Content-Type: multipart/mixed; boundary="-------------0011200713866" This is a multi-part message in MIME format. ---------------0011200713866 Content-Type: text/plain; name="00-459.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-459.keywords" generalised Brownian motion, quantum white noise ---------------0011200713866 Content-Type: application/x-tex; name="gbm.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="gbm.tex" \documentclass[10pt]{article} \usepackage{amsmath, latexsym, amssymb, amscd, psfrag, epsfig} \newcommand{\bi}{\begin{itemize}} \newcommand{\ei}{\end{itemize}} \newcommand{\U}{\mathcal{U}} \newcommand{\C}{\mathbb{C}} % complex numbers \newcommand{\R}{\mathbb{R}} % real numbers \newcommand{\N}{\mathbb{N}} % natural numbers \newcommand{\Z}{\mathbb{Z}} % integers \newcommand{\Dsum}{\bigoplus} % direct sum \newcommand{\dsum}{\oplus} % direct sum \newcommand{\Tens}{\bigotimes} % tensor product \newcommand{\tens}{\otimes} % tensor product \newcommand{\lsymm}{\ell_\mathrm{\small symm}^2} %little l2 symm. \newcommand{\symm}{\mathrm{S}} %symmetric group \newcommand{\e}{\epsilon} \newcommand{\F}{\mathcal{F}} % Fock space \newcommand{\G}{\Gamma} % Field algebra \newcommand{\K}{{\mathcal K}} \newcommand{\h}{\mathcal{H}} \newcommand{\A}{{\mathcal A}} %free field alg \newcommand{\m}{\mathbf} \newcommand{\br}{\left\langle} \newcommand{\ke}{\right\rangle} \newcommand{\lng}{\langle} \newcommand{\rng}{\rangle} \newcommand{\Ra}{\longrightarrow} \newcommand{\ra}{\rightarrow} \newcommand{\ti}{\tilde} \newcommand{\om}{\omega} \newcommand{\mi}{\bigl| \bigr.} \newcommand{\zk}{\z{K}} \newtheorem{theorem}{Theorem}[section] \newcommand{\definitie}{\noindent\textbf{Definition.} } \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \def\square{{\vcenter{\vbox{\hrule height.4pt \hbox{\vrule width.4pt height1.45ex \kern1.45ex \vrule width.4pt} \hrule height.4pt}}}} \def\qed{\hfill$\square$} \newcommand{\barint}{\hbox{$\int$\kern-0.75\intwidth\vrule width 0.5\intwidth height 2.4pt depth -2pt\kern0.25\intwidth}} \newlength\intwidth \setbox0=\hbox{$\int$} \intwidth=\wd0 %\endinput \numberwithin{equation}{section} \begin{document} \title{Generalised Brownian Motion\\ and Second Quantisation} \author{\normalsize M\u ad\u alin Gu\c t\u a \footnote{E-mail: guta@sci.kun.nl}\and \normalsize Hans Maassen \footnote{E-mail: maassen@sci.kun.nl}} \date{} \maketitle \begin{center} {\rm Mathematisch Instituut}\\ {\rm Katholieke Universiteit Nijmegen} \\ {\rm Toernooiveld 1, 6526 ED Nijmegen}\\ {\rm The Netherlands}\\ {\rm fax}:+31 24 3652140 \end{center} %\newpage \begin{abstract}\noindent A new approach to the generalised Brownian motion introduced by M. Bo\.zejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species $V$ gives rise to an endofunctor $\mathcal{F}_V$ of the category of Hilbert spaces with contractions mapping a Hilbert space $\mathcal{\h}$ to a symmetric Hilbert space $\mathcal{F}_V(\mathcal{\h})$ with the same symmetry as the species $V$. A general framework for annihilation and creation operators on these spaces is developed and shown to give vacuum expectations as prescribed by Bo\.zejko and Speicher. The existence of the second quantisation as functor from Hilbert spaces to von Neumann algebras with completely positive maps is investigated. For a certain one parameter interpolation between the classical and the free Brownian motion it is shown that the ``field algebras'' $\Gamma(\K)$ are type $\mathrm{II}_1$ factors when $\K$ is infinite dimensional. \end{abstract} %\newpage \section{Introduction}\label{sec.introduction} In non-commutative probability theory one is interested in finding generalisations of classical probabilistic concepts such as independence and processes with independent stationary increments. Motivated by a central limit theorem result and by the analogy with classical Brownian motion, M. Bo\.zejko and R. Speicher proposed in \cite{Boz.Sp.1} a class of operator algebras called ``generalised Brownian motions'' and investigated an example of interpolation between the classical \cite{Simon} and the free motion of Voiculescu \cite{Voi.Dy.Ni.}. A better known interpolation is provided by the ``$q$-deformed commutation relations'' \cite{Boz.Ku.Spe., Boz.Sp.2, Boz.Sp.3, Fiv., Fr.Bur., Grb., Maa.vLee., Zag.}. Such an operator algebra is obtained by performing the GNS representation of the free tensor algebra $\mathcal{A}(\K)$ over an arbitrary infinite dimensional real Hilbert space $\K$, with respect to a ``Gaussian state'' $\tilde{\rho}_{\mathbf{t}}$ defined by the following ``pairing prescription'': \begin{equation} \tilde{\rho}_{\mathbf{t}}(\om(f_1)\dots \om(f_n))= \left\{\begin{array}{ll} 0 & \textrm{if $n$ odd} \\ \underset{\mathcal{V}\in\mathcal{P}_2(n)}{\sum} \mathbf{t}(\mathcal{V}) \underset{(k,l)\in \mathcal{V}}{\prod}\br f_k,f_l\ke & \textrm{if $n$ even} \end{array}\right. \end{equation} where $f_i\in\K, \omega(f_i)\in \mathcal{A}(\K)$ and the sum runs over all pair partitions of the ordered set $\{1,2,\dots ,n\}$. The functional is uniquely determined by the complex valued function $\mathbf{t}$ on pair partitions. Classical Brownian motion is obtained by taking $\K=\mathrm{L}^2(\R_+)$ and $B_s:= \om(\mathbf{1}_{[0,s)})$ with the constant function $\mathbf{t}(\mathcal{V})=1$ on all pair partitions; the free Brownian motion \cite{Voi.Dy.Ni.} requires $\mathbf{t}$ to be 0 on crossing partitions and 1 on non-crossing partitions. %\noindent If one considers complex Hilbert spaces, the analogue of a Gaussian state is called a Fock state. We show that the GNS representation of the free algebra $\mathcal{C}(\h)$ of creation and annihilation operators with respect to a Fock state $\rho_{\mathbf{t}}$ can be described in a functorial way inspired by the notions of tensorial species of Joyal \cite{Joyal, Joyal.2}: the representation space has the form \begin{equation} \F_{\mathbf{t}}(\h):=\Dsum_{n=0}^\infty \frac{1}{n!} V_n \tens_s \h^{\tens n} \end{equation} where $V_n$ are Hilbert spaces carrying unitary representations of the symmetric groups $\mathrm{S}(n)$ and $\tens_s$ means the subspace of the tensor product containing vectors which are invariant under the double action of $\mathrm{S}(n)$. The creation operators have the expression: \begin{equation} a^*_{\mathbf{t}}(h) ~v\tens_s (h_0\tens\dots\tens h_{n-1})= (j_n v)\tens_s (h_0\tens\dots\tens h_{n-1}\tens h_n) \end{equation} where $j_n:V_n\to V_{n+1}$ is an operator which intertwines the action of $\mathrm{S}(n)$ and $\mathrm{S}(n+1)$. %\noindent In Section \ref{sec.semigroup} we connect these Fock representations with positive functionals on a certain algebraic object $ \mathcal{B}\mathcal{P}_2(\infty)$ which we call the $^*$-semigroup of ``broken pair partitions''. The elements of this $^*$-semigroup can be described graphically as segments located between two vertical lines which cut through the graphical representation of a pair partition. In particular, the pair partitions are elements of $\mathcal{B}\mathcal{P}_2(\infty)$. We show that if $\rho_{\mathbf{t}}$ is a Fock state then the function $\mathbf{t}$ has a natural extension to a positive functional $\hat{\mathbf{t}}$ on $ \mathcal{B}\mathcal{P}_2(\infty)$. The GNS-like representation with respect to $\hat{\mathbf{t}}$ provides the combinatorial data $(V_n, j_n)_{n=0}^\infty$ associated to $\rho_{\mathbf{t}}$. The representation of $\mathcal{A}(\K)$ with respect to a Gaussian state $\tilde{\rho}_{\mathbf{t}}$ is a $^*$-algebra generated by ``fields'' $\omega_{\mathbf{t}}(f)$. Monomials of such fields can be seen as moments, with the corresponding cumulants being a generalisation of the Wick products known from the $q$-deformed Brownian motion \cite{Boz.Ku.Spe.}. Using generalised Wick products we prove that any Gaussian state $\tilde{\rho}_{\mathbf{t}}$ extends to a Fock state $\rho_{\mathbf{t}}$ on the algebra of creation and annihilation operators $\mathcal{C}(\K_\C)$ (see section \ref{gen.wick.products}). %\newline %\noindent Second quantisation is a functor from the category of real Hilbert spaces with contractions to the category of (non-commutative) probability spaces. For any real Hilbert space $\K$ we consider first the $^*$-algebra $\tilde{\Gamma}_{\mathbf{t}}(\K)$ of generalised Wick products. In a standard manner we associate to it a von Neumann algebra $\Gamma_{\mathbf{t}}(\K)$ and investigate the possibility of making $\Gamma_{\mathbf{t}}$ into a functor of second quantisation. A necessary condition is that the function $\mathbf{t}$ has the multiplicative property, a form of statistical independence. A sufficient condition is the separating property of the vacuum. By modifying the definition of the ``field algebras'' we obtain another second quantisation functor denoted $\Delta_{\mathbf{t}}$ this time for \textit{all} multiplicative (positive definite) functions $\mathbf{t}$. In the last section we develop a useful criterion, in terms of the spectrum of a characteristic contraction, for factoriality of the algebras $\Gamma_\mathbf{t}(\ell^2(\Z))$ in the case when the vacuum state $\rho_{\mathbf{t}}$ is tracial. We then apply it to a particular example of positive definite function $\mathbf{t}_q$ where $0\leq q<1$, which interpolates between the bosonic and free cases and has been introduced in \cite{Boz.Sp.1} (see \cite{Gu.Maa.} for another proof of the positivity). We conclude that that $\Gamma_\mathbf{t}(\ell^2(\Z))$ is a type $\mathrm{II}_1$ factor. Further generalisation of this criterion to factors of type $\mathrm{III}$ will be investigated in a forthcoming paper \cite{Gu.Maa.2}. \section{Definitions and description of the Fock representation} \label{SecRepresentations} The generalised Brownian motions \cite{Boz.Sp.1} are representations with respect to special \textit{gaussian} states on free algebras over real Hilbert spaces. We start by giving all necessary definitions and subsequently we will analyse the structure of the \textit{Fock representations} which are intimately connected with the generalised Brownian motion (see section \ref{gen.wick.products}) . \begin{definition}{\rm Let $\K$ be a real Hilbert space. The algebra $\mathcal{A}(\K)$ is the free unital $^*$-algebra with generators $\om (h)$ for all $h \in \K$, divided by the relations: \begin{equation} \om(af+bg)=a\om (f)+ b\om (g), \qquad \om (f)=\om (f)^* \end{equation} for all $f,g\in\K$ and $a,b\in\R$.} \end{definition} \begin{definition}{\rm Let $\h$ be a complex Hilbert space. The algebra $\mathcal{C}(\h)$ is the free unital $^*$-algebra with generators $a(h)$ and $a^*(h)$ for all $h \in \h$, divided by the relations: \begin{equation} a^*(\lambda f+\mu g)=\lambda a^*(f)+ \mu a^*(g),\qquad a^*(f)= a(f)^* \end{equation} for all $f,g\in\h$ and $\lambda, \mu\in\C$.} \end{definition} \noindent We notice the existence of the canonical injection from $\mathcal{A}(\K)$ to $\mathcal{C}(\K_\C)$ \begin{equation} \om (h)\mapsto a(h)+ a^*(h) \end{equation} where $\K_\C$ is the complexification of the real Hilbert space $\K$. On the algebras defined above we would like to define positive linear functionals by certain pairing prescriptions for which we need some notions of pair partitions. \begin{definition}{\rm Let $S$ be a finite ordered set. We denote by $\mathcal{P}_2(S)$ is the set of pair partitions of $S$, that is $\mathcal{V}\in\mathcal{P}_2(S)$ if $\mathcal{V}$ consists of $\frac{1}{2}n$ disjoint ordered pairs $(l,r)$ with $lN$ for all $1\leq q \leq n $ and some fixed big enough $N\in \N$ . \\ The map $j_n$ is defined as the restriction of $a^*_{\mathbf{t}}(e_{n+1})$ to $V_n$: \begin{displaymath} j_n\prod_{k=1}^{2p+n}a^{\sharp_k}_{\mathbf{t}}(e_{i_k})\Omega_{\mathbf{t}}= a^*_{\mathbf{t}}(e_{n+1})\prod_{k=1}^{2p+n}a^{\sharp_k}_{\mathbf{t}}(e_{i_k})\Omega_{\mathbf{t}}. \end{displaymath} Obviously, the image of $j_n$ lies in $V_{n+1}$. \noindent The state $\rho_{\mathbf{t}}$ is invariant under unitary transformations $U\in\mathcal{U}(\h)$: \begin{displaymath} \rho_{\mathbf{t}}(\prod_{k=1}^n a^{\sharp_k}(e_{i_k})) =\rho_{\mathbf{t}}(\prod_{k=1}^n a^{\sharp_k}(Ue_{i_k})). \end{displaymath} Thus \begin{equation}\label{def.secquant.Hilbert} \F_{\mathbf{t}}(U):\prod_{k=1}^n a^{\sharp_k}_{\mathbf{t}}(e_{i_k})\Omega_{\mathbf{t}}\mapsto \prod_{k=1}^n a^{\sharp_k}_{\mathbf{t}}(Ue_{i_k})\Omega_{\mathbf{t}} \end{equation} is unitary and $\F_{\mathbf{t}}(U_1)\F_{\mathbf{t}}(U_2)=\F_{\mathbf{t}}(U_1U_2)$ for two unitaries $U_1, U_2$. The action on the algebra of creation and annihilation operators is \begin{equation}\label{defF_t} \F_{\mathbf{t}}(U)a^\sharp_{\mathbf{t}}(f)\F_{\mathbf{t}}(U^*)=a^\sharp_{\mathbf{t}}(Uf). \end{equation} \noindent Considering unitaries which act by permuting the basis vectors $\{e_1,\dots ,e_n\}$ and leave all the others invariant we obtain a unitary representation of $\symm (n)$ on $V_n$. The intertwining property (\ref{intertwining}) follows immediately from the definition of $j_n$. Having the ``combinatorial data'' $(V_n,j_n)$, we can construct the triple $(\F_V(\h),\mathcal{C}_{V,j}(\h), \Omega_V )$ according to equations (\ref{defsymm}, \ref{defcr}, \ref{defann}). Similarly to $\F_{\mathbf{t}}(U)$ we have the unitary \begin{eqnarray}\label{defsecquant} \F_{V}(U) :\F_{V}(\h) & \ra & \F_{V}(\h)\nonumber\\ v\tens_s (h_0\tens\dots h_{n-1}) &\mapsto & v\tens_s (U h_0\tens\dots\tens U h_{n-1}) \end{eqnarray} for $U\in\U(\h), v\in V_n$. We call $F_{V}(U)$ the \textit{second quantisation} of $U$ at the Hilbert space level. Its action on operators is: \begin{equation}\label{defF_V} \F_V(U)a^\sharp_{V,j}(f)\F_V(U^*)=a^\sharp_{V,j}(Uf). \end{equation} \noindent Analogously to $V_n$ we define for any finite subset $\{i_1,\dots ,i_n\}\subset \N$ the linear subspace $V(i_1,\dots ,i_n)$ of $\F_{\mathbf{t}}(\h)$ spanned by applying to the vacuum $\Omega_{\mathbf{t}}$ monomials $\prod_{k=1}^{2p+n} a^{\sharp_k}_{\mathbf{t}}(e_{j_k})$ for which the colours $(j_k)_{k=1}^{2p+n}$ satisfy conditions similar to i), ii) but now with $\{i_1,\dots ,i_n\}$ instead of $\{1,\dots ,n\}$. For a unitary $U$ which permutes the basis vectors, $Ue_i=e_{u(i)}$ we get \begin{equation} \F_{\mathbf{t}}(U)V(i_1,\dots ,i_n)=V(u(i_1),\dots ,u(i_n)). \end{equation} One can check by calculating inner products that any two such spaces are either orthogonal or coincide. Similarly, we define the following subspaces of $\F_{V}(\h)$ \begin{equation} \tilde{V}(i_1,\dots ,i_n):= \overline{\mathrm{lin}\{v\tens_s(e_{i_1}\tens\dots\tens e_{i_n}): \qquad v\in V_n\}} \end{equation} \noindent which are also orthogonal for different sets of ``colours'' $\{i_1,\dots ,i_n\}$.\\ 2. We proceed by proving the equality of the states $\rho_{\mathbf{t}}$ and $\rho_{V,j}$. \noindent As $\rho_{V,j}$ is a Fock state by Theorem \ref{prop.combinatorics}, we need only verify that the positive definite function $\mathbf{t}$ we have started with and the one derived from $\rho_{V,j}$ coincide. By definition there is an isometry \begin{eqnarray} T_n: V_n & \ra & \F_{V,j}(\h)\nonumber\\ v & \mapsto & v\tens_s (e_1\tens\dots\tens e_n). \end{eqnarray} Furthermore for any unitary $U\in\U(\h)$ which permutes the basis vectors such that $Ue_k=e_{i_k} $, the operator \begin{displaymath} T(i_1,\dots ,i_n ): V(i_1,\dots ,i_n ) \ra \tilde{V}(i_1,\dots ,i_n ) \end{displaymath} \noindent defined by \begin{equation}\label{defT} T(i_1,\dots ,i_n ):=\F_{V}(U)T_n \F_{\mathbf{t}}(U^*) \end{equation} depends only on the set $\{i_1,\dots ,i_n\}$. Finally, the definitions of $j_n, a^\sharp_{V,j}(f)$ amounts to the fact that the following diagram commutes \begin{equation} \begin{CD} V_n @>T_n>> \tilde{V}_n\\ @V{a^*_{\mathbf{t}}(e_{n+1})}VV @VV{a^*_{V,j}(e_{n+1})}V\\ V_{n+1} @>T_{n+1}>> \tilde{V}_{n+1} \\ \end{CD} \end{equation} \noindent and by acting from the left and from the right with the appropriate second quantisation operators and using (\ref{defT}, \ref{defF_t}, \ref{defF_V}) we obtain \begin{equation} \begin{CD} V(i_1,\dots ,i_n) @>T(i_1,\dots ,i_n)>> \tilde{V}(i_1,\dots ,i_n)\\ @V{a^*_{\mathbf{t}}(e_{i_{n+1}})}VV @VV{a^*_{V,j}(e_{i_{n+1}})}V\\ V (i_1,\dots ,i_{n+1}) @>T(i_1,\dots ,i_{n+1})>> \tilde{V}(i_1,\dots ,i_{n+1})\\ \end{CD} \end{equation} with a similar diagram for the annihilation operators. This is sufficient for proving the equality $\rho_{\mathbf{t}}(\prod_{k=1}^{2n} a^{\sharp_k}_{\mathbf{t}}(e_{i_k}))= \rho_{V,j}(\prod_{k=1}^{2n} a^{\sharp_k}_{V,j}(e_{i_k}))$ for monomials containing $n$ pairs of creation and annihilation operators of $n$ different colours.\\ 3. Finally we prove that $\Omega_{V,j}$ is cyclic vector for $\mathcal{C}_{V,j}(\h)$. \noindent The space $\F_V(\h)$ has a decomposition with respect to occupation numbers \begin{displaymath} \F_V(\h)=\Dsum_{\{n_1,\dots ,n_k\}}\F_V(n_1, \dots ,n_k) \end{displaymath} with \begin{equation}\label{eq.occunumb} \F_V(n_1, \dots ,n_k)= \overline{\mathrm{lin}\{v\tens_s(\underbrace{e_1\tens\dots\tens e_1}_{n_1} \tens\dots\tens\underbrace{e_k\tens\dots\tens e_k}_{n_k}, v\in V_{n_1+\dots +n_k}\}}. \end{equation} We recall that $\tilde{V_n}=\F_V(\underbrace{1,\dots ,1}_{n})$ is spanned by linear combinations of vectors of the form \begin{displaymath} \prod_{k=1}^{2p+n}a^{\sharp_k}_{V,j}(e_{i_k})\Omega_{V}= v\tens_s(e_1\tens\dots\tens e_n) \end{displaymath} with monomials satisfying the conditions i) and ii). By replacing the creation operators $(a^*(e_k))_{k=1}^n$ appearing in the monomial, with the sequence containing $n_i$ times the creator $a^*(e_i$) for $i\in\{1,\dots p\}$ and $\sum_{i=1}^{p} n_i=n$ we obtain a set of vectors which are dense in $\F_V(n_1,\dots ,n_p)$ and this completes the proof of the cyclicity of the vacuum. Putting together 1., 2. and 3. we conclude that the representations $(\F_{\mathbf{t}}(\h),\mathcal{C}_{\mathbf{t}}(\h), \Omega_{\mathbf{t}} )$ and $(\F_V(\h),\mathcal{C}_{V,j}(\h), \Omega_V )$ are unitarily equivalent for infinite dimensional $\h$. The case $\h$ finite dimensional follows by restriction of the previous representations to the appropriate subspaces. \qed \section{The $^*$-semigroup of broken pair partitions} \label{sec.semigroup} The content of the last two theorems can be summarised by the following fact: there exist a bijective correspondence between positive definite functions on pair partitions $\mathbf{t}$, and ``combinatorial data'' $(V_n, j_n)_{n=0}^\infty$. This suggests that the positivity of $\mathbf{t}$ can be characterised in a simpler way by regarding $\mathbf{t}$ as a positive functional on an algebraic object containing $\mathcal{P}_2(\infty)$ as a subset. Theorem 1 of \cite{Boz.Sp.1} shows that a positive definite function on pair partitions $\mathbf{t}$ restricts to positive definite functions on the symmetric groups $\symm (n)$ for all $n\in\N$ through the embedding \begin{equation} \symm (n)\ni\tau\mapsto \mathcal{V}_\tau\in\mathcal{P}_2(n) \end{equation} given by \begin{equation} \mathcal{V}_\tau:=\{(i,2n+1-\tau(i)):~i=1,\dots ,2n\}. \end{equation} However $\mathbf{t}$ is not determined completely by its restriction and thus one would like to find another algebraic object which completely encodes the positivity requirement. We will show that this is the $^*$-semigroup of \textit{broken pair partitions} which we denote by $\mathcal{B}\mathcal{P}_2(\infty)$ and will be described below. Pictorially, the elements of the semigroup are segments obtained by sectioning pair partitions with vertical lines. \begin{definition}\label{def.semigroup} Let $X$ be an arbitrary finite ordered set and $(L,P,R)$ a disjoint partition of $X$. We consider all the triples $(\mathcal{V}, f_l,f_r)$ where $\mathcal{V}\in\mathcal{P}_2(P)$ and \begin{equation} f_l:L\rightarrow\{1,\dots , |L|\}, \qquad f_r:R\rightarrow\{1,\dots , |R|\} \end{equation} are bijections. Any order preserving bijection $\alpha :X\to Y$ induces an obvious map \begin{equation} (\mathcal{V}, f_{l},f_{r})\to (\alpha\circ\mathcal{V}, f_{l}\circ \alpha^{-1}, f_{r}\circ \alpha^{-1}) \end{equation} where $\alpha\circ\mathcal{V}:=\{(\alpha(a),\alpha(b)): (a,b)\in\mathcal{V}\}$. This defines an equivalence relation; an element $d$ of $\mathcal{B}\mathcal{P}_2(\infty)$ is an equivalence class of triples $(\mathcal{V}, f_l,f_r)$ under this equivalence relation. \end{definition} \noindent We have the following pictorial representation: an element $d$ is given by a diagram containing a sequence of $l+r+2n$ points displayed horizontally with $2n$ of them connected into $n$ pairs, $l$ points are connected with other $l$ points vertically ordered on the left side (left legs) and $r$ points are connected with $r$ points vertically ordered on the right (right legs). An example is given in Figure \ref{diagram}. In this case we have $X=\{1,\dots,5\}$, $\mathcal{V}=\{(1,4)\}$, the left legs are connecting the points labeled 2 and 5 on the horizontal to the the points on the left side which are ordered vertically and labeled by 1 and 2. Similarly for the right legs. Usually we will label the ordered set of horizontal points will be of the form $\{n, n+1,\dots n+m\}$. %\begin{figure}[h]\label{diagram} %\begin{center} %\includegraphics[width=4cm]{diagram.eps} %\end{center} %\end{figure} \noindent The product of two diagrams is calculated by drawing the diagrams next to each other and joining the right legs of the left diagram with the left legs of the right diagram which are situated at the same level on the vertical. Figure \ref{products} illustrates an example. %\begin{figure}[h]\label{products} %\begin{center} %\includegraphics[width=10cm]{product.eps} %\end{center} %\end{figure} \noindent More formally if $d_i=(\mathcal{V}_i, f_{l,i}, f_{r,i})$ for $i=1,2$ with the notations from Definition \ref{def.semigroup}, then $d_1\cdot d_2=(\mathcal{V}, f_{l}, f_{r})$ with \begin{equation} \mathcal{V}=\mathcal{V}_1\cup\mathcal{V}_2\cup \{(f_{r,1}^{-1}(i),f_{l,2}^{-1}(i)):i\leq\mathrm{min}(|R_1|,|L_2|)\}, \end{equation} $f_{l}$ is defined on the disjoint union $L_1+(L_2\setminus f_{l,2}^{-1}(\{1,\dots,\mathrm{min}(|R_1|,|L_2|)\} )$ by \begin{displaymath} \left\{ \begin{array}{ll} f_l(a)=f_{l,1}(a) & \textrm{for}~ a\in L_{1}\\ f_l(b)=f_{l,2}(b)+|L_1| & \textrm{for}~ b\in L_2\setminus f_{l,2}^{-1}(\{1,\dots,\mathrm{min}(|R_1|,|L_2|)\} \end{array}\right. \end{displaymath} and similarly for $f_{r}$. The product does not depend on the chosen representatives for $d_i$ in their equivlence class and is associative. The diagrams with no legs are the pair partitions, thus $\mathcal{P}_2(\infty)\subset\mathcal{B}\mathcal{P}_2(\infty)$. \noindent The involution is given by mirror reflection (see Figure \ref{adjoint}). If $d=(\mathcal{V},f_l, f_r)$ then $d^*=(\mathcal{V}^*,f_r,f_l)$ with the underlying set $X^*$ obtained by reversing the order on $X$ and \begin{equation} \mathcal{V}^* :=\{(b,a):(a,b)\in\mathcal{V}\} \end{equation} is the adjoint of $\mathcal{V}$. It is easy to check that \begin{displaymath} (d_1\cdot d_2)^*=d_2^*\cdot d_1^*. \end{displaymath} %\begin{figure}[h]\label{adjoint} %\begin{center} %\includegraphics[width=10cm]{adjoint.eps} %\end{center} %\end{figure} Let $\mathbf{t}$ be a linear functional on pair partitions. We extend it to a function $\hat{\mathbf{t}}$ on $\mathcal{B}\mathcal{P}_2(\infty)$ defined as \begin{equation}\label{def.t.hat} \hat\mathbf{{t}}(d)=\left\{\begin{array}{ll} \mathbf{t}(d) & \textrm{if $d\in\mathcal{P}_2(\infty) $} \\ 0 & \textrm{otherwise}. \end{array}\right. \end{equation} \begin{theorem} The function $\mathbf{t}$ on pair partitions is positive definite if and only if $\hat{\mathbf{t}}$ is postive on the $^*{-}$semigroup $\mathcal{B}\mathcal{P}_2(\infty)$. \end{theorem} \noindent \textit{Proof.} The main ideas are already present in the proof of Proposition \ref{th.GNSrep}. A GNS-type of construction associates to the pair $(\mathcal{B}\mathcal{P}_2(\infty), \hat{\mathbf{t}})$ a cyclic representation $\chi_{\mathbf{t}}$ of $\mathcal{B}\mathcal{P}_2(\infty)$ on a Hilbert space $V$ with cyclic vector $\xi\in V$. We have $\br\xi,\chi_{\mathbf{t}}(d)\xi\ke=\hat\mathbf{{t}}(d)$. We denote by $\mathcal{B}\mathcal{P}_2^{(n,0)}$ the set of diagrams with $n$ left legs and no right legs. Then using \begin{equation} \br\chi_{\mathbf{t}}(d_1)\xi,\chi_{\mathbf{t}}(d_2)\xi\ke_V= \hat{\mathbf{t}}(d_1^*\cdot d_2) \end{equation} we obtain: \noindent 1. the representation space $V$ is of the form \begin{equation} V=\Dsum_{n=0}^\infty V_n \qquad\mathrm{where}\qquad V_n=\overline{\mathrm{lin} \{\chi_{\mathbf{t}}(d)\xi:d\in\mathcal{B}\mathcal{P}_2^{(n,0)}\} } \end{equation} \noindent 2. on $\mathcal{B}\mathcal{P}_2^{(n,0)}$ there is an obvious action of $\symm (n)$ by permutations of the positions of the left ends of the legs. Figure \ref{transposition} shows the action of the transposition $\tau_{1,2}$. %\begin{figure}[h]\label{transposition} %\psfrag{a}{$\tau_{1,2}$} %\begin{center} %\includegraphics[width=10cm]{permutinglegs.eps} %\end{center} %\end{figure} \noindent This induces a unitary representation of $\symm (n)$ on $V_n$ as \begin{equation} \tau(d_1)^*\cdot\tau(d_2)=d_1^*\cdot d_2 \end{equation} for all $d_1,d_2\in\mathcal{B}\mathcal{P}_2^{(n,0)}$ and $\tau\in\symm (n)$. \noindent 3. let $d_0\in\mathcal{B}\mathcal{P}_2^{(1,0)}$ be the ``left hook'' (the diagram with no pairs). Then $j:=\chi_{\mathbf{t}}(d_0)$ is an operator on $V$ whose restriction $j_n$ to $V_n$ maps it into $V_{n+1}$ and satisfies the intertwining condition (\ref{intertwining}) with respect ot the representations of the symmetric groups on $V_n$ and $V_{n+1}$. \noindent Using the data $(V_n,j_n)$ we construct the triple $(\F_V(\h),\mathcal{C}_{V,j}(\h), \Omega_V )$. According to Proposition \ref{prop.combinatorics} there exists a positive definite function on pair partitions $\mathbf{t}'$ such that $\rho_{V,j}=\rho_{\mathbf{t}'}$. We have to prove that $\mathbf{t}$, which is the restriction of $\hat\mathbf{t}$ to $\mathcal{P}_2(\infty)$ coincides with $\mathbf{t}'$. \noindent Any pair partition $\mathcal{V}$ can be written in a ``standard form'' (see Figure \ref{standardform}): \begin{equation}\label{eq.standard.form} \mathcal{V}= (d_0^*)^{p_m}\cdot\pi_{m-1}( \dots\pi_2( d_0^{k_2} \cdot(d_0^*)^{p_1}\cdot\pi_1(d_0^{k_1}))) \end{equation} where the permutations $\pi_i$ are uniquely defined by the requirement that any two lines connecting two pairs in the associated graphic intersect minimally and at the rightmost possible position. %\begin{figure}[htbp]\label{standardform} %\begin{center} %\includegraphics[width=10cm]{decomposition.eps} %\end{center} %\end{figure} \noindent Let $\prod_{k=1}^{2n}a_{V,j}^{\sharp_k}(e_{i_k})$ be a monomial containing $n$ creation operators and $n$ annihilation operators such that by pairing creators with annihilators of the same colour on their right side, we generate a pair partition $\mathcal{V}$. The definitions (\ref{defcr}), (\ref{defann}) of the creation and annihilation operators give their expressions in terms of the operator $j, j^*$ and the unitary representations of the permutation groups on the spaces $V_n$. By using the intertwining property (\ref{intertwining}) we can pass all permutations to the left of the $j$-terms and obtain: \begin{eqnarray}\nonumber \mathbf{t}'(\mathcal{V}) & = & \br\Omega_V,\prod_{k=1}^{2n}a_{V,j}^{\sharp_k}(e_{i_k})\Omega_V\ke \\ &= &\br\xi,(j^*)^{p_m}\cdot U(\pi_{m-1}) \dots U(\pi_2)\cdot j^{k_2} \cdot (j^*)^{p_1}\cdot U(\pi_1)\cdot j^{k_1}\xi\ke_V \nonumber\\ &= & \br\xi,\chi_{\mathbf{t}}(\mathcal{V}) \xi\ke_V = \hat\mathbf{{t}}(\mathcal{V}) \nonumber \end{eqnarray} Conversely, starting from a positive definite function $\mathbf{t}$ we construct the representation $(V, \chi_{\mathbf{t}}(\mathcal{B}\mathcal{P}_2(\infty)),\xi)$ through applying Theorem \ref{th.GNSrep} and thus $\hat{\mathbf{t}}$ is positive on $\mathcal{B}\mathcal{P}_2(\infty)$. \qed \section{Generalised Wick products} \label{gen.wick.products} As argued in the introduction, the representations of the ``field algebras'' $\A(\K)$ with respect to Gaussian states $\tilde{\rho}_{\mathbf{t}}$ give rise to (noncommutative) processes called generalised Brownian motions \cite{Boz.Sp.1} for $\K$ (infinite dimensional) real Hilbert space. In all known examples such representations appear as restrictions to the subalgebra $\A(\K)$ of Fock representations of the algebra of creation and annihilation operators $\mathcal{C}(\K_\C)$ with respect to the state $\rho_{\mathbf{t}}$. We will prove that this is always the case, thus answering a question put in \cite{Boz.Sp.1}. \noindent Let \begin{equation} \mathbf{t}: \mathcal{P}_2(\infty)\ra \C \end{equation} be such that $\tilde{\rho}_{\mathbf{t}}$ is a Gaussian state on $\A(\K)$ for $\K$ infinite dimensional Hilbert space. Let $(\tilde{\mathcal{F}}_{\mathbf{t}}(\K), \tilde{\pi}_{\mathbf{t}}(\A(\K)), \tilde{\Omega}_{\mathbf{t}})$ be the GNS-triple associated to $(\mathcal{A}(\K),\tilde{\rho}_{\mathbf{t}})$. The $^*$-algebra $\tilde{\pi}_{\mathbf{t}}(\A(\K))$ is generated by the operators $\omega_{\mathbf{t}} (f):=\tilde{\pi}_{\mathbf{t}}(\omega (f))$ for all $f\in \K$. \begin{lemma} The operators $\omega_{\mathbf{t}} (f)$ are essentially selfadjoint. \end{lemma} \noindent \textit{Proof.} By definition $\omega_{\mathbf{t}} (f)$ is symmetric as operator defined on the dense domain $D:=\tilde{\pi}_{\mathbf{t}}(\mathcal{A}(\K))\tilde{\Omega}_{\mathbf{t}}$. By Theorem $\mathrm{VIII}.3$ in \cite{Reed.Simon}, $\omega_{\mathbf{t}} (f)$ is essentially selfadjoint if and only if $\mathrm{Ran}(\omega_{\mathbf{t}} (f)\pm i)$ is dense in $\tilde{\mathcal{F}}_{\mathbf{t}}(\K)$. Let $B\in\mathcal{A}(\K)$ such that for all $A\in\mathcal{A}(\K)$ we have \begin{equation} \br (\omega_{\mathbf{t}} (f)+i)\pi_{\mathbf{t}}(A)\Omega_{\mathbf{t}}, \pi_{\mathbf{t}}(B)\Omega_{\mathbf{t}}\ke=0 \end{equation} By choosing $A=\omega_{\mathbf{t}}(f)B$ we get $\rho_{\mathbf{t}}(B^*\om(f) B)=0$ and $\rho_{\mathbf{t}}(B^*\om(f)^2 B)=0$. Choose now $A=B$, then $\rho_{\mathbf{t}}(B^*B)=0$ thus $\pi_{\mathbf{t}}(B)\Omega_{\mathbf{t}}=0$. \qed In analogy to (\ref{def.secquant.Hilbert}) for any orthogonal operator $O\in\mathcal{O}(\K)$ there exists a unitary \begin{equation} \tilde{\F}_{\mathbf{t}}(O): \prod_{k=1}^n \om_{\mathbf{t}}(f_k)\tilde{\Omega}_{\mathbf{t}} \ra \prod_{k=1}^n \om_{\mathbf{t}}(Of_k)\tilde{\Omega}_{\mathbf{t}} \end{equation} and $\tilde{\F}_{\mathbf{t}}(O_1)\tilde{\F}_{\mathbf{t}}(O_2)= \tilde{\F}_{\mathbf{t}}(O_1\cdot O_2)$ for $O_1,O_2\in\mathcal{O}(\K)$. This induces an action on the $^*$-algebra $\tilde{\pi}_{\mathbf{t}}(\A(\K))$: \begin{equation} \tilde{\Gamma}_{\mathbf{t}}(O): X\mapsto \tilde{\F}_{\mathbf{t}}(O)X \tilde{\F}_{\mathbf{t}}(O^*). \end{equation} \noindent Certain operators play a similar role to that of the Wick products in quantum field theory \cite{Simon,Str.Wigh.} or for the $q$-deformed Brownian motion \cite{Boz.Ku.Spe., Boz.Sp.2}. \begin{definition}\label{def.Wick.prod} {\rm Let $\{P, F\}$ be a partition of the ordered set $\{1,\dots ,2p+n\}$ with $|P|=2p$ and $|F|=n$. Let $\mathcal{V}=\{(l_1,r_1),\dots ,(l_{p},r_{p})\} \in\mathcal{P}_2(P)$ and $\mathbf{f}:F\ra\K$. \\ For every $\mathcal{V}'=\{(l'_1,r'_1),\dots ,(l'_{p'},r'_{p'})\}\in\mathcal{P}_2(P')$ with $P'\subset F$ we introduce \begin{equation} \eta_\mathbf{f}(\mathcal{V}')= \prod_{i=1}^{p'}\br \mathbf{f}(l'_{i}),\mathbf{f}(r'_{i})\ke. \end{equation} The \textit{generalised Wick product} associated to $(\mathcal{V},\mathbf{f})$ is the operator $\Psi(\mathcal{V},\mathbf{f})$ determined recursively by \begin{eqnarray} \Psi(\mathcal{V},\mathbf{f}) & + & \sum_{\emptyset\neq P'\subset F}~~\sum_{\mathcal{V}'\in \mathcal{P}_2(P')} \eta_\mathbf{f}(\mathcal{V}')\cdot \Psi(\mathcal{V}\cup \mathcal{V}',\mathbf{f}\upharpoonright_{F\setminus P'} )= M(\mathcal{V},\mathbf{f}) \nonumber\\ M(\mathcal{V},\mathbf{f}) & := & \mathop{\mathrm{w{-}lim}}_{n\ra \infty}\prod_{k=1}^{2p+n} \om_{\mathbf{t}}(f_{k,n}) \label{eq.Wick.prod} \end{eqnarray} where $f_{k,n}:=\mathbf{f}(k)$ for $k\in F$ and $f_{l_i,n}= f_{r_i,n}=e_{np+i}$ for $i=1,\dots, p$ with $(e_l)_{l\in\mathbb{N}}$ a set of normalised vectors, orthogonal to each other and to the vectors $(\mathbf{f}(k))_{k=1}^n$. } \end{definition} \noindent \textbf{Remarks.} 1) The right side of the last equation needs some clarifications. All operators appearing in (\ref{eq.Wick.prod}) are defined on the domain $D$. The operator $M(\mathcal{V},\mathbf{f})$ is defined on $D$ by its matrix elements. If $\psi_i=\prod_{a=1}^{m_i}\om_{\mathbf{t}}(g_a^{(i)})\tilde{\Omega}_{\mathbf{t}}$ for $i=1,2$ are vectors in $D$ then from the definition of the Gaussian state follows immediately that \begin{equation} \lim_{n\ra\infty} \br\psi_1,\prod_{k=1}^{2p+n}\om_{\mathbf{t}}(f_{k,n})\psi_2\ke= \br\psi_1,\prod_{k=1}^{2p+n}\om_{\mathbf{t}}(f_{k,n_0})\psi_2\ke \end{equation} \noindent for $n_0\in\N$ large enough such that the subspaces $\K_1$ and $\K_2$ of $\K$ spanned by $(e_{n_0+j})_{j\geq 0}$ respectively by $(g_a^{(i)})_{a=1}^{m_i}$ for $ i=1,2$ are orthogonal to each other. The limit does not depend on the choice of the vectors $(e_i)_{i\in\mathbb{N}}$ (as long as they are normal and orthogonal to each other) but depends only on their positions in the monomial which are determined by the pair partition $\mathcal{V}$. Thus $M(\mathcal{V},\mathbf{f})$ is well defined. \noindent 2) If the vectors $(\mathbf{f}(k))_{k=1}^n$ are orthogonal on each other then $\eta_\mathbf{f}(\mathcal{V}')=0$, thus $\Psi(\mathcal{V},\mathbf{f})=M(\mathcal{V},\mathbf{f})$. \noindent 3) The dense domain $D$ is spanned by the vectors of the form $\Psi(\mathcal{V},\mathbf{f})\tilde{\Omega}_{\mathbf{t}}$. Indeed let $\psi= \prod_{k=1}^{n}\om_{\mathbf{t}}(\mathbf{f}(k))\tilde{\Omega}_{\mathbf{t}}$; then \begin{equation} \psi=\Psi(\emptyset,\mathbf{f})\tilde{\Omega}_{\mathbf{t}}+ \sum_{\emptyset\neq P'\subset F}~\sum_{d'\in \mathcal{P}_2(P')} \eta_f(\mathcal{V}')\cdot \Psi(\mathcal{V}',\mathbf{f}\upharpoonright_{F\setminus P'} ) \tilde{\Omega}_{\mathbf{t}} \end{equation} with $F=\{1,\dots ,n\}$. \noindent 4) The choice for $\{1,\dots ,2p+n\}$ as the underlying ordered set is not essential. It is useful to think of $\Psi(\mathcal{V},\mathbf{f})$ in terms of an arbitrary underlying finite ordered set $X$, where $\mathcal{V}\in\mathcal{P}_2(A)$, $A\subset X$, $\mathbf{f}: X \setminus A\ra \K$. For example we can consider the set $X=\{0\}$ and $\mathbf{f}(0)=h$, then $\Psi(\emptyset, \mathbf{f})=\om_{\mathbf{t}}(h)$. \noindent The relation between $M(\mathcal{V},\mathbf{f})$ and $\Psi(\mathcal{V},\mathbf{f})$ is similar to the one between moments and cumulants. \begin{lemma}\label{lemma.moments.cumulants} Let $\Psi(\mathcal{V},\mathbf{f}),$ $M(\mathcal{V},\mathbf{f})$ be as in Definition \ref{def.Wick.prod}. The equations (\ref{eq.Wick.prod}) can be inverted into: \begin{equation} \Psi(\mathcal{V},\mathbf{f})=M(\mathcal{V},\mathbf{f})+ \sum_{\emptyset\neq P'\subset F}~\sum_{\mathcal{V}'\in\mathcal{P}_2(P')} (-1)^{\frac{|P'|}{2}} \eta_\mathbf{f}(\mathcal{V}')\cdot M(\mathcal{V}\cup \mathcal{V}',\mathbf{f}\upharpoonright_{F\setminus P'}). \end{equation} \end{lemma} \noindent \textit{Proof.} Direct application of the definition. \qed \noindent Let $X$ be an ordered set. Let $\{P,F\}$ be a partition of $X$ into disjoint sets and consider a pair $(\mathcal{V}\in\mathcal{P}_2(P),\mathbf{f}:F\ra \K)$. Then for $X^*$ as underlying set we define the pair $(\mathcal{V}^*, \mathbf{f}^*)$ where $\mathcal{V}^*\in\mathcal{P}_2(X^*)$ contains the same pairs as $\mathcal{V}$ but with the reversed order and $\mathbf{f}^*=\mathbf{f}$. \begin{lemma}\label{lemma.adjoint} With the above notations the following relation holds: \begin{equation} \Psi(\mathcal{V},\mathbf{f})^*=\Psi(\mathcal{V}^*,\mathbf{f}^*). \end{equation} \end{lemma} \noindent \textit{Proof.} Apply Lemma \ref{lemma.moments.cumulants} and use $M(\mathcal{V},\mathbf{f} )^*=M(\mathcal{V}^*,\mathbf{f}^*)$ which follows directly from Definition \ref{def.Wick.prod}. \qed \noindent For two ordered sets $X$ and $Y$ we define their concatenation $X+Y$ as the disjoint union with the original order on $X$ and $Y$ and with $x