(\mathbb{N},\sum_{n=1}^{\infty}\frac{1}{n}\delta_{n})$: for
$t\in\mathbb{R}$, $z\in\mathbb{C}$ with $|z|<1$, and

$z_{t}=\frac{z+\tanh t}{1+z\tanh t},\quad f_{t}(n)=z_{t}^{n}$

one has

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}z_{t}^{n}=nz_{t}^{n-1}(1-z_{t}^{2})=(\mathrm{k}^{+}-\mathrm{k}^{-})f_{t}(n)+\delta_{n,1}.$

The special case $z=-\tanh s$ with fixed $s\in\mathbb{R}$ shows that
$g_{t}(n)=(\tanh(t-s))^{n}$ satisfies the same differential equality,
therefore
$\partial_{t}(f_{t}-g_{t})=(\mathrm{k}^{+}-\mathrm{k}^{-})(f_{t}-g_{t})$ and
$f_{t}-g_{t}=\exp(t(\mathrm{k}^{+}-\mathrm{k}^{-}))(f_{0}-g_{0})$. In the
previous equality $g_{0}$ depends on $s$ and the equality holds true for every
$s$, in particular, $s=-t$. For this choice $g_{0}(n)=(-\tanh t)^{n}$ and
$g_{t}(n)=0$. Thus, we get

$\mathrm{e}^{t(\mathrm{k}^{+}-\mathrm{k}^{-})}(f_{0}-g_{0})(n)=f_{t}(n)=z_{t}^{n}$

which is essentially the desired equality.

##  Appendix B Generalized Lebesgue measure and $SL(2,\mathbb{R})$

Here we explain how our representations with the Gamma random measure relate
to representations of the special linear group with the multiplicative
measure. Let us first briefly describe the construction, following Tsilevich,
Vershik and Yor [57]. We assume that the measure $\alpha$ on $\mathbb{X}$ is
finite so that the Gamma random measure with law $\rho_{\alpha}$ has almost
surely finite total mass, $\mu(\mathbb{X})<\infty$. The measure
$\mathscr{L}_{\alpha}$ is by definition the $\sigma$-finite measure on
$\mathbf{M}$ that is absolutely continuous with respect to $\rho_{\alpha}$,
with Radon-Nikodym derivative

$\frac{\mathrm{d}\mathscr{L}_{\alpha}}{\mathrm{d}\mathscr{\rho}_{\alpha}}(\mu)=\exp(\mu(\mathbb{X})).$

The definition is motivated by a simple observation in the univariate case:
The Lebesgue measure on $\mathbb{R}_{+}$ is absolutely continuous with respect
to the Gamma distribution $\mathrm{e}^{-x}\mathrm{d}x$ with parameter $1$,
with Radon-Nikodym derivative $\exp(x)$.

The $SL(2,\mathbb{R})$ current group over $\mathbb{X}$ consists of the bounded
measurable maps taking values in $SL(2,\mathbb{R})$, the real $2\times 2$
matrices with determinant $1$. The lower triangular subgroup consists of the
elements $g:\mathbb{X}\to SL(2,\mathbb{R})$ of the form

$g(x)=\begin{pmatrix}a(x)^{-1}&0\\\ b(x)&a(x)\end{pmatrix}$

with bounded measurable maps $a,b:\mathbb{X}\to\mathbb{R}$. The map $a$ is
bounded away from zero. For $\mu\in\mathbf{M}$, the dilated measure
$M_{a^{2}}\mu$ is given by $M_{a^{2}}\mu(B)=\int_{B}a^{2}(x)\mu(\mathrm{d}x)$.
For $F\in L^{2}(\mathbf{M},\mathscr{L}_{\alpha})$ and $g$ as above, define

$\mathcal{U}_{g}F(\mu)=\exp\Bigl{(}\int|\log a|\mathrm{d}\alpha+\mathrm{i}\int
ab\mathrm{d}\mu\Bigr{)}F(M_{a^{2}}\mu).$

The map $g\mapsto\mathcal{U}_{g}$ is a unitary representation of the
triangular subgroup of the $SL(2,\mathbb{R})$ current group ([57, Section 5]
and references therein). The action of the full group is more complicated; in
the univariate case it requires, in addition to the multiplication and
dilation operators, an integral operator [57].

To establish the connection with our representation from Section 3.3, we first
notice that the spaces are unitarily equivalent. The map $F\mapsto\tilde{F}$
with $\tilde{F}(\mu)=\exp(\mu(\mathbb{X})/2)F(\mu)$ is a unitary isomorphism
from $L^{2}(\mathbf{M},\rho_{\alpha})$ onto
$L^{2}(\mathbf{M},\mathscr{L}_{\alpha})$. In
$L^{2}(\mathbf{M},\rho_{\alpha})$, the unitary $\mathcal{U}_{g}$ becomes

$U_{g}F(\mu)=\exp\Bigl{(}\int|\log a|\mathrm{d}\alpha+\mathrm{i}\int
ab\,\mathrm{d}\mu+\frac{1}{2}\int(1-a^{2})\mathrm{d}\mu\Bigr{)}F(M_{a^{2}}\mu)$

For $a(x)\equiv 1$, this is simply multiplication with
$\exp(\mathrm{i}\mu(b))$ hence, in view of (18), this is the operator
$\exp(\mathrm{i}(K^{+}(b)+K^{-}(b)+2K^{0}(b)))$. For $b(x)\equiv 0$ and
$a(x)=\exp(t\varphi(x))$ with real-valued bounded $\varphi$, $U_{g}$ maps $F$
to

$F_{t}(\mu)=\exp\Bigl{(}t\int\varphi\mathrm{d}\alpha+\frac{1}{2}\int(1-\mathrm{e}^{2t\varphi})\mathrm{d}\mu\Bigr{)}F(M_{\exp(2t\varphi)}\mu).$

An explicit computation for polynomial $F$ gives

$\displaystyle\partial_{t}F_{t}(\mu)$
$\displaystyle=2\int\varphi(x)\frac{\delta
F_{t}}{\delta\mu(x)}(\mu)\mu(\mathrm{d}x)+\bigl{(}\alpha(\varphi)-\mu(\varphi)\bigr{)}F_{t}(\mu)$
$\displaystyle=\bigl{(}K^{-}(\varphi)-K^{+}(\varphi)\bigr{)}F_{t}(\mu)$

and hence, the unitary $U_{g}$ for $g$ the diagonal matrix with diagonal
entries $\exp(-t\varphi(x))$, $\exp(t\varphi(x))$, corresponds to
$\exp(t(K^{-}(\varphi)-K^{+}(\varphi))$.

To conclude, let $k^{\pm},k^{0}$ be the $2\times 2$ matrices from Section 2.1.
The correspondence

$\displaystyle\mathrm{i}(k^{+}+k^{-}+2k^{0})$
$\displaystyle\mapsto\begin{pmatrix}0&0\\\ 1&0\end{pmatrix},$ $\displaystyle
k^{-}-k^{+}\mapsto\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix},$
$\displaystyle\mathrm{i}(k^{+}+k^{-}-2k^{0})$
$\displaystyle\mapsto\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$

extends to an isomorphism from the real Lie algebra $su(1,1)$ onto
$sl(2,\mathbb{R})$.

Thus, the representations with the multiplicative measure
$\mathscr{L}_{\alpha}$ agree with our representations with the Gamma random
measure. We pass from one to the other by (1) mapping
$L^{2}(\mathbf{M},\mathscr{L}_{\alpha})$ to $L^{2}(\mathbf{M},\rho_{\alpha})$,
and (2) exploiting the isomorphism between the real Lie algebra $su(1,1)$ and
$sl(2,\mathbb{R})$.

#### Acknowledgements

S.F. acknowledges financial support from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy–
EXC-2047/1 – 390685813. S.J. and S.W. were supported under Germany’s
excellence strategy EXC-2111-390814868. S.F. and S.W. thank the Hausdorff
Institute for Mathematics (Bonn) for its hospitality during the Junior
Trimester Program Stochastic modelling in life sciences funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - EXC-2047/1 - 390685813. The authors thank F. Redig for
his insights at the beginning of the project. S.W. thanks D. Spanò for helpful
discussions.

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