数理解析研究所講究録 1227 巻 2001 年 124-139

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A construction of interacting Fock spaces
derived from quantum decomposition
and its applications
Yukihiro Hashimoto 1 (橋本行洋)
Graduate school of Polymathematics, Nagoya University(名古屋大学大学院多元数理科学研究科)

1. Introduction

Random walks on discrete graphs have been discussed in algebraic probability theory as they
provide us arich resource of many new ideas. Recently we introduce the idea of quantum
be adiscrete group with alength
decomposition in the category of discrete groups [12]. Let$a\inG$. Suppose that an element
has the property:
function$G$$|\cdot$$Garrow \mathrm{N}\cup\{0\}$$|$$|ag|=|g|\pm 1$ ,$\forall g\in G$.

Then we naturally come to a‘decomposition’$a=a^{+}+a^{-}$

where $a^{+}g=ag$ if $|ag|=|g|+1$ (resp. $a^{-}g=ag$ if $|ag|=|g|-1$ ) and $a^{+}g=0$ if $|ag|\neq|g|+1$(resp. $a^{-}g=0$ if $|ag|\neq|g|-1$ ). The idea traces back to the Hudson-Parthasarathy theory [14]on quantum analysis of stochastic evolutions. Our approach of the quantum decomposition has
advantage in quantum analysis of set partition statistics. We construct asequence of one-mode
interacting Fock spaces associated with a filtration of the group. Then we directly observe an
intrinsic Fock space structure associated with discrete Laplacians on large Cayley graphs. We
show stochastic convergence of such asequence of interacting Fock spaces and obtain afully
quantum version of the de Moivre-Laplace theorem for the discrete Laplacians (Theorem 4.1).
In this paper we present various applications of quantum decomposition to the limit theorems
for the discrete Laplacians on Cayley graphs. In Section 5.2, we see that the Haagerup statesgive rise to atransform on set partition statistics through their coherent expression. Hence we
explain the reason for the appearance of the free Poisson law in the context of the central limit
theorem under the Haagerup states and reproduce the result in [10]. We observe the asymptotic
behavior of products of free elements in Section 5.3 and the anti-commutation of free elements
in Section 5.4. The idea of quantum decomposition is applied to more general discrete graphs.
As an instance, we illustrate aresult on the Hamming graph in Section 6.

2. Quantum decomposition in adiscrete group:
and aset
Let be adiscrete group equipped with alength function. Throughout this paper we adopt the conventionof generatorsfor simplicity. We say that is compatible with respect to the lengthand assume
iffunction,(A1)
for any$g|=|g|\pm 1$ for any(A2)
and $g\in G$ .
Then we introduce a‘quantum’ decomposition of :$G$$|\cdot|$$\Sigma$$g_{-\alpha}\neq g_{\alpha}$$|\cdot$$|$$|g_{\alpha}|=1$$|g_{\alpha}\cdot$$g_{\alpha}\in\Sigma$$g_{\alpha}\in\Sigma$$g_{\alpha}$(2.1)

1JSPS Research Fellow$Garrow \mathrm{N}_{0}:=\mathrm{N}\mathrm{U}\{0\}$$\{g_{-\alpha}=g_{\alpha}^{-1}\}$$\Sigma:=\{g_{\alpha}|\alpha\in \mathrm{Z}^{\mathrm{x}}\}$$g_{\alpha}=g_{\alpha}^{+}+g_{\alpha}^{-}$.

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We take the left regular representationby$\delta_{g_{\alpha}g}$

(2.2)

Here

$\pi(g_{\alpha}^{+})\delta_{g}=\{$$\pi$: $Garrow l^{2}(G)$ . Let

if $|g_{\alpha}g|=|g|+1$ ,otherwise ,,0,$\pi(g_{\alpha}^{\pm})$

be bounded operators defined$\delta_{g_{\alpha}g}$$\pi(g_{\alpha}^{-})\delta_{g}=\{$

if $|g_{\alpha}g|=|g|-1$ ,otherwise .

,0,denotes the characteristic function of asinglet. Then we see that,andfor $f\in l^{2}(G)$ .
Afamily of subsets $\Sigma^{(N)}:=\{g_{\alpha}||\alpha|=1, \ldots, N\}\subset\Sigma$ gives afiltration of, whereis the subgroup generated by. For each $g\in G^{(N)}$ put$\{g\}$$\delta_{g}$$\pi(g_{\alpha}^{-})$$\pi(g_{\alpha}^{+})^{*}=\pi(g_{-\alpha}^{-})$$\pi(g_{\alpha})=\pi(g_{\alpha}^{+})+$$||\pi(g_{\alpha}^{\pm})f||_{l^{2}}\leq||f||_{l^{2}}$$G:G^{(1)}\subset G^{(2)}\subset$$G^{(N)}$$\ldots\subset G$$\Sigma^{(N)}$$\omega_{+}^{(N)}(g):=\{(g_{\alpha}, x)\in\Sigma^{(N)}\cross G^{(N)}|\pi(g_{\alpha}^{+})\delta_{x}=\delta_{g}\}$,$\omega_{-}^{(N)}(g):=\{(g_{\alpha}, x)\in\Sigma^{(N)}\cross G^{(N)}|\pi(g_{\alpha}^{-})\delta_{x}=\delta_{g}\}$

.

(2.2)

We note that
implies $|g|=|x|\pm 1$ and $\#\omega_{+}^{(N)}(g)+\#\omega_{-}^{(N)}(g)=2N$ by assumption(A2). With these notations we assume(A3) for each, there exist
and $C_{n}>0$ such that$\pi(g_{\alpha}^{\pm})\delta_{x}=\delta_{g}$$n\in \mathrm{N}$$\#$(A4) for each$\omega_{n}\in \mathrm{N}${ $g\in G^{(N)}||g|=n$ and$n\in \mathrm{N}$$\#\omega_{+}^{(N)}(g)\neq\omega_{n}$}$\leq C_{n}(2N)^{n-1}$,,$\sup_{N\in \mathrm{N}}\sup_{g\in G(N)|g|=n},\#\omega_{+}^{(N)}(g)=:W_{n}<\infty$

and $\lim\sup_{narrow\infty}W_{n}^{1/n}<\infty$ .
(A5) if $|g|=n$ and $g\in G^{(N)}$ , there exists an -tuple$n$,$(g_{\alpha_{1}}, \ldots, g_{\alpha_{n}})\in(\Sigma^{(N)})^{n}$$\delta_{g}=\pi(g_{\alpha_{1}}^{+})\cdots\pi(g_{\alpha_{n}}^{+})\delta_{e}$such that.

Canonical examples are given in section 5.

3. Asymptotic interacting Fock space associated with adiscrete group
For each subgroup, we construct aone-mode interacting Fock space [2] as follows. The
vacuum vector is defined by. Avector $v\in l^{2}(G^{(N)})$ is called -homogeneous if it has
aform of$G^{(N)}$$|0\rangle$(3.1)$:=\delta_{e}$$n$$v= \sum_{|g\in G^{(N)},g|=n}v(g)\pi(g)|0\rangle$

We define asequence of homogeneous vectors,$v(g)\in \mathrm{C}$$|n\rangle^{(N)}\in l^{2}(G)$.

for each$|n \rangle^{(N)}:=(\frac{1}{\sqrt{2N}})^{n}\sum_{|g\in G^{(N)},g|=n}\pi(g)|0\rangle$$n$and $N$ , by.

Then the operators$a_{N}^{+}$$a_{\overline{N}}$$:=$$\pi(g_{-N}^{+})+\cdots+\pi(g_{-1}^{+})+\pi(g_{1}^{+})+\cdots+\pi(g_{N}^{+})$$:=$$\pi(g_{-N}^{-})+\cdots+\pi(g_{-1}^{-})+\pi(g_{1}^{-})+\cdots+\pi(g_{N}^{-})$

behave like acreation and an annihilation, respectively

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Lemma 3.1. For any $n\geq 0$ ,$\frac{a_{N}^{+}}{\sqrt{2N}}|n\rangle^{(N)}=\omega_{n+1}|n+1\rangle^{(N)}+v_{n+1}(\frac{1}{\sqrt{N}})$$\frac{a_{\overline{N}}}{\sqrt{2N}}|n+1\rangle^{(N)}=|n\rangle^{(N)}+v_{n}(\frac{1}{N})$

,

and

,

$\frac{a_{\overline{N}}}{\sqrt{2N}}|0\rangle$

$=0$

,

where $v_{n}(N^{m})$ is an -homogeneous vector such that $||v_{n}(N^{m})||_{l^{2}}=O(N^{m})$ .
Proof. By definition we have
$n$

(3.2)

$\frac{a_{N}^{+}}{\sqrt{2N}}|n\rangle^{(N)}=(\frac{1}{\sqrt{2N}})^{n+1}\sum_{|\alpha|=1,\ldots,N}\pi(g_{\alpha}^{+})\sum_{|x\in G^{(N)},x|=n}\pi(x)|0\rangle$.

By virtue of assumption (A5), the right hand side becomes(3.3)$=( \frac{1}{\sqrt{2N}})^{n+1}\sum_{|g\in G^{(N)},g|=n+1_{(g_{\alpha\prime}x}}\sum_{)\in\omega_{+}^{(N)}(g)}\pi(g_{\alpha}^{+})\pi(x)|0\rangle$$=( \frac{1}{\sqrt{2N}})^{n+1}\sum_{|g\in G^{(N)},g|=n+1}\#\omega_{+}^{(N)}(g)\pi(g)|0\rangle$,$=\omega_{n+1}|n+1\rangle^{(N)}+v$

where

$v$

is an $(n+1)$ -homogeneous vector$v=( \frac{1}{\sqrt{2N}})^{n+1}\sum_{g\in G^{(N)},|g|=n+1}(\#\omega_{+}^{(N)}(g)-\omega_{n+1})\pi(g)|0\rangle$.

It follows from assumptions (A3) and (A4) that $||v||^{2}=O(1/N)$ . The second relation is obtained
similarly. The last one is proved by definition.
it holds thatand N
Lemma 3.2. For each n$\square$$\geq 0$$\geq 1$$\langle n|n\rangle^{(N)}=\frac{1}{(\omega_{n})!}+O(\frac{1}{N})$

,

.
where
Proof. Note that $(2N)^{n}\langle n|n\rangle^{(N)}=\#\{g\in G^{(N)}||g|=n\}$ . Then, the relation (3.2)(3.3) andassumptions (A3) (A4) imply that$(\omega_{n})!=\omega_{1}\omega_{2}\cdots$ $\omega_{n}$$2N\#\{g\in G^{(N)}||g|=n\}=\omega_{n+1}\#\{g\in G^{(N)}||g|=n+1\}+O(N^{n})$ .

Hence the assertion follows.
be aone-dimensional complex vector space equipped with apre-scalar
Definition 3.3. Let
product,,,$\square$$\Gamma_{n}^{(N)}$$(z|w)_{n}^{(N)}:=\overline{z}w\langle\Phi(n)|\Phi(n)\rangle^{(N)}$

where$|\Phi(n)\rangle^{(N)}$$z$$w\in \mathrm{C}$stands for anumber vector defined by$|\Phi(n)\rangle^{(N)}:=(\omega_{n})!|n\rangle^{(N)}$The completion of the orthogonal sum$\oplus_{n}\{\Gamma_{n}^{(N)}, (\cdot |\cdot)_{n}^{(N)}\}$.

is denoted by$\Gamma(G^{(N)})$

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By Lemmas 3.1 and 3.2, we have for any $n\geq 0$ ,$\frac{a_{N}^{+}}{\sqrt{2N}}|\Phi(n)\rangle^{(N)}=|\Phi(n+1)\rangle^{(N)}+v_{n}(\frac{1}{\sqrt{N}})$,$\frac{a_{\overline{N}}}{\sqrt{2N}}|\Phi(n+1)\rangle^{(N)}=\omega_{n+1}|\Phi(n)\rangle^{(N)}+v_{n}(\frac{1}{N})$,$\frac{a_{\overline{N}}}{\sqrt{2N}}|\Phi(0)\rangle^{(N)}=0$

and $\langle\Phi(n)|\Phi(n)\rangle^{(N)}=(\omega_{n})!+O(1/N)$ . With eachwith parametersinteracting Fock space
Definition 3.4. For
we define$\Gamma(G^{(N)})$

we thereby associate aone-mode. (See [2] for definitions.)$G^{(N)}$$\{\lambda_{0}:=1, \lambda_{n}:=(\omega_{n})!\}$$\lambda\in \mathrm{C}$$| \mathcal{E}(\lambda)\rangle^{(N)}:=\sum_{n=0}^{\infty}\lambda^{n}|n\rangle^{(N)}=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{(\omega_{n})!}|\Phi(n)\rangle^{(N)}$,converges. We callacoherent vector.
in an asymptotic sense, in fact,
It is shown that acoherent vector is an eigenvector of
in a neighborhood of 0we have
Lemma 3.5. For A

whenever$|\mathcal{E}(\lambda)\rangle^{(N)}$$\langle \mathcal{E}(\lambda)|\mathcal{E}(\lambda)\rangle=\sum_{n=0}^{\infty}|\lambda|^{2n}/(\omega_{n})!$$a_{\overline{N}}$$\in \mathrm{C}$$\frac{a_{\overline{N}}}{\sqrt{2N}}|\mathcal{E}(\lambda)\rangle^{(N)}=\lambda|\mathcal{E}(\lambda)\rangle^{(N)}+v(\frac{1}{N})$.

Proof. Assumption (A4) yields the estimate:.

$|| \frac{a_{\overline{N}}}{\sqrt{2N}}|\mathcal{E}(\lambda)\rangle^{(N)}-\lambda|\mathcal{E}(\lambda)\rangle^{(N)}||=O(\frac{1}{N})$$\square$Lemma 3.6. Let $v\in l^{2}(G^{(N)})$ be an n-homogeneous vector. Then,$(n+1)-$ and $(n-1)$ -homogeneous vectors $with$ the norm estimates:$a_{N}^{+}v$

Proof. Suppose that$v$$||a_{N}^{+}v||_{l^{2}}^{2}\leq 2NW_{n}||v||_{l^{2}}^{2}$,$||a_{N}^{-}v||_{l^{2}}^{2}\leq 2NW_{n}||v||_{l^{2}}^{2}$.

and$a_{\overline{N}}v$

are respectively

is given as in (3.1). Then,$||a_{N}^{+}v||^{2}= \sum_{|g\in G^{(N)},g|=n+1}||(\sum v(x)\pi(g_{\alpha}^{+})\pi(x))|0\rangle(g_{\alpha},x)\in\omega_{+}^{(N)}(g)||^{2}$$= \sum_{|g\in G^{(N)},g|=n+1}|\sum v(x)(g_{\alpha},x)\in\omega_{+}^{(N)}(g)|^{2}\leq\sum_{|g\in G^{(N)},g|=n+1}\#\omega_{+}^{(N)}(g)(\sum|v(x)|^{2})(g_{\alpha},x)\in\omega_{+}^{(N)}(g)$,$\leq W_{n}\sum_{|x\in G(N),x|=n}\#\omega_{-}^{(N)}(x)|v(x)|^{2}\leq 2NW_{n}||v||^{2}$

where $.\#\omega_{-}^{(N)}(x)\leq 2N$ is taken into account. Assumption(A3)implies the inequality for$a_{\underline{\overline{N}}}$

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4. Interacting Fock space in the limit
We construct an interacting Fock space for the limit ofand
as $Narrow\infty$ . Let
denote the one-mode interacting Fock space with parameters. Byis the completion of the orthogonal sum of one-dimensional space
definition
equipped with apre-scalar product$a_{N}^{+}$$\Gamma(G)$$a_{\overline{N}}$$\{\lambda_{0}:=1, \lambda_{n}:=(\omega_{n})!\}$$\Gamma(G)$$\Gamma_{n}:=\mathrm{C}|\Phi(n)\rangle$$(z|w)_{n}:=\overline{z}w\langle\Phi(n)|\Phi(n)\rangle$

,

$z$

where
stands for anumber vector with norm
annihilation
are uniquely determined by$|\Phi(n)\rangle$,$w\in \mathrm{C}$,$\sqrt{\lambda_{n}}=\sqrt{(\omega_{n})!}$. The creation$a^{+}$

and$a^{-}$$a^{+}|\Phi(n)\rangle=|\Phi(n+1)\rangle$

For A$\in \mathrm{C}$,$a^{-}|\Phi(n+1)\rangle=\omega_{n+1}|\Phi(n)\rangle$,$a^{-}|\Phi(0)\rangle=0$.

, we define acoherent vector(4.1)$| \mathcal{E}(\lambda)\rangle:=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{(\omega_{n})!}|\Phi(n)\rangle$

whenever
For $u= \sum u_{n}|\Phi(n)\rangle\in\Gamma(G)$ , we put$\langle \mathcal{E}(\lambda)|\mathcal{E}(\lambda)\rangle=\sum_{n=0}^{\infty}|\lambda|^{2n}/(\omega_{n})!$(4.2)

converges. By definition,$u^{(N)}:= \sum u_{n}|\Phi(n)\rangle^{(N)}\in\Gamma(G^{(N)})$

Theorem 4.1. Let $m\geq 1$ and$\epsilon_{1}$,$\ldots$,$\epsilon_{m}\in\{\pm\}$$a^{-}|\mathcal{E}(\lambda)\rangle=\lambda|\mathcal{E}(\lambda)\rangle$.

. Then, for any $u\in\Gamma(G)$ and$n\in \mathrm{N}_{0}$$\lim_{Narrow\infty}\langle u^{(N)}|(\frac{a_{N}^{\epsilon_{1}}}{\sqrt{2N}})\cdots(\frac{a_{N}^{\epsilon_{m}}}{\sqrt{2N}})|\Phi(n)\rangle_{\Gamma()}^{(N)}c(N)=\langle u|a^{\epsilon_{1}}\cdots a^{\epsilon_{m}}|\Phi(n)\rangle_{\Gamma(G)}$Proof. By definition, there exists aconstantindices of the product such that$\omega_{n}(\epsilon_{1}, \ldots, \epsilon_{m})$holds.

depending on$a^{\epsilon_{1}}\cdots a^{\epsilon_{m}}|\Phi(n)\rangle=\omega_{n}(\epsilon_{1}, \ldots, \epsilon_{m})|\Phi(l)\rangle$

we have.

$n\in \mathrm{N}$and the,where $l=n+\#\{i|\epsilon_{i}=+\}-\#\{i|\epsilon_{i}= -\}$ . Then, combining Lemmas 3.1, 3.2 and 3.6 we
see that$( \frac{a_{N}^{\epsilon_{1}}}{\sqrt{2N}})\cdots(\frac{a_{N}^{\epsilon_{m}}}{\sqrt{2N}})|\Phi(n)\rangle^{(N)}=\omega_{n}(\epsilon_{1}, \ldots, \epsilon_{m})|\Phi(l)\rangle^{(N)}+|v\rangle^{(N)}$wherefollows.

$|v\rangle^{(N)}$,is an -homogeneous vector whose norm is of order $1/\sqrt{N}$ . Then the assertion$l$$\square$

5. Applications to limit theorems in algebraic probability theory
5.1. Central limit theorems on the vacuum state. Since Theorem 4.1 is nothing but
ageneral form of limit theorems in algebraic probability theory, it particularly leads to thealgebraic central limit theorem with respect to the vacuum state:$\langle 0|\cdot|0\rangle$(5.1)$\lim_{Narrow\infty}\langle 0|(\frac{a_{N}^{+}+a_{N}^{-}}{\sqrt{2N}})^{m}|0\rangle_{\Gamma()}^{(N)}c(N)=\langle\Phi(0)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle_{\Gamma(G)}$.

(resp. $F$ ) be afree abelian group (resp. afree grouP) generated by, whereaccording to our convention. Each element $g\in A$ (resp. $g\in F$ ) has acanonicalexpression in terms of :

Let

$A$

$\Sigma:=\{g_{\alpha}\}$$g_{\alpha}^{-1}=g_{-\alpha}$$\Sigma$$g=g_{\alpha_{1}}^{\epsilon_{1}}\cdots g_{\alpha_{m}}^{\epsilon_{m}}$

,

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,

$0<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{m}$(resp.

$\epsilon_{i}\in \mathrm{Z}$$|\alpha_{1}|\neq|\alpha_{2}|\neq\cdots\neq|\alpha_{m}|$,$\epsilon_{i}\in \mathrm{Z}$),and areduced length function is defined by$|g|=|\epsilon_{1}|+\cdots+|\epsilon_{m}|$.

Conditions (AI)(A2)(A5) are obvious. As for (A3)(A4), we employ some combinatorial arguments. In fact, we see for $g\in A$ (resp. $g\in F$ ) with $|g|=n$ ,$\#\omega_{+}^{(N)}(g)\leq n$, $\#\{g\in A||g|=n, \#\omega_{+}^{(N)}(g)<n\}\leq(2N)^{n-1}$ ,$2N-n\leq\#\omega_{-}^{(N)}(g)\leq 2N-1$ ,(resp. $\#\omega_{+}^{(N)}(g)=1$ , $\#\omega_{-}^{(N)}(g)=2N-1$ ).

Hence we have $\omega_{n}=n$ (resp.
), which implies that the interacting Fock space in thelimit is the one-mode boson (resp. free) Fock space. In fact, by (5.1) the limit distributionof the field operator
under the vacuum state is determined by its $2m$-th
moments, which is well known:$\omega_{n}=1$$(a_{N}^{+}+a_{N}^{-})/\sqrt{2N}$$d\mu$$\frac{(2m)!}{2^{m}m!}$for $G=A$ ,,$\langle 0|(a^{+}+a^{-})^{2m}|0\rangle_{\Gamma(G)}=\{$$\frac{(2m)!}{(m+1)!m!}$,for $G=F$.

In other words,$\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}dx$

for $G=A$ ,,$d\mu(x)=\{$$\frac{1}{2\pi}\chi[-2,2]\sqrt{4-x^{2}}dx$,

for $G=F$.

5.2. Limit theorems for Haagerup states. It is well known $([4][6])$ that the Haagerup
functionon afinitely generated free groupis positive definite.
In the previous paper [10], the limit distribution of the field operator
under
the Haagerup state was obtained by way of Fourier transform of the Bessel function. The
result was thought as an analogy of acentral limit theorem for avacuum state, though thelimit distribution had peculiar properties. In fact, the limit distribution coincides with the free
Poisson law [22] up to translation. It is noticeable that limit behaviors of the field operators
under avacuum state are completely described by pair-partition statistics [3], while the Poissonlaws are induced from all set partition statistics [14][19][17]. The next result shows that theHaagerup states give rise to atransform of set partition statistics through acoherent expressionof the Haagerup functions.
Let us first consider ageneral interacting Fock space with parameters. For $n\geq$, let:
be a shift operator on the polynomial ring in ’s defined by. As,
is well defined.
$F^{(N)}$$\varphi_{\gamma}(g)=\gamma^{|g|}(0\leq\gamma<1)$$(a_{N}^{+}+a_{N}^{-})/\sqrt{2N}$$\Gamma$$0$$\sigma_{n}$$\mathrm{C}[\omega_{1}, \omega_{2}, \ldots]arrow \mathrm{C}[\omega_{1}, \omega_{2}, \ldots]$$\sigma_{n}(\omega_{i})=\omega_{i+n}$$\langle\Phi(p)|a^{\epsilon_{1}}\cdots a^{\epsilon_{m}}|\Phi(q)\rangle_{\Gamma}\in \mathrm{C}[\omega_{1}, \omega_{2}, \ldots]$$\{\lambda_{n}=(\omega_{n})!\}$$\omega_{i}$$\sigma_{n}(\langle\Phi(p)|a^{\epsilon_{1}}\cdots a^{\epsilon_{m}}|\Phi(q)\rangle_{\Gamma})$130
Proposition 5.1. For an interacting Fock spaceby (4. f), we have a recurrence decomposition$(\mathrm{I}^{\ovalbox{\tt\small REJECT}}, \{\mathrm{A}_{n}\})$

and a coherent vector$|\mathrm{f}(\mathrm{A}))_{\mathrm{B}}$given$\langle \mathcal{E}(\lambda)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle_{\Gamma}=\langle\Phi(0)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle_{\Gamma}$$+ \sum_{n=1}^{\infty}\sum_{l=0}^{\lfloor(m-1)/2\rfloor}\sigma_{n}(\langle\Phi(0)|(a^{+}+a^{-})^{2l}|\Phi(0)\rangle_{\Gamma})$.

$\frac{1}{\omega_{n}}\langle\Phi(n)|\lambda a^{+}a^{-}|\Phi(n)\rangle_{\Gamma}\cdot$$\frac{\lambda^{n-1}}{(\omega_{n-1})!}\langle\Phi(n-1)|(a^{+}+a^{-})^{m-2l-1}|\Phi(0)\rangle_{\Gamma}$.

Proof. By definition, the left hand side is a linear combination ofwhere $n\geq 0$ . The term for $n=0$ appears in the first term of the right hand side. Suppose$n\geq 1$ . If
, there exists aunique satisfying$\langle\Phi(n)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle$$l$$\langle\Phi(n)|a^{\epsilon_{1}}\cdots a^{\epsilon_{m}}|\Phi(0)\rangle\neq 0$

(5.2)
(5.3)

$a^{\epsilon_{2l+2}}\cdots$$a^{\epsilon_{m}}|\Phi(0)\rangle\in \mathrm{C}|\Phi(n-1)\rangle$$a^{\epsilon_{k}}\cdots a^{\epsilon_{2l}}|\Phi(n)\rangle\in\oplus_{p=n}^{\infty}\mathrm{C}|\Phi(p)\rangle$,$a^{\epsilon_{21+1}}=a^{+}$

for any,$1\leq k\leq 2l$.

It follows from (5.3) that.

$\langle\Phi(n)|a^{\epsilon_{1}}\cdots a^{\epsilon_{2l}}|\Phi(n)\rangle=\sigma_{n}(\langle\Phi(0)|a^{\epsilon_{1}}\cdots a^{\epsilon_{21}}|\Phi(0)\rangle)\langle\Phi(n)|\Phi(n)\rangle$

Note that$\langle\Phi(n)|a^{\epsilon_{2l+1}}|\Phi(n-1)\rangle=\langle\Phi(n)|a^{+}a^{-}|\Phi(n)\rangle/\omega_{n}$. Then we have adecomposition$(\omega_{n})!\langle\Phi(n)|a^{\epsilon_{1}}\cdots a^{\epsilon_{m}}|\Phi(0)\rangle$$=\sigma_{n}$($\langle\Phi(0)|a^{\epsilon_{1}}\cdots a^{\epsilon_{2l}}|\Phi(0)\rangle$)$\langle\Phi(n)|a^{+}a^{-}|\Phi(n)\rangle\cdot$$\cdot$$\langle\Phi(n-1)|a^{\epsilon_{21+2}}\cdots a^{\epsilon_{m}}|\Phi(0)\rangle$,

which leads to$( \omega_{n})!\langle\Phi(n)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle=\sum_{l=0}^{\lfloor(m-1)/2\rfloor}\sigma_{n}(\langle\Phi(0)|(a^{+}+a^{-})^{2l}|\Phi(0)\rangle)$.
Thus the assertion follows.
Let us return to the special case ofwithLemma 5.2. For any A$\in \mathrm{C}$$\langle\Phi(n)|a^{+}a^{-}|\Phi(n)\rangle\cdot$$\langle\Phi(n-1)|(a^{+}+a^{-})^{m-2l-1}|\Phi(0)\rangle$.
$\square$$\Gamma(F)$$|\lambda|<1$.

,$\varphi_{\frac{\lambda}{\sqrt{2N}}}(g)=\langle \mathcal{E}(\lambda)|\pi(g)|0\rangle_{\Gamma(F^{(N)}}^{(N)})$.
holds for any g
Proof. If $|g|=n$ , by definition we have$\in F^{(N)}$.

$\varphi_{\frac{\lambda}{\sqrt{2N}}}(g)=(\frac{\lambda}{\sqrt{2N}})^{n}=\sum_{k=0}^{\infty}\lambda^{k}\langle k|\pi(g)|0\rangle^{(N)}=\langle \mathcal{E}(\lambda)|\pi(g)|0\rangle^{(N)}$The assumption$|\lambda|<1$

guarantees that$|\mathcal{E}(\lambda)\rangle\in\Gamma(G^{(N)})$is defined.

$\square$

It follows from Theorem 4.1 that the moments of the field operator in the limit are given by.

$\lim_{Narrow\infty}\varphi_{\frac{\lambda}{\sqrt{2N}}}((\frac{a_{N}^{+}+a_{N}^{-}}{\sqrt{2N}})^{m})=\langle \mathcal{E}(\lambda)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle_{\Gamma(F)}$131
Since the parameters A.
$\langle^{\ovalbox{\tt\small REJECT}}\mathrm{j}\ovalbox{\tt\small REJECT}(0\ovalbox{\tt\small REJECT}(a"+(2^{-)^{2l}|^{\ovalbox{\tt\small REJECT}}\mathrm{I}^{\ovalbox{\tt\small REJECT}}(0)\rangle}.$are identically equal to 1, we have
Denoting by
the vacuum projection, we rewrite the factor${}^{\mathrm{t}}\mathrm{s}$$(\mathrm{r}_{n}(\langle^{\ovalbox{\tt\small REJECT}})\ovalbox{\tt\small REJECT}(\mathrm{O}\ovalbox{\tt\small REJECT}(()"+a\ovalbox{\tt\small REJECT}^{-})^{\ovalbox{\tt\small REJECT}}|^{\ovalbox{\tt\small REJECT}}\mathrm{D}(0)\rangle)\ovalbox{\tt\small REJECT}$$P_{0}$$\langle\Phi(n)|a^{+}a^{-}|\Phi(n)\rangle=1=\langle\Phi(0)|P_{0}|\Phi(0)\rangle$.

Then it follows from Proposition 5.1 that
(5.4)$\langle \mathcal{E}(\lambda)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle=\langle\Phi(0)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle$$+ \sum_{l=0}^{\lfloor(m-1)/2\rfloor}\langle\Phi(0)|(a^{+}+a^{-})^{2l}|\Phi(0)\rangle$.

$\langle\Phi(0)|\lambda P_{0}|\Phi(0)\rangle\cdot\langle \mathcal{E}(\lambda)|(a^{+}+a^{-})^{m-2l-1}|\Phi(0)\rangle$,

while an expansion of $(a^{+}+a^{-}+\lambda P_{0})^{m}$ gives$\langle\Phi(0)|(a^{+}+a^{-}+\lambda P_{0})^{m}|\Phi(0)\rangle=\langle\Phi(0)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle$$+ \sum_{l=0}^{\lfloor(m-1)/2\rfloor}\langle\Phi(0)|(a^{+}+a^{-})^{2l}|\Phi(0)\rangle$.

$\langle\Phi(0)|\lambda P_{0}|\Phi(0)\rangle\cdot\langle\Phi(0)|(a^{+}+a^{-}+\lambda P_{0})^{m-2l-1}|\Phi(0)\rangle$.

By induction we then obtain the following
Corollary 5.3. In the free Fock space$\{\Gamma(F), \{\lambda_{n}\equiv 1\}\}$, the identity$\langle \mathcal{E}(\lambda)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle_{\Gamma(F)}=\langle\Phi(0)|(a^{+}+a^{-}+\lambda P_{0})^{m}|\Phi(0)\rangle_{\Gamma(F)}$holds, where$P_{0}$

is the vacuum projection.

Remarks.
(1) Since in the free Fock space, the vacuum projection $P_{0}=I-a^{+}a^{-}$ , the random variable$a^{+}+a^{-}+\lambda P_{0}$ is nothing but the free
Poisson one [19], that is,$a^{+}+a^{-}+ \lambda P_{0}=\lambda+\frac{1}{\lambda}-(\sqrt{\lambda}a^{+}-\frac{1}{\sqrt{\lambda}})(\sqrt{\lambda}a^{-}-\frac{1}{\sqrt{\lambda}})$.

(2) Corollary 5.3 describes aGaussian-Poisson transform in terms of coherent vectors, while
Oravecz [16] recently introduced aGaussian-Poisson transform called the F-transform
for an arbitrary symmetric measure from the viewpoint of orthogonal polynomials.
(3) With the recurrence formula (5.4) one obtains afunctional identity for the moment
generating functions. Put $F(t)=1+ \sum_{n=1}^{\infty}F_{m}t^{m}$ withand$C(t)=(1-\sqrt{1-4t^{2}})/2t^{2}$ , which is the moment generating function of the normalizedsemi-circle law. Then we have the identity $F(t)=C(t)+\lambda tC(t)F(t)$ , from which thelimit distribution is computed easily:$F_{m}=\langle \mathcal{E}(\lambda)|(a^{+}+a^{-})^{m}|\Phi(0)\rangle$$d \mu(x)=\max\{0$ ,.

$\frac{1}{2}(1-\frac{1}{\lambda^{2}})\}\delta_{\lambda+\frac{1}{\lambda}}(x)+\frac{1}{2\pi}\chi_{[-2,2]}\frac{\sqrt{4-x^{2}}}{\lambda^{2}+1-\lambda x}dx$

This coincides with the result in [10] up to translation in the case of$\lambda\in[0,1]$.

132
5.3. Multiplication of free elements. Corollary 5.3 implies that under the Haagerup state,$S_{N}/\sqrt{2N}=\pi(g_{1})+\pi(g_{1}^{-1})+\cdots+\pi(g_{N})+\pi(g_{N}^{-1})$ is decomposed into acreation, anin the limit. In this subsection we present acentral limit theorem
andannihilation
associated with amultiplication of free elements, and again we obtain the same decomposition
as Corollary 5.3.
, where ’s are generator of order 2. Note,
Let be afree product ofin the sense ofthat ’s are free from each other with respect to the vacuum state
Voiculescu [22]. Let us consider aset of products of free elements{ :=\^a aj $(i\neq j)$ },. We use anew
generated bywhich are not free from each other, and asubgroup:with respect to E2 different from the one introduced in Section
length function
productany
has areduced expression in terms of ’s,
5.1. Since$a^{+}$$a^{+}a^{-}$$a^{-}$$F$$\mathrm{Z}/2\mathrm{Z}$$F=*_{i=1}^{\infty}\mathrm{Z}/2\mathrm{Z}\sigma_{i}$$\sigma_{i}$$\langle 0|\cdot|0\rangle$$\sigma_{i}$$\Sigma_{2}:=$$w_{ij}$$F_{2}\subset F$$\Sigma_{2}$$F_{2}arrow \mathrm{N}\circ$$|\cdot|_{2}$$w_{i_{1}j_{1}}\cdots$ $w_{i_{m}j_{m}}$$\sigma_{i}$$w_{i_{1}j_{1}}\cdots w_{i_{m}j_{m}}=\sigma_{k_{1}}\cdots\sigma_{k_{l}}$where$k_{1}\neq k_{2}\neq\cdots\neq k_{l}$, its length is defined by(5.5)$|w_{i_{1}j_{1}}\cdots w_{i_{m}j_{m}}|_{2}:=l/2$

Then we see for any

and$g=\sigma_{k_{1}}\cdots$ $\sigma_{k_{m}}\in F_{2}$

(5.6)

,

$|w_{\dot{l}j}g|_{2}=\{$$|g|_{2}+1$,$|g|_{2}-1$,$|g|_{2}$$w_{ij}$.

, the following three cases occur:

if I ,
if $(i,j)=(k_{2}, k_{1})$ ,.
if $j=k_{1}$ and$j$,$k_{1}$$i\neq k_{2}$for any $g\in F_{2}$ .
It follows from $|w_{ij}|_{2}=1$ and (5.6) that
The existence of the third case is anoticeable difference from our discussions before. Concerning the new case, we extend our argument. The condition (A2) in Section 2is replacedto$g|=|g|$ or $|g|\pm 1$ for any(A2)’
and $g\in G$ .
$|g|_{2}\in \mathrm{N}\circ$$g_{\alpha}\in\Sigma$$|g_{\alpha}\cdot$

In accordance with (A2)’ we have a‘quantum’ decomposition of
(5.7)

where

$g_{\alpha}=g_{\alpha}^{+}+g_{\alpha}^{-}+g_{[mathring]_{\alpha}}$$g_{\alpha}^{\pm}$

are defined by (2.1) and$g_{[mathring]_{\alpha}}$$\delta_{g_{\alpha}g}$$(F2, \Sigma 2)$$\Sigma_{2}^{(N)}$$F_{2}^{(N)}$

(5.8)

:

,

is given by$\pi(g_{[mathring]_{\alpha}})\delta_{g}=\{$Let us return to the example. For
the subgroup generated by
, and
by
for

$g_{\alpha}$0,,

if $|g_{\alpha}g|=|g|$ ,otherwise.

. We put $\Sigma_{2}^{(N)}:=\{w_{ij}|1\leq i\neq j\leq N\}$ .
denotesby (2.3) where we take
we define$F_{2}^{(N)}$$g\in F_{2}^{(N)}$$G^{(N)}$$\omega_{\pm}^{(N)}(g)$$\omega_{0}^{(N)}(g)$$\omega_{\mathrm{o}}^{(N)}(g):=\{(w_{ij}, x)\in\Sigma_{2}^{(N)}\cross F_{2}^{(N)}|\pi(g_{[mathring]_{\alpha}})\delta_{x}=\delta_{g}\}$and
By definition, we havewith $|g|_{2}=n\geq 1$ and for large $N$ ,to see that for$\omega_{+}^{(N)}(e)$$=\omega_{0}^{(N)}(e)$$=\emptyset$$\omega_{-}^{(N)}(e)=\Sigma_{2}^{(N)}$.

for the unit

$g\in F_{2}^{(N)}$$\#\omega_{+}^{(N)}(g)=1$, $\#\omega_{-}^{(N)}(g)=(N-1)^{2}$ and $\#\omega_{\mathrm{o}}^{(N)}(g)=N-2$ .

$e\in F_{2}$. It is easy

133
) $(g)+\#’\ovalbox{\tt\small REJECT}’)(g)+\#\mathrm{w}6^{N)}(g)\ovalbox{\tt\small REJECT}$

Also note that
3we define anumber vector$”\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$*’\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT})}\ovalbox{\tt\small REJECT}$$N(N$1). According to Section$| \Phi(n)\rangle^{(N)}:=(\frac{1}{\sqrt{\#\Sigma_{2}^{(N)}}})^{n}\mathit{9}\in F^{(N)}\sum_{(2),|g|_{2}=n}\pi(g)|0\rangle$,

and define canonical operators corresponding to the decomposition (5.7),$a_{N}^{\epsilon}:= \frac{1}{\sqrt{\#\Sigma_{2}^{(N)}}}\sum_{1\leq i\neq j\leq N}\pi(w_{ij}^{\epsilon})$where$\epsilon=\pm,$$\circ$,. Then by similar arguments in Section 3, we have,$a_{N}^{+}| \Phi(n)\rangle^{(N)}=|\Phi(n+1)\rangle^{(N)}+v_{n+1}(\frac{1}{N})$$a_{N}^{-}| \Phi(n+1)\rangle^{(N)}=|\Phi(n)\rangle^{(N)}+v_{n}(\frac{1}{N^{2}})$(5.9)$a_{N}^{\mathrm{o}}| \Phi(n)\rangle^{(N)}=|\Phi(n)\rangle^{(N)}+v_{n}(\frac{1}{N})$$a_{N}^{-}|0\rangle^{(N)}=a_{N}^{\mathrm{o}}|0\rangle^{(N)}=0$,,

for $n\geq 1$ ,

,

where
is ahomogeneous vector defined in Section 3. We also construct an asymptotic
by way of similar manner in Section 3.
one-mode Fock space,andbe the creation, the annihiLet
be the one-mode free Fock space,lation and the vacuum projection respectively, used in the previous subsection. It follows from
(5.9) that we have a’quantum’ central limit theorem associated with
as follows.
. Then for any $u\in\Gamma(F)$ and,, weTheorem 5.4. Let $m\geq 1$ and ,have$v_{n}$$\Gamma(F_{2}^{(N)})$$a^{+}$$\Gamma(F)$$a^{-}$$a^{\mathrm{o}}:=P\circ$$(F_{2}^{(N)}, \Sigma_{2}^{(N)})$$\epsilon_{1}$$\ldots$$\epsilon_{m}\in\{\pm, 0\}$$n\in \mathrm{N}_{0}$,$\lim_{Narrow\infty}\langle u^{(N)}|(\frac{a_{N}^{\epsilon_{1}}}{\sqrt{\#\Sigma_{2}^{(N)}}})\cdots(\frac{a_{N}^{\epsilon_{m}}}{\sqrt{\#\Sigma_{2}^{(N)}}})|\Phi(n)\rangle_{\mathrm{r}(F_{2}^{(N)})}^{(N)}=\langle u|a^{\epsilon_{1}}\cdots a^{\epsilon_{m}}|\Phi(n)\rangle_{\Gamma(F)}$where we use the notation (5.7),
Particularly we obtain aclassical central limit theorem.

Corollary 5.5.
.

$\lim_{Narrow\infty}\langle 0|(\frac{1}{\sqrt{N(N-1)}}\sum_{1\leq i\neq j\leq N}\pi(w_{ij}))^{m}|0\rangle_{\mathrm{r}(F_{2}^{(N)})}^{(N)}=\langle 0|(a^{+}+a^{-}+a^{\mathrm{o}})^{m}|0\rangle_{\Gamma(F)}$5.4. Anti-commutation of semi-circular elements. Throughout this paragraph, we use, we obtain various kinds of ‘quantum’
the notations in Section 5.3. By taking subsets ofdecomposition. See [11] for detail, where combinatorial moments are calculated by way of pathcounting on graphs. Here we observe one of the decompositions, which leads us an analysis ofanti-commutations of semi-circular elements.
for $0<\gamma<1$ ,Let us take asubset$\Sigma_{2}^{(N)}$$\Sigma_{2,\gamma}^{(N)}\subset\Sigma_{2}^{(N)}$$\Sigma_{2,\gamma}^{(N)}:=${$w_{ij} \in\Sigma_{2}^{(N)}|1\leq i\leq\max\{1$, $\gamma N\}<j\leq N$ or $1 \leq j\leq\max\{1$ , $\gamma N\}<i\leq N$ }.

134\yen denote the subgroup of \yen generated by$\mathrm{p}\ovalbox{\tt\small REJECT} 7$). Since (5.7) holds, we again obtain adecomposition(5.10)$w_{ij}=w_{ij}^{+}+w_{ij}^{-}+w_{ij}^{\mathrm{o}}$.

According to the decomposition, we define operators(5.11)’$a_{N}^{\epsilon}:= \frac{1}{\sqrt{v}}\sum_{w_{ij}\in F_{2\gamma}^{(N)}}\pi(w_{ij}^{\epsilon})|$. The purpose of this paragraph is to establish
where
and
alimit theorem on the operators (5.11) for aconstant . General cases are discussed in [11].
, we have
It is easy to see that for any$\epsilon=\pm,$$\circ$$v:=\#\Sigma_{2,\gamma}^{(N)}\sim 2\gamma(1-\gamma)N^{2}$$\gamma$$g\in F_{2,\gamma}^{(N)}$(5.12)$2(\gamma N-1)((1-\gamma)N-1)\leq\#\omega_{-}^{(N)}(g)$$\leq 2\gamma(1-\gamma)N^{2}$.

However, to determine, we need adelicate argument as follows. Let
and$I:=[1, \gamma N]\cap \mathrm{N}$ and $J:=(\gamma N, N]\cap \mathrm{N}$ . For each $n>0$ , let us decompose the setof elementswith $|g|_{2}=n$ into
disjoint subsets,$\#\omega_{+}^{(N)}(g)$$\#\omega_{\mathrm{o}}^{(N)}(g)$$\mathcal{V}_{n}$$2^{2n}$$g\in F_{2,\gamma}^{(N)}$$\mathcal{V}_{n}:=\{g\in F_{2,\gamma}^{(N)}||g|_{2}=n\}=\prod_{\eta_{1\prime\cdots\prime}\eta_{2n}\in\{I,J\}}\Delta(\eta_{1}, \ldots, \eta_{2n})$where$\Delta(\eta_{1}, \ldots, \eta_{2n})’ \mathrm{s}$are given by$\Delta(\eta_{1}, \ldots, \eta_{2n}):=\{\sigma_{i_{1}}\cdots\sigma_{i_{2n}}\in \mathcal{V}_{n}|i_{1}\in\eta_{1}, \ldots, i_{2n}\in\eta_{2n}\}$.

With the notations above, we see(5.13)$\#\omega_{+}^{(N)}(g)=1$, for$g\in\Delta(I, J, \eta_{3}, \ldots, \eta_{n})\cup\Delta(J, I, \eta_{3}, \ldots, \eta_{2n})$,$\#\omega_{+}^{(N)}(g)=0$, for$g\in\Delta(I, I, \eta_{3}, \ldots, \eta_{n})\cup\Delta(J, J, \eta_{3}, \ldots, \eta_{2n})$,$(1-\gamma)N-1\leq\#\omega_{\mathrm{o}}^{(N)}(g)\leq(1-\gamma)N$$\gamma N-1\leq\#\omega_{\mathrm{o}}^{(N)}(g)\leq\gamma N$

According to the decomposition of$\mathcal{V}_{n}$

, for

, for

$g\in\Delta(I, \eta_{2}, \ldots, \eta_{2n})$$g\in\Delta(J, \eta_{2}, \ldots, \eta_{2n})$, the number vector$| \Phi(n)\rangle^{(N)}:=(\frac{1}{\sqrt{v}})^{n}\sum_{g\in \mathcal{V}_{n}}\pi(g)|0\rangle$

has an orthogonal decomposition into$2^{2n}$homogeneous vector$| \Phi(n)\rangle^{(N)}=\sum_{\eta_{1\prime}\ldots,\eta_{2n}\in\{I,J\}}|\eta_{1}$,$\ldots$,$\mathrm{s}$$\eta_{2n}\rangle^{(N)}$

,

.

,

135
where
$\oplus_{\eta_{1}}$,$|\eta_{1}$,$\ldots$,. Then we define aFock space. It follows from (5.12) and (5.13), we have$\eta_{2n}\rangle^{(N)}:=\sum_{g\in\Delta(\eta_{1},\ldots,\eta_{2n})}\pi(g)|0\rangle/\sqrt{v}$$.,\eta_{2n}\in\{I_{\backslash }J\}|\eta 1$,$\ldots$,$\eta_{2n}\rangle^{(N)}$$a_{N}^{+}(|\eta_{1}, \ldots, \eta_{2n}\rangle^{(N)})=|I$$a_{N}^{-}|I$, ,$a_{N}^{-}|I$, ,$a_{N}^{\mathrm{o}}|I$, ,$a_{N}^{\mathrm{o}}|I$, ,(5.14)$J$$I$$\eta_{3}$$J$$a_{N}^{\mathrm{o}}|J$$I$$\eta_{3}$$\eta_{3}$, ,$a_{N}^{\mathrm{o}}|J$$\eta_{3}$

$I$

, ,
$J$

$\eta_{3}$$\eta_{3}$,$\ldots$, ,$J$$\eta_{1}$,,$\eta_{2n}\rangle^{(N)}=a_{N}^{-}|J$$\ldots$

, ,
$I$

,

, ,

$\eta_{2n}\rangle^{(N)}+|J$ $I$$\eta_{3}$,$\ldots$,$\ldots$$\eta_{2n}\rangle^{(N)}=\frac{1}{2}|\eta_{3}$,$\eta_{2n}\rangle^{(N)}=0$$\ldots$,$\eta_{2n}\rangle^{(N)}=a_{N}^{-}|J$ $J$,$\ldots$,$\eta_{2n}\rangle^{(N)}=\sqrt{\frac{\gamma}{2(1-\gamma)}}|J$ $J$,$\ldots$,$\eta_{2n}\rangle^{(N)}=\sqrt{\frac{\gamma}{2(1-\gamma)}}|J$ $I$,,$\eta_{2n}\rangle^{(N)}=\sqrt{\frac{1-\gamma}{2\gamma}}|I$$\ldots$,,$\eta_{2n}\rangle^{(N)}=\sqrt{\frac{1-\gamma}{2\gamma}}|I$$\ldots$$\eta_{3}$

, ,

, ,

, ,
$I$

$\eta_{3}$

, ,
$J$

$\ldots$

,

,

, ,

,

$\eta_{1}$$\eta_{3}$$\eta_{3}$$\eta_{3}$,$\ldots$,$\ldots$-linear$\eta_{2n}\rangle^{(N)}+v_{n}(\frac{1}{\sqrt{v}})$,,$\eta_{2n}\rangle^{(N)}+v_{n}(\frac{1}{\sqrt{v}})$,,$\ldots$,$\eta_{2n}\rangle^{(N)}+v_{n}(\frac{1}{\sqrt{v}})$,,$\ldots$,$\eta_{2n}\rangle^{(N)}+v_{n}(\frac{1}{\sqrt{v}})$.

$a^{-}$$\Gamma^{2}:=\oplus_{n=0}^{\infty}\Gamma^{2}(n)$$\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{|\eta_{1}, \ldots, \eta_{2n}\rangle|\eta_{i}\in\{I, J\}\}$

equipped with a canonical inner product,specified bytors
and$a^{+}$$\eta_{2n}\rangle^{(N)}+v_{n}(\frac{1}{v})$,,In view of the relations (5.14), we prepare atw0-mode free Fock space,$\Gamma^{2}(n):=\mathrm{C}$,$\eta_{2n}\rangle^{(N)}+v_{n}(\frac{1}{\sqrt{v}})$,$\ldots$,,,$\Gamma(F_{2,\gamma}^{(N)}):=$, where,$\langle\eta_{1}, \ldots, \eta_{2n}|\eta_{1}’, \ldots, \eta_{2n}’\rangle=\delta_{\eta_{1},\eta_{1}’}\cdots$$\delta_{\eta_{2n},\eta_{\acute{2}n}}$, and opera-$a^{\mathrm{o}}$$a^{+}(|\eta_{1}, \ldots, \eta_{2n}\rangle)=|I$$a^{-}|I$, ,$a^{-}|I$

, ,

$J$

$I$

$a^{\mathrm{o}}|I$, ,$a^{\mathrm{o}}|I$

, ,

$J$

$\eta_{3}$$\eta_{3}$$\eta_{3}$

, ,
$J$

$\eta_{1)}\ldots$,$\ldots$,$\eta_{2n}\rangle=a^{-}|J$,$\ldots$,$\eta_{2n}\rangle=a^{-}|J$

, ,
$I$

, ,
$J$

,

$\eta_{3}$$\eta_{3}$, ,$\eta_{2n}\rangle+|J$ $I$,$\ldots$,$\ldots$,$\ldots$,$\eta_{2n}\rangle=\sqrt{\frac{\gamma}{2(1-\gamma)}}|J$,$\ldots$,$\eta_{2n}\rangle=\sqrt{\frac{\gamma}{2(1-\gamma)}}|J$,,$\eta_{2n}\rangle=\sqrt{\frac{1-\gamma}{2\gamma}}|I$$\ldots$,,$\eta_{2n}\rangle=\sqrt{\frac{1-\gamma}{2\gamma}}|I$$\ldots$$\eta_{1}$,$\ldots$,$\eta_{2n}\rangle=\frac{1}{2}|\eta_{3}$,$\eta_{2n}\rangle=0$

, ,
$J$

$\eta_{3}$,,$\eta_{2n}\rangle$$\ldots$,,$\eta_{2n}\rangle$

,

,

,

$\ldots$,$\eta_{2n}\rangle$,,$\ldots$,$\eta_{2n}\rangle$,(5.15)$I$, ,$a^{\mathrm{o}}|J$$I$,$a^{\mathrm{o}}|J_{)}J$$\eta_{3}$$\eta_{3}$$\eta_{3}$

, ,

, ,
$I$

, ,
$J$

$I$

$\eta_{3}$$\eta_{3}$$\eta_{3}$,$\ldots$,$\eta_{2n}\rangle$,$\ldots$,$\eta_{2n}\rangle$,.

Summing uP the arguments above, we have established acentral limit theorem associatedwith the decomposition (5.10) as follows.
,
Theorem 5.6. Let $m\geq 1$ and ,. Then for any constant $0<\gamma<1$ ,,and ,we have$\epsilon_{1}$$\eta_{1}$$\ldots$$\ldots$$u\in\Gamma^{2}$$\epsilon_{m}\in\{\pm, \circ\}$$\eta_{n}\in\{I, J\}_{f}$,$\lim_{Narrow\infty}\langle u^{(N)}|(\frac{a_{N}^{\epsilon_{1}}}{\sqrt{v}})\cdots(\frac{a_{N}^{\epsilon_{m}}}{\sqrt{v}})|\eta_{1}, \ldots, \eta_{n}\rangle_{\mathrm{r}(F_{2,\gamma}^{(N)})}^{(N)}=\langle u|a^{\epsilon_{1}}\cdots a^{\epsilon_{m}}|\eta_{1}, \ldots, \eta_{n}\rangle_{\Gamma^{2}}$$w/iere$we use the notation (4.2). The operators$a_{f}^{+}a^{-}$

and$a^{\mathrm{o}}$are specified by (5.15)

136
In particular, combinatorial moments with respect to the vacuum state are given by.16)$\lim_{Narrow\infty}\langle 0|(\frac{1}{\sqrt{v}}.\cdot\sum_{w_{j}\in\Sigma_{2,\gamma}^{(N)}}\pi(w_{ij}))^{m}|0\rangle_{\mathrm{r}(F_{2_{1}\gamma}^{(N)})}^{(N)}=\langle 0|(a^{+}+a^{-}+a^{\mathrm{o}})^{m}|0\rangle_{\Gamma^{2}}$.

Remarks.
(1) By acombinatorial argument, we see that the combinatorial moments (5.16) are given
as the number of closed walks on ainduced subgraph of aweighted binary tree. (Theweights $\lambda=1/\sqrt{2}$ are given in the figure below.) Indeed $m$-th combinatorial momentbeis given by the number of $m$-step walks which leave and return to . Let
By0.
arrivingattowithout$m$the number of -step walks which leave and returnthe self-similarity of the graph, we see for $m\geq 2$ ,$F_{m}$$0$$\mathit{0}$$f_{m}$$z$$z$$f_{m}$$= \sum_{k=0}^{m-2}(f_{k}+\frac{F_{k}}{\sqrt{2}})f_{m-k-2}$$F_{m}$

$=$

,

$m-2$
$\sum f_{k}F_{m-k-2}$

,

$k=0$

where $f\mathrm{o}=F_{0}=1$ . Then the moment generating function $F(t)= \sum_{m}F_{m}t^{m}$ and$f(t)= \sum_{m}f_{m}t^{m}$ satisfy$f(t)-1$$=t^{2}(f(t)+ \frac{F(t)}{\sqrt{2}})f(t)$$F(t)-1$$=t^{2}F(t)^{2}$

,

,

hence
$t^{2}F(t)^{3}+t^{2}F(t)^{2}-2F(t)+2=0$.

The Cauchy transform $G(t)$ of the distribution associated with the combinatorial m0ments (5.16) is given as asolution of$tG(t)^{3}+G(t)^{2}-2tG(t)+2=0$ .

(2) The limit distribution associated with (5.16) coincides with the one in Examples 1.5(1.16) and (1.17) of [15], up to the variance, where the anti-commutation $ab+ba$ ofsemi-circular elements , which are free from each other is observed. Indeed we
$a$

$b$

$\mathrm{s}\mathrm{e}$

137
that
$\frac{1}{\sqrt{v}}\sum_{w_{ij}\in\Sigma_{2,\gamma}^{(N)}}\pi(w_{ij})\approx(\frac{\sigma_{1}+\cdots+\sigma_{\gamma N}}{\sqrt{\gamma N}})(\frac{\sigma_{\gamma N+1}+\cdots+\sigma_{N}}{\sqrt{(1-\gamma)N}})$$+( \frac{\sigma_{\gamma N+1}+\cdots+\sigma_{N}}{\sqrt{(1-\gamma)N}})(\frac{\sigma_{1}+\cdots+\sigma_{\gamma N}}{\sqrt{\gamma N}})$.

which is nothing but the anti-commutation of semi-circular elements in the limit $Narrow$.
$\infty$

6. Commutative association scheme and its quantum decomposition
It is quite natural to apply our approach of the quantum decomposition to isotropic random
walks on homogeneous graphs. In fact, it will be seen that any adjacency operator of acertain
graph is decomposed into asum of a‘creation’, an ‘annihilation’ and afunction of ‘number’operators. Such adecomposition is motivated by the standard theory of quantum stochastic
evolutions [14]. This idea is useful for quantum limit theorems associated with random walks ongraphs, particularly, on distance regular graphs. Indeed, in [21] [9] we carried out our idea on theHamming scheme as follows. Let $F$ be aset of $n+1$ points. A Hamming graph $H(d, n+1)=F^{d}$defined byis adistance regular graph with adistance$\partial(x, y)$$\partial(x, y):=\#\{i|x_{i}\neq y_{i}\}$$y=(y_{1}, \ldots, y_{d})\in H(d, n+1)$ . Let $A_{(d,n+1)}$ be the adjacency matrix offor $x=$the Hamming graph $H(d, n+1)$ . The adjacency matrix acts on the finite dimensional complex
Hilbert space $H$ spanned by $\{|x\rangle|x\in H(d, n+1)\}$ with the inner product uniquely determinedby$(x_{1}, \ldots, x_{d}))$$\langle x|y\rangle=\delta_{xy}$,$x$, $y\in H(d.n+1)$ .

The action is given by$A|x \rangle=\sum|y\rangle$,$x\sim y$where $x\sim y$ means that is adjacent to . We fix an arbitrary point $\mathit{0}\in H(d, n+1)$ as an
origin. Then we have apartition
$x$

$y$

$H(d, n+1)=\cup^{d}H_{k}k=0$

where $H_{k}:=\{x\in H(d, n+1)|\partial(0, x)=k\}$ . We define number vectors$| \Phi(k)\rangle:=\frac{1}{\sqrt{\# H_{k}}}\sum_{x\in H_{k}}|x\rangle$

for $k=1$ ,, . The orthonormal system$d$$\ldots$$\{|\Phi(k)\rangle\}$in $H$ gives a one-mode Fock space$\Gamma_{(d,n+1)}:=\oplus^{d}|\Phi(k)\rangle k=1^{\cdot}$

By the definition of the adjacency matrix, we have
(6.1)

$A_{(d,n+1)}|x \rangle=\sum_{a\in H_{+}^{(d,n+1)}}|a\rangle+\sum_{b\in H_{-}^{(d,n+1)}}|b\rangle+\sum_{c(x)(x)\in H_{o}^{(d,n+1)}(x)}|c\rangle$

138
Here we put for$\epsilon=+,$ $-$and ,$\circ$$H_{\epsilon}^{(d,n+1)}(x):=\{y\sim x|\partial(y, 0)=\partial(x, \mathit{0})+\eta\}$with $\eta=+1,$ -1 and 0respectively. The equation (6.1) gives raise to anatural decompositionof the adjacency matrix $A(d,n+1)$ :$A_{(d,n+1)}=A_{(d,n+1)}^{+}+A_{(d,n+1)}^{-}+A_{(d,n+1)}^{\mathrm{o}}$,where each factor is determined by.

$A_{(d,n+1)}^{\epsilon}|x \rangle=\sum_{y\in H_{e}^{(d.n+1)}(x)}|y\rangle$

It follows from the distance regularity of $H(d, n+1)$ that as ,
have
$n$

$\frac{A_{(d,n+1)}^{+}}{\sqrt{nd}}|\Phi(k)\rangle$$darrow\infty$with $n/darrow\tau\geq 0$ we$\sqrt{(k+1)}|\Phi(k+1)\rangle$$\approx$$\frac{A_{(d,n+1)}^{-}}{\sqrt{nd}}|\Phi(k)\rangle$$\approx$$\sqrt{k}|\Phi(k-1)\rangle$$\frac{A_{(d,n+1)}^{\mathrm{o}}}{\sqrt{nd}}|\Phi(k)\rangle$$\approx$$k\sqrt{\tau}|\Phi(k)\rangle$.

As aconsequence, the asymptotic behavior of $A(d,n+1)$ is described in terms of aone-mode
Boson Fock space
and the canonical creation and annihilation operators
defined by$B^{\pm}$$\Gamma_{b}=\oplus \mathrm{C}|\Psi(k)\rangle$$B^{+}|\Psi(k)\rangle=\sqrt{k+1}|\Psi(k+1)\rangle$

and$B^{-}|\Psi(k)\rangle=\sqrt{k}|\Psi(k-1)\rangle$

Theorem 6.1. Let,, we have$0,1$$\tau\geq 0$, $m\geq 1$ and$\epsilon_{1}$,$\ldots$,$k\geq 1$,$B^{-}|\Psi(0)\rangle=0$.

, $\epsilon_{m}\in\{+, -, 0\}$ be given. Then for any$k$, $l=$$\ldots$$\lim_{d,narrow\infty,n/darrow\tau}\{\Phi(k)|\frac{A_{(d,n+1)}^{\epsilon_{1}}}{\sqrt{nd}}\cdots\frac{A_{(d,n+1)}^{\epsilon_{m}}}{\sqrt{nd}}|\Phi(l)\}_{\Gamma_{(d,n+1)}}=\langle\Psi(k)|B^{\epsilon_{1}}\cdots B^{\epsilon_{m}}|\Psi(l)\rangle_{\Gamma_{b}}$.

In particular, we have a classical central limit theorem$\lim_{d,narrow\infty,n/darrow\tau}\langle\Phi(k)|(\frac{A_{(d,n+1)}}{\sqrt{nd}})^{m}|\Phi(l)\rangle_{\Gamma_{(d,n+1)}}=\langle\Psi(k)|(\sqrt{\tau}B^{+}B^{-}+B^{+}+B^{-})^{m}|\Psi(l)\rangle_{\Gamma_{b}}$As aresult, we reproduce one of Hora’s results [13] on the asymptotic distribution of eigen
values of the adjacency matrix associated with aHamming graph.
Acknowledgement. The author would like to thank Professor N. Obata for his constantencouragement

139
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$q$

$q$

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$\mathrm{n}\mathrm{a}\mathrm{g}\mathrm{o}\mathrm{y}\mathrm{a}-\mathrm{u}.\mathrm{a}\mathrm{c}$. jP