\input amstex
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\nologo
\def\cA{{\Cal A}}
\def\cH{{\Cal H}}
\def\cK{{\Cal K}}
\def\bC{{\Bbb C}}
\def\bN{{\Bbb N}}
\def\bQ{{\Bbb Q}}
\def\bR{{\Bbb R}}
\def\bT{{\Bbb T}}
\def\bZ{{\Bbb Z}}
\def\fT{{\frak T}}
\def\ol{\overline}
\topmatter
\title Ergodic theorems for quantum Kronecker flows\endtitle
\author S\l awomir Klimek and Andrzej Le\'sniewski \endauthor
\leftheadtext{S. KLIMEK and A. LE\'SNIEWSKI}%
\address Department of Mathematics, IUPUI, Indianapolis, IN 46202,USA
\endaddress
\email sklimek\@math.iupui.edu \endemail
\address Lyman Laboratory of Physics, Harvard University, Cambridge,
MA 02138, USA \endaddress
\email alesniew\@husc.harvard.edu \endemail
%\cvol{}
%\cvolyear{}
%\cyear{}
\subjclass Primary 47-06; Secondary 47A35, 47D45\endsubjclass
\abstract We discuss examples of ergodic theorems in quantum dynamics.
Our focus is on infinite dimensional quantum dynamical systems arising
from quantization of Kronecker type flows.\endabstract
\thanks The first author was supported in part by NSF
Grant DMS--9500463. The second author was supported in part by NSF
Grant DMS--9424344.
\endthanks
\endtopmatter
\document
\head 1. Quantum ergodic theorems\endhead
In this talk we present some results on the quantum and semiclassical
ergodic theory for a family of quantum dynamical systems called
quantum Kronecker flows. These quantum dynamical systems arise as
quantization of finite and infinite dimensional classical Kronecker
flows. They play an important role in quantum field theory, see e.g.
\cite {KL} where this point is emphasized. Also, special cases of these
systems have a natural interpretation in the context of number theory.
For a background on quantum maps and quantum ergodic theory, see
our contribution ``Quantum maps" to this volume. Our presentation is
pedagogical; we explain the ideas, formulate the results, and outline
some of the proofs. Complete discussion may be found in the literature
cited.
\subhead 1.1. Quantum ergodicity\endsubhead
We introduce some notation. Let $\cA$ be a ($\bC^*$- or von
Neumann) algebra of operators acting on a Hilbert space $\cH$, and let
$\rho_t$ be a one-parameter group of $*$-automorphisms of $\cA$. We will
assume that
$$
\rho_t(a)=e^{itH}ae^{-itH},\ \ \ a\in\cA ,
$$
where $H$ is a self-adjoint, possibly unbounded operator on $\cH$.
Furthermore, we assume that there is a $\rho_t$-invariant state $\tau$
on $\cA$. The main object of ergodic theory is the time average of an
observable,
$$
:=\lim_{T\to\infty}{1\over T}\int_0^T\rho_t(a)\,dt\ ,
\tag 1.1.1
$$
where, the limit in (1.1.1) is taken with respect to a suitable
(usually $\tau$ dependent) topology. Under suitable assumptions, this
limit exists and is $\rho_t$-invariant. The dynamical system
$(\cA,\rho_t,\tau)$ is then called ergodic if the following statement
holds (ergodic theorem):
$$
=\tau(a)I\ .
\tag 1.1.2
$$
Here is an example.
\example{Kronecker flow}
Consider the following dynamics on a quantum torus, namely the
quantum Kronecker flow. Recall that the algebra $\cA$ of observables
on a quantum torus is defined as the universal ($\bC^*$- or von Neumann)
algebra generated by two unitary generators $U,V$ satisfying the relation
$$
UV=e^{i\lambda}VU\ .
$$
Here $\lambda=2\pi h$, where $h$ is Planck's constant. One can think
of the elements of $\cA$ as series of the form $a=\sum a_{n,m}U^nV^m$.
A natural trace on $\cA$ is simply given by $\tau(a)=a_{0,0}$. The
quantum Kronecker flow is defined on generators by:
$$
\rho_t(U):=e^{2\pi i\omega_1t}U,\ \
\rho_t(V):=e^{2\pi i\omega_2t}V,
$$
in analogy with the classical situation. It extends to a one-parameter
group of automorphisms of $\cA$. If we assume that $\omega_1,\omega_2$
are linearly independent over $\bZ$ then (1.1.2) holds with the limit
taken in the operator norm sense, see \cite {KLMR}.
\endexample
\subhead 1.2. Semiclassical ergodicity\endsubhead
In many important quantum mechanical problems the scenario of the
previous section does not apply. Let us assume that the spectrum of $H$
is purely discrete:
$$
H\phi_n=\lambda_n\phi_n\ .
$$
Now, if $\cA$ contains compact operators, then we cannot expect
an ergodic theorem of the type explained in the previous section.
Indeed, for a continuous function $f$ vanishing at infinity,
$$
\;=f(H).
$$
To account for that one can reasonably expect the following variant of
the ergodic theorem:
$$
=\tau(a)I+C_a\ ,\tag 1.2.1
$$
where $C_a$ is compact. If this is the case, then we have
$$
\aligned
(\phi_n,a\phi_n)
&=(\phi_n,\phi_n)\\
&=\tau(a)+(\phi_n,C_a\phi_n)\\
&\to\tau(a),\ \text{as}\ n\to\infty\ .
\endaligned\tag 1.2.2
$$
In physics, $H$ is the Hamiltonian of a system, and the limit
$\lambda_n\to\infty$ is the semiclassical limit. For this reason,
we refer to the above type of ergodic behavior as {\it semiclassical
ergodicity}.
It is very difficult to establish (1.2.1) in examples of quantum
dynamical systems. A weaker version of (1.2.1) has been studied
extensively in the literature, see \cite{Z} and references therein.
The starting point is equation (1.2.2),
$$
\tau(a)=\lim_{n\to\infty}(\phi_n,a\phi_n).
$$
Consider a possibility where the above limit exists for almost all
subsequences only. More precisely, assume there is a subset
$S\subset \bN$ of density $1$ such that
$$
\tau(a)=\lim_{n\in S,\ n\to\infty}(\phi_n,a\phi_n).
$$
Alternatively, by a well known lemma in ergodic theory, the above equation
can be rephrased as
$$
\tau(a)=\lim_{E\to\infty}\tau_E(a):=\lim_{E\to\infty}{1\over\#\{\lambda_n\leq
E\}}\sum_{\lambda_n\leq E} (\phi_n,a\phi_n).\tag 1.2.3
$$
The ergodic theorem which holds within this framework reads:
$$
=\tau(a)I+C_a,\ \ \text{and}\ \ \lim_{E\to\infty}\tau_E(C_a^*C_a)=0
\tag 1.2.4
$$
More precisely, the following theorem due to Zelditch \cite {Z} gives
sufficient conditions for (1.2.4) to hold.
\proclaim{Theorem 1.2.1}
Let $\pi_\tau$ the GNS representation of $\cA$ associate with $\tau$.
Assume that (i) the algebra $\tau(\cA)$ is commutative, and (ii)
$\tau(\rho_t)$ is ergodic (in the classical sense). Then the system
$(\cA,\rho_t,\tau)$ is semiclassically ergodic i.e. (1.2.4) holds.
\endproclaim
Here is an example of this scenario.
\example{Geodesic flow}
Consider the following quantum version of the geodesic flow on a Riemann
surface $M$ with constant negative curvature. The geodesic flow on such a
manifold is known to be ergodic (in fact, mixing). Consider the Hilbert
space $\cH:=L^2(M,d\mu)$ of functions on $M$ square integrable
with respect to the measure $\mu$ defined by the metric. Take the
corresponding Laplace operator $\Delta$ and set $H:=\sqrt{-\Delta}$.
Let $\cA$ be a completion of the algebra of order $0$ pseudodifferential
operators on $M$, and define the trace $\tau$ on $\cA$ by
$$
\tau(a):=\int_{ST^*M}\sigma(a)\,d\nu.
$$
Here $ST^*M$ is the unit sphere in the cotangent bundle of $M$,
$\sigma(a)$ is the principal symbol of $a\in\cA$, and $\nu$ is
the normalized Liouville measure on $ST^*M$. The results of
\cite {C}, \cite {S} state that (1.2.4) holds in this case.
In fact, as remarked in \cite {Z}, the algebra $\cA$ becomes
the commutative algebra of functions on $ST^*M$ in the GNS representation
with respect to $\tau$, and $\rho_t$ becomes the geodesic flow
on $M$. One can then apply theorem 1.2.1 to yield the results
of \cite {C} and \cite {S}.
\remark{Remark} It was conjectured in \cite {RS} that the remainder term
$C_a$ in (1.2.4) is compact so that the limit in (1.2.2) exists
for every subsequence. If true, this would imply that no
``scars" exist for such systems.
\endremark
\endexample
\head 2. Quantum Kronecker flow\endhead
Another example of semiclassical ergodicity is a variant of quantized
Kronecker flow discussed below. It is a quantization of the classical
Kronecker flow different from the one consider in Section 1.1.
Interesting technical aspects appear when the number of degrees of
freedom is infinite. It turns out that the proof of ergodicity in this
case requires an analysis of a generalized partition problem of
number theory. To explain this point we take a historical approach
and first discuss the Hardy-Ramanujan solution of the partition problem.
The main question in that theory is to obtain an asymptotic expansion of
the Laplace transform of certain functions. This is typically done by
using the method of stationary phase.
\subhead 2.1. Method of stationary phase\endsubhead
We begin by reviewing the method of stationary phase. The argument
presented here, while very simple, clearly illustrates
the main ideas of the method.
Let $f$ be a function on $[-N,N]$ with a simple maximum at
$x=0$ and $f''(0)<0$. We consider the problem of finding the asymptotic
behavior of the integral $\int_{-N}^Ne^{Ef(x)}\,dx$ as $E\to\infty$.
\proclaim{Proposition 2.1.1} Under the above assumptions,
$$
\int_{-N}^Ne^{Ef(x)}\,dx=\sqrt{{2\pi\over-f''(0)E}}\,e^{Ef(0)}\,
(1+o(1)),\ \text{as}\ E\to\infty.\tag 2.1.1
$$
\endproclaim
\demo{Proof}
For large $E$, the main contribution to the integral comes from
the vicinity of the critical point. To see it precisely, we write:
$$
\int_{-N}^Ne^{Ef(x)}\,dx=\int_{|x|<\delta}e^{Ef(x)}\,dx+
\int_{\delta<|x|0$. Now, the idea is to choose $\sigma$ so that
the integrand of (2.2.4) has a critical point at $t=0$, and use the
stationary phase method to get the asymptotic of $p(n)$ as $n\to\infty$.
Expanding the exponent of (2.2.4) to the second order around $t=0$ and
assuming that the stationary phase method can be justified, we obtain
$$\aligned
p(n)&={1\over 2\pi}\int_{-\pi}^{\pi}e^{\phi(\sigma_n)+n\sigma_n}
e^{-\phi''(\sigma_n)t^2/2}\,dt\;\big(1+o(1)\big)\\
&={e^{\phi(\sigma_n)+n\sigma_n}\over\sqrt{2\pi\phi''(\sigma_n)}}
\;\big(1+o(1)\big)\ ,
\endaligned
$$
where $\sigma_n$ satisfies $\phi'(\sigma_n)+n=0$. All we need to
do now is to wok out the asymptotics of $\sigma_n$, $\phi(\sigma_n)$
and $\phi''(\sigma_n)$. As an example, we will find the asymptotic
behavior of $\sigma_n$. To this end, we approximate the derivative
of $\phi(s)$ for small $s$,
$$\aligned
\phi'(s)&=-\sum_{k=1}^\infty{ke^{-ks}\over1-e^{-ks}}\\
&=-{1\over s^2}\sum_{k=1}^\infty{kse^{-ks}\over1-e^{-ks}}\,s\\
&\approx-{1\over s^2}\int_0^\infty{xe^{-x}\over1-e^{-x}}\,dx\\
&={-\pi^2\over 6s^2}\ .
\endaligned
$$
Consequently, $\sigma_n={\pi/\sqrt{6n}}$, as $n\to\infty$. Similar
arguments give the asymptotics of $\phi(\sigma_n)$ and $\phi''(\sigma_n)$.
\quad\qed
\enddemo
\subhead 2.3. Kronecker flow\endsubhead
In this section we define a non-commutative variant of
the infinite dimensional Kronecker flow. We then sketch the proof
in \cite{KL} of semiclassical ergodicity of that flow.
The proof is based on the ideas of Theorem 1.2.1.
\definition{Definition 2.3.1}
A countable ordered subset $\Omega$ of $\bR$ is
called a {\it Kronecker system} if it satisfies conditions below:
\roster
\item $\Omega$ consists of positive numbers.
\item Let $\omega_n$, $n=1,\ldots$, be the elements of $\Omega$
listed in increasing order. Then $\omega_n\to\infty$, as $n\to\infty$.
\item The elements of $\Omega\ $ are algebraically independent
over $\bZ$.
\endroster\enddefinition
To prove ergodicity, assumption 2 will be considerably strengthened later.
Furthermore, define $\bN[\Omega]:=\{\sum n_i\omega_i,\ 0\leq n_i\in\bZ\}$,
the additive group of finite linear combinations of elements of $\Omega$
with non-negative integer coefficients. Note that, as a consequence of
assumption 3, all numbers $\sum n_i\omega_i$ are different, so that
$\bN[\Omega]$ can be viewed as subset of $\bR$.
We set $\cA$ to be the (reduced) $\bC^*$-algebra of the semigroup
$\bN[\Omega]$. Recall that
it is defined as the algebra generated by the (left) regular
representation of $\bN[\Omega]$ in $\cH:=l^2(\bN[\Omega])$.
It turns out that the $\bC^*$-algebra $\cA$ can be identified with
an infinite tensor product of standard Toeplitz $\bC^*$-algebras
$\fT_\omega$,
$$
\cA=\bigotimes_{\omega\in\Omega}\fT_\omega\ ,
$$
where $\fT_\omega$ is a copy of the (reduced)
$\bC^*$-algebra of the semigroup $\bN$.
For $\eta\in\bN[\Omega]$, let $e(\eta)$ denote the canonical basis
element in $l^2(\bN[\Omega])$. Let $H$ be the
unbounded, self-adjoint operator in $\cH$ defined by
$$
He(\eta)=\eta e(\eta),
$$
and let $\rho_t(a):=e^{itH}ae^{-itH},\ a\in\cA$. By $N(E)$ we denote
the counting function for the eigenvalues of $H$ i.e.
$$
N(E):=\#\{\eta\in\bN[\Omega], \eta\leq E\}.
$$
The dynamical system $(\cA,\rho_t)$ is the quantum Kronecker flow
which we will study in this section. The $\rho_t$-invariant state
$\tau$ will be defined later in (1.2.3). The relation of $(\cA,\rho_t)$
to the classical Kronecker flow is best described in the language of
Toeplitz operators to be discussed next.
Let $X$ denote the infinite cartesian product of unit circles,
$X=\prod_{\omega\in\Omega}S^1$, equipped with the Tikhonov
topology. Let $\mu$ be the normalized Haar measure on $X$. The
classical Kronecker flow $\alpha_t$ on $X$ is given by
$$
\alpha_t\big(\prod_{\omega\in\Omega}e^{ix_\omega}\big)=
\prod_{\omega\in\Omega}e^{ix_\omega+it\omega}\ .
$$
As a consequence of our assumptions on $\Omega$, this flow is ergodic.
In quantum theory, one associates with a function $f\in C(X)$
an operator. This can be done in the following way.
Let $L^2_+(X,\mu)\subset L^2(X,\mu)$ be the subspace of
$L^2(X,\mu)$ consisting of functions whose Fourier transformations
involve non-negative frequencies only, and let $P$ be the orthogonal
projection onto $L^2_+(X,\mu)$. Notice that the Fourier transformation
gives a natural isomorphism $L^2_+(X,\mu)\simeq l^2(\bN[\Omega])=\cH$.
Every $f\in C(X)$ defines a {\it Toeplitz operator} $T(f)$ on
$L^2_+(X,\mu)\simeq\cH$ by
$$
T(f)=PM(f)P,
$$
where $M(f)$ is the operator of multiplication by $f$ on $L^2(X,\mu)$.
It is not difficult to see that $\cA$ coincides with the $C^*$-algebra
generated by the Toeplitz operators. Moreover, we have
$$
\rho_t(T(f))=T(\alpha_t(f))\ ,
$$
which justifies the view that the dynamical system $(\cA,\rho_t)$ is a
non-commutative analog of the ``classical" Kronecker flow.
We set
$$
\tau_E(x):={1\over N(E)}\sum_{\bN[\Omega]\ni\eta\leq E}
(e(\eta),ae(\eta)),
$$
and study $\tau(a):=\lim_{E\to\infty}\tau_E$, the semiclassical
limit. Simple calculation shows that for every $f\in C(X)$,
$$
\tau_E(T(f))=\int f(x)\,d\mu(x).
$$
To proceed further, we need to know the structure of $\cA$.
Let $\Cal{I}$ be the commutator ideal of $\cA$. The structure of the
standard Toeplitz algebra implies that the quotient $\cA/\Cal{I}$ is
isomorphic to $C(X)$. The quotient map $\pi:\cA\to C(X)$ is called the
symbol map, and $\pi(T(f))=f$. The crucial observation here is that if
$\omega_n$ grow sufficiently fast then $\tau$ is zero on the
commutator ideal $\Cal{I}$. To see that we analyze $\tau_E(a)$
for $a\in\Cal{I}$.
Again, the structure of the standard Toeplitz algebra implies that
$\Cal{I}$ is generated by the operators of the form
$$
a=a_{\omega_1}\otimes a_{\omega_2}\otimes\ldots\otimes a_{\omega_N}
\otimes I\otimes I\ldots,
$$
where at least one of the operators $a_{\omega_1}\ldots a_{\omega_N}$,
say $a_{\omega_k}$, is compact. By a density argument, it is no loss
of generality to assume that $a_{\omega_k}$ is a finite rank operator
whose range is spanned by finitely many elements of the canonical basis.
Then
$$
\tau_E(a)\leq {\Vert a\Vert \over N(E)}
\#\{n_\omega\geq 0\ ,\ n_{\omega_k}\leq M : \sum n_\omega\omega\leq E\}.
$$
But $\#\{n_\omega\geq 0\ ,\ n_{\omega_k}\leq M : \sum n_\omega\omega\leq E\}
=N(E)-N(E-\omega_k(M+1))$ by a simple counting argument. The {\it main
difficulty} is to show that, under additional assumptions,
$$
{N(E)-N(E-\omega_k(M+1))\over N(E)}\longrightarrow 0,\ \text{as}\ E\to 0.
\tag 2.3.1
$$
Assume that (2.3.1) is true. Then
$$
\tau(a)=\lim_{E\to\infty}\tau_E(a)=\int\pi(a)(x)\,d\mu(x),
$$
for all $a\in\cA$. In other words, the state $\tau$ is trivial on
the commutator ideal, and it coincides with the Lebesgue integral on
the abelian quotient. Moreover, let $\pi_\tau$ be the GNS representation of
$\cA$ with respect to the state $\tau$. Then,
$$
\pi_\tau(\cA)\simeq C(X),
$$
and for every $a\in\cA$ we have
$$
\pi_\tau(\rho_t(a))=\alpha_t(\pi_\tau(a)).
$$
Theorem 1.2.1 implies now the semiclassical ergodicity.
\remark{Remark 2.3}
The ratio ${N(E)-N(E-\omega_k(M+1))\over N(E)}$ does not always go
to zero as $E\to\infty$. For example if
$\Omega=\{\log p,\ p\ \text{prime}\}$, see \cite{BC}, then it is easy to
see that $N(E)\sim e^E$ and (2.3.1) is not true.
\endremark
Notice that (2.3.1) is equivalent to:
$$
N(E+1)=N(E)(1+o(1))\ \text{as}\ E\to\infty\ .\tag 2.3.2
$$
Also let us remark that $N(E)$ is the number of solutions of
$$
\omega_1n_1+\omega_2n_2+\omega_3n_3+\ldots\leq E
$$
in nonnegative integers $(n_i)$. This is strikingly similar to
the formula for the number of partitions $p(n)$. It also
suggests to use the Hardy-Ramanujan argument to obtain the asymptotic
behavior of $N(E)$, and from there deduce (2.3.2).
In order to apply the method of Theorem 2.2.1 we will need additional
assumptions on the set $\Omega$. Specifically, assume that for every $s>0$,
$$
\theta(s):=\sum_{n=1}^\infty e^{-s\omega_n}<\infty .
$$
This implies that the following $\zeta$-type function (see (2.2.2))
$$
\zeta_\Omega(s):=\prod_{n=1}^\infty(1-e^{-s\omega_n})^{-1}
$$
converges for all $s>0$. Expanding each term of $\zeta_\Omega(s)$
in a power series and multiplying out the terms, we can express
$\zeta_\Omega(s)$ as the following Lebesgue-Stieltjes integral:
$$
\zeta_\Omega(s)=1+\int_0^\infty e^{-sx}\,dN(x) ,
$$
where, as above, $N(x)$ is the counting function for the
eigenvalues of $H$. Equivalently, we can write this formula as
$$
{e^{\phi(s)}\over s}=\int_0^\infty e^{-sx}(N(x)+1)\,dx ,\tag 2.3.3
$$
where $\phi(s):=-\sum_{n=1}^\infty\log(1-e^{-s\omega_n})$, see (2.2.3).
As before, the right hand side of (2.3.3) is essentially the
Laplace transform of $N(x)$, while the left hand side is
controllable and depends on the behavior of $\omega_n$. The standard
lore says that the behavior of $N(x)$, as $x\to\infty$,
can be extracted from the behavior of its Laplace transform
as $s\to 0$.
Informally, one proceeds as follows. Taking the inverse Laplace transform
of both sides of (2.3.3) yields:
$$
N(x)+1={1\over 2\pi}\int_\bR{e^{\phi(\sigma+it)+x(\sigma+it)}
\over \sigma+it}\,dt\ ,
$$
where $\sigma>0$ is arbitrary. Now choose $\sigma=\sigma_x$ such that the
function
$$
t\mapsto\phi(\sigma+it)+x(\sigma+it)
$$
has a critical point at $t=0$. This leads to the following condition on
$\sigma_x$:
$$
\phi'(\sigma_x)+x=0 .
$$
At this point we can use the method of stationary phase to get the
asymptotic of $N(x)$ as $x\to \infty$:
$$\aligned
N(x)&={1\over 2\pi\sigma_x}e^{\phi(\sigma_x)+x\sigma_x}
\int_\bR e^{-\phi''(\sigma_x)t^2/2}\,dt\,(1+o(1))\\
&={e^{\phi(\sigma_x)+x\sigma_x}\over\sqrt{2\pi\sigma_x^2\phi''(\sigma_x)}}
(1+o(1))\ .
\endaligned\tag 2.3.4
$$
The last step is to establish (2.3.2) by analyzing (2.3.4).
It is unlikely that the above strategy will work in such a generality.
The main obstacle is the difficulty to control the other critical
points of $\phi(\sigma+it)+x(\sigma+it)$. Such a control is possible,
however, under the assumption of polynomial growth of $\omega_n$.
The following theorem was proved in \cite {KL}.
\proclaim{Theorem 2.3}
Let $\omega_n=An^\alpha(1+\mu_n)$, where
$A>0$ and $\alpha\geq 1$ are constant, and where $\mu_n=o(1)$,
as $n\rightarrow\infty$. Then (2.3.1) is true and consequently
for every $a\in\cA$,
$$
w-\lim_{T\to\infty}{1\over T}\int_0^T\rho_t(a)\,dt=\tau(a)I+C_a,
$$
where $C_a$ is in the weak closure of $\cA$, and
$$
\lim_{E\to\infty}\tau_E(C_a^*C_a)=0\ .
$$
\endproclaim
\refstyle{A}
\widestnumber\key{KLMR}
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\enddocument