数理解析研究所講究録
第 1859 巻 2013 年 1-8

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Arcsine Law as Classical Limit
Hayato Saigo*
Nagahama Institute of Bio-Science and Technology
Nagahama 526-0829, Japan

Abstract
Recently we have proved that the Arcsine Law appears as theClassical Limit of quantum harmonic oscillators, in the framework ofalgebraic probability thoery. In the present paper we discuss how to
generalize the result by means of the notion of interacting Fock spaces,
which associates algebraic probability theory and the theory of orthogonal polynomials of probability measures. As an application we showthat the Classical Limit for interacting Fock spaces corresponnding to-Gaussians and the exponential distribution are the Arcsine Law.
$q$

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Introduction

Let us consider the time-avereged distribution of position for a 1-dimensionalclassical harmonic oscillator. It is easy to see that the distribution (after
standardization) has the form
$x$$\mu_{A_{8}}(dx)=\frac{1}{\pi}\frac{dx}{\sqrt{2-x^{2}}} (-\sqrt{2}<x<\sqrt{2})$.

The distribution
is called the (normalized) Arcsine Law, which also
plays lots of crucial roles both in pure and applied probability theory. The
n-th moment
is given by$\mu_{As}$$M_{n}:= \int_{\mathbb{R}}x^{n}\mu_{As}(dx)$$M_{2m+1}=0, M_{2m}= \frac{1}{2^{m}}[Matrix].$

The moment problem for the Arcsine law is determinate, that is, the moment
sequence
characterizes. In [4] we have proved that the Arcsine$\{M_{n}\}$$\mu_{As}$*Lmail: h-saigoh@nagahama-i-bio.ac.jp

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Law appears as the Classical Limit of quantum harmonic oscillator, in theframework of algebraic probability thoery (also known as “noncommutative
probability theory” or “quantum probability theory”).
The purpose of this paper is to develop the result. and viewpoint in [4].
Section 2 and 3 are devoted to review basic notions in algebraic probabity
and the “Quantum-Classical Correspondence” for quantum harmonic oscillators proved in [4] In section 4 we discuss how to generalize the resultand viewpoint by means of “interacting Fock spaces [1]”, which associates
algebraic probability theory and the theory of orthogonal polynomials. In
the last section we show that the Classical Limit for the interacting Fockspace corresponding to Laguerre polynomials again becomes the Arcsine law(after standardization).

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Quantum Harmonic OscillatorLet
be $a*$ -algebra. We call a linear map
satisfies$\mathcal{A}$$\varphi$:$\mathcal{A}arrow \mathbb{C}$a state on$\mathcal{A}$if it$\varphi(1)=1, \varphi(a^{*}a)\geq 0.$

A pairof $a*$ -algebra and a state on it is called an algebraic probability, an element
space. Here we adopt a notation for a state :and a probability distribution on$(\mathcal{A}, \varphi)$$X\in \mathcal{A}$$\mathcal{A}arrow \mathbb{C}$$\varphi$$\mathbb{R}.$$\mu$

Notation 2.1. We use the notationall$X\sim_{\varphi}\mu$when$\varphi(X^{m})=\int_{\mathbb{R}}x^{m}\mu(dx)$for$m\in \mathbb{N}.$

Remark 2.2. Existence of$\mu$

for $X$ which satisfies$X\sim_{\varphi}\mu$always holds.

Definition 2.3 (Quantum harmonic oscillator). quantum harmonic oscillator is a triple
is a Hilbert space
where
, and
with inner product given by
are operators defined
as follows:$A$$(\Gamma(\mathbb{C}), a, a^{*})$$\Gamma(\mathbb{C})$$\langle\Phi_{n},$$\Gamma(\mathbb{C})$$\Phi_{m}\rangle=\delta_{n,m}$$a,$$:=\oplus_{n=0}^{\infty}\mathbb{C}\Phi_{n}$$a^{*}$$a\Phi_{0}=0, a\Phi_{n}=\sqrt{n}\Phi_{n-1}(n\geq 1)$$a^{*}\Phi_{n}=\sqrt{n+1}\Phi_{n+1}.$

be the -algebra generated by , andbe the state defined as. Thenis an algebraic probability space. It is
well known thatLet$\varphi_{n}(\cdot)$$\mathcal{A}$$:=\langle\Phi_{n},$$*$$(\cdot)\Phi_{n}\rangle$$a$$\varphi_{n}$$(\mathcal{A}, \varphi_{n})$$a+a^{*}$

represents the “position” and that$a+a^{*} \sim_{\varphi 0}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^{2}}dx.$

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That is, in $n=0$ case, the distribution of position is Gaussian.
On the other hand, the asymptotic behavior of the distributions of position as tends to infinity is quite nontrivial.
$n$3

Quantum-Classical Correspondence

Then a question arises: Is it possible to see whether and in what meaning the.
“Quantum-Classical Correspondence” holds for harmonic oscillators? This
question, which is related to fundamental problems in Quantum theory andasymptotic analysis [2], was analyzed in [4] from the viewpoint of noncommutative algebraic probability with quite a simple combinatorial argument.
The folloing is the main result in [4]:Theorem 3.1. Let$\mu_{N}$be a probability distribution on$\mathbb{R}$such that$\frac{a+a^{*}}{\sqrt{2N}}\sim_{\varphi_{N}}\mu_{N}.$Then$\mu_{N}$weakly converges to$\mu_{As}.$

Proof. We only have to prove moment convergence because it is known thatmoment convergence implies weak convergence when the moment problem
for the limit distribution is determinate.
First we can easily prove that$\varphi_{N}((\frac{a+a^{*}}{\sqrt{2N}})^{2m+1})=\langle\Phi_{N}, (\frac{a+a^{*}}{\sqrt{2N}})^{2m+1}\Phi_{N}\rangle=0$

sincewhen $N\neq M.$
To consider the moments of even degrees, we introduce the following
notations:$\langle\Phi_{N},$$\bullet$$\Lambda^{2m}$$\Phi_{M}\rangle=0$$:=${maps from {1, 2,$\ldots,$$2m\}$

to$\{1,$ $*\}$},$\bullet\Lambda_{m}^{2m}:=\{\lambda\in\Lambda^{2m};|\lambda^{-1}(1)|=|\lambda^{-1}(*)|=m\}.$Note that the cardinality
equals tobecause the choice of isequivalent to the choice of $m$ elements which consist the subset
from$2m$ elements in$2m\}.$
1, 2,
It is clear that for any$|\Lambda_{m}^{2m}|$$\lambda$$(\begin{array}{l}2mm\end{array})$$\lambda^{-1}(1)$$\{$$\ldots,$$\lambda\not\in\Lambda_{m}^{2m}$$\langle\Phi_{N}, a^{\lambda_{1}}a^{\lambda_{2}}\cdots a^{\lambda_{2m}}\Phi_{N}\rangle=0$

since$\langle\Phi_{N},$$\Phi_{M}\rangle=0$

when $N\neq M.$

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On the other hand, for any$\lambda\in\Lambda_{m}^{2m}$the inequality$N\cdots(N-m+1)\leq\langle\Phi_{N},$ $a^{\lambda_{1}}a^{\lambda_{2}}\cdots a^{\lambda_{2m}}\Phi_{N}\rangle\leq(N+1)\cdots(N+m)$

holds when $N$ is sufficiently large, because the minimum is achieved when$(1\leq i\leq m)$$\lambda_{i}=\{$$*1,$$(m+1\leq i\leq 2m)$

and the maximum is achieved when$\lambda_{i}=\{$$*,$$(1\leq i\leq m)$1,$(m+1\leq i\leq 2m)$by the definition ofUsing the inequality above we have$a,$$a^{*}$$\frac{1}{N^{m}}\langle\Phi_{N}, a^{\lambda_{1}}a^{\lambda_{2}}\cdots a^{\lambda_{2m}}\Phi_{N}\ranglearrow 1 (Narrow\infty)$.

and then$\varphi_{N}((\frac{a+a^{*}}{\sqrt{2N}})^{2m})=\langle\Phi_{N}, (\frac{a+a^{*}}{\sqrt{2N}})^{2m}\Phi_{N}\rangle$$= \frac{1}{2^{m}}\sum_{\lambda\in\Lambda^{2\mathfrak{m}}}\frac{1}{N^{m}}\langle\Phi_{N}, a^{\lambda_{1}}a^{\lambda_{2}}\cdots a^{\lambda_{2m}}\Phi_{N}\rangle$$= \frac{1}{2^{m}}\sum_{\lambda\in\Lambda_{m}^{2m}}\frac{1}{N^{m}}\langle\Phi_{N}, a^{\lambda_{1}}a^{\lambda_{2}}\cdots a^{\lambda_{2m}}\Phi_{N}\rangle$$arrow\frac{1}{2^{m}}|\Lambda_{m}^{2m}|=\frac{1}{2^{m}}(\begin{array}{l}2mm\end{array}) (Narrow\infty)$.
$\square$

As we have stated, the Arcsine law as the time-averaged distribution
for classical harmonic oscillators emerges from the distributions for quantum harmonic oscillators. This is nothing but a noncommutative algebraic
realization of Quantum-Classical Correspondence for harmonic oscillators.
In other words, the Arcsine Law appears as the classical limit ofquantum harmonic oscillators. The “time averaged” nature is deeply related to the notion of Bohr’s “complementarity”for energy and time. Startingfrom energy eigenstates, one cannot obtain the classical harmonic oscillator
itself but time averaged distribution of it.

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4Generalization: Interacting Fock space
In this section we discuss a generalization of Theorem 3.1.

Definition 4.1 (Jacobi sequence).
sequencequence if it satisfies one of the conditions below:$A$$\bullet$is called a Jacobi se-$\{\omega_{n}\}$(finite type) There exist a number $m$ such thatfor $n\geq m$ ;$\omega_{n}>0$

for $n<m$ and$\omega_{n}=0$$\bullet$.(infinite type)$\omega_{n}>0$

for all$n.$

Definition 4.2 (Interacting, Fock space). Let
be a Jacobi sequence. Aninteracting Fock space
is a triple
whereis a Hilbertspacewith inner product given by, and
are operators defined as follows:$\{\omega_{n}\}$$(\Gamma(\mathbb{C}), a, a^{*})$$\Gamma_{\{\omega_{n}\}}$$\Gamma(\mathbb{C})$$a,$$\Gamma(\mathbb{C})$$:=\oplus_{n=0}^{\infty}\mathbb{C}\Phi_{n}$$\langle\Phi_{n},$$\Phi_{m}\rangle=\delta_{n,m}$$a^{*}$$a\Phi_{0}=0, a\Phi_{n}=\sqrt{\omega_{n}}\Phi_{n-1}(n\geq 1)$$a^{*}\Phi_{n}=\sqrt{\omega_{n+1}}\Phi_{n+1}.$

As before, letdefined as$\mathcal{A}$be the -algebra generated by , andbe the state. Thenis an algebraic probability$*$$\varphi_{n}(\cdot);=\langle\Phi_{n},$$a$$(\cdot)\Phi_{n}\rangle$$\varphi_{n}$$(\mathcal{A}, \varphi_{n})$space.
The following result proved in [5] is a generalization of Theorem 3.1:

Theorem 4.3. Let
fying the condition$\Gamma_{\{\omega_{n}\}}:=(\Gamma(\mathbb{C}), a, a^{*})$

be an interacting Fock space satis-$\lim_{narrow\infty}\frac{\omega_{n+1}}{\omega_{n}}=1$

and

be a probability distribution on$\mu_{N}$$\mathbb{R}$such that$\frac{a+a^{*}}{\sqrt{2\omega_{N}}}\sim_{\varphi_{N}}\mu_{N}.$.Then$\mu_{N}$weakly converges to$\mu_{As}.$The theorem above has an interpretation in terms of orthogonal polynomials. To see this we explain the relation between interacting Fock spaces,
probability measures and orthogonal polynomials.
Let be a probability measure on
having finite moments. (For the
rest of the present paper, we always assume that all the moments are finite.)
Then the space of polynomial functions is contained in the Hilbert space. A Gram-Schmidt procedure provides orthogonal polynomials which
only depend on the moment sequence.
$\mu$$L^{2}(\mathbb{R}, \mu)$$\mathbb{R}$6be the monic orthogonal polynomials ofLetequals to . Then there exist sequences
the degree ofsuch that
Jacobi sequence$\{p_{n}(x)\}_{n=0,1},\cdots$$n$$p_{n}$$\mu$such that
and$\{\alpha_{n}\}_{n=0,1},\cdots$$\{\omega_{n}\}_{n=1,2},\cdots$$xp_{n}(x)=p_{n+1}(x)+\alpha_{n+1}p_{n}(x)+\omega_{n}p_{n-1}(x) (p_{-1}(x)\equiv0)$

if is symmetric, i.e., $\mu(-dx)=\mu(dx)$ .
:It is known that there exist an isometry
which we obtain$\alpha_{n}\equiv 0$

.

$\mu$

$U$

$\Gamma_{\{\omega_{n}\}}arrow L^{2}(\mathbb{R}, \mu)$through$a+a^{*}+a^{o}\sim_{\varphi_{N}}|P_{N}(x)|^{2}\mu(dx)$denotes theandis an operator defined by
wherenormalized orthogonal polynomial \’of degree $n[3]$ . Then Theorem 4.3 implies
the following:$a^{o}\Phi_{n};=\alpha_{n+1}\Phi_{n}$$a^{o}$

Theorem 4.4. Let
Jacobi sequence$\mu$$\{\omega_{n}\}$$P_{n}$

be a symmmetric measure such that the corresponting

satisfies$\lim_{narrow\infty}\frac{\omega_{n+1}}{\omega_{n}}=1$

Then the measure

converge toSince$\mu_{n}$defined as$\mu_{n}(dx);=|P_{n}(\sqrt{2\omega_{n}}x)|^{2}\mu(\sqrt{2\omega_{n}}dx)$

weakly$\mu_{As}.$-Gaussians” $(0\leq q\leq 1,$ $q=1$ is Gaussian and $q=0$ is Wigner$q$$:=1+q+q^{2}+\cdots+q^{n-1}$ , satisfy
Semicircle Law) , corresponding tois turned out to be the Classical Limit of these
the condition above,measures.
In the next section we discuss the Classical Limit for the case of exponential distribution as an example of asymmetric measure.
$\omega_{n}=[n]_{q}$$\cdot$$\mu_{As}$Example: Exponential-Laguerre case

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Let

$\mu$

be the exponential distribution, i.e., $\mu(dx):=e^{-x}dx(x>0)$ .Then$xl_{n}(x)=l_{n+1}(x)+(2n+1)l_{n}(x)+n^{2}l_{n-1}(x) (l_{-1}(x)\equiv 0)$ ,holds, where denotes the Laguerre polynomial of n-th degree modified to. Asforbe monic. Let us consider the interacting Fock space
we have discussed,$l_{n}$$\Gamma_{\{\omega_{n}\}}$$a+a^{*}+a^{o}\sim_{\varphi_{N}}|L_{N}(x)|^{2}e^{-x}dx(x>0)$$\omega_{n}=n^{2}$

.

where
denotes the usual (normalized) Laguerre polipomial of n-th order.
$:=|L_{n}(nx)|^{2}ne^{-nx}dx$
Then we can calculate the “Limit moment” of$(x>0)$ in the spirit of the proof of Theorem 3.1 and Theorem 4.3 [5].
$L_{n}$$\mu_{n}(dx)$7
Proposition 5.1.
$\lim_{Narrow\infty}\varphi_{N}((\frac{a+a^{*}+a^{o}}{N})^{m})=\sum_{\iota}2^{m-2l}(\begin{array}{ll} mm -2l\end{array}) (\begin{array}{l}2ll\end{array}).$The right hand side of the proposition above is simplified as follows.

Lemma 5.2.
$\sum_{l}2^{m-2l}(\begin{array}{ll} mm -2l\end{array}) (\begin{array}{l}2ll\end{array})=(\begin{array}{l}2mm\end{array}).$Proof. Consider two sets of maps
$L:=\{f:marrow 4;|f^{-1}(0)|=|f^{-1}(1)|\}$$R :=\{\tilde{f}:2\cross marrow 2;|\tilde{f}^{-1}(0)|=|\tilde{f}^{-1}(1)|\},$

where $m$ $:=\{0,1,2, \ldots, m-1\}$ . Since we can construct an isomorphism
between and $R,$ $|L|=|R|$ . This is what to be proved. (This proof is
obtained in discussion with Hiroki Sako).
$L$$\square$

It is easy to show that$(\begin{array}{l}2mm\end{array})=\int_{0}^{4}x^{m}\frac{1}{\pi}\frac{dx}{\sqrt{4-(x-2)^{2}}},$and hence we obtain the following theorem[5].
Theorem 5.3. Let
be the normalized Laguerre polynomial of n-th degree.
Then $\mu_{n}(dx);=|L_{n}(nx)|^{2}ne^{-nx}dx(x>0)$ weakly converge to$L_{n}$$\frac{1}{\pi}\frac{dx}{\sqrt{4-(x-2)^{2}}}(0<x<4)$.

That is, the Classical Limit of “Laguerre oscillator” is also the
Arcsine Law (just translated and dilated).

Acknowledgments
The author would like to thank Prof. Izumi Ojima and Mr. Kazuya Okamura
for discussions on Quantum-Classical Correspondence. He deeply appreciates

Hiroki Sako for collaboration.

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References[1] L. Accardi and M. Bozejko, Interacting Fock spaces and Gaussianizationof probability measures, Infin. Dimens. Anal. Quantum Probab. Relat.
Top. 1 (1998), 663-670.
[2] H. Ezawa, Asymptotic analysis (in Japanese) Iwanami-Shoten, 1995.

[3] A. Hora and N. Obata, Quantum Probability and Spectml Analysi\’{s}of Graphs, Theoretical and Mathematical Physics (Springer, Berlin
Heiderberg-Verlag 2007).
[4] H. Saigo, A new look at the Arcsine law and “Quantum-Classical Correspondence”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15,
no. 3 (2012), 1250021.
[5] H. Saigo, in preparation.