\document
\magnification = 1200
\documentstyle{amsppt}
%\input{amssymb.def}
%\input{amssymb.tex}
\font\aa=cmti7 at 7pt
\baselineskip=16pt
\pageheight{19.2cm}
\pagewidth{12.9cm}
\hcorrection{0.5cm}
\topmatter
\title
Quantization of a class of piecewise affine transformations on the torus
\endtitle
\rightheadtext{Quantization of toral maps}
\author S. De Bi\`{e}vre${}^{\,(1)}$,
M. Degli Esposti${}^{\,(2)}$, R. Giachetti${}^{\,(3)}$,
\endauthor
\affil{
${}^{\,(1)}$ UFR de Math\'{e}matiques et
Laboratoire de Physique Th\'{e}orique et Math\'{e}matique
Universit\'{e} Paris VII - Denis Diderot 75251 Paris Cedex 05 France,\\
debievre\@mathp7.jussieu.fr\\ \\
${}^{\,(2)}$ Dipartimento di Matematica, Universit\`a di Bologna,\\
Porta di Piazza San Donato 5, 40127 Bologna, Italy,\\
desposti\@dm.unibo.it\\ \\
${}^{\,(3)}$ Dipartimento di Fisica, Universit\`a di Firenze,\\
Largo E. Fermi 2, 50125 Firenze, Italy,\\
giachetti\@fi.infn.it}
\endaffil
\thanks{The authors would like to thank Prof. Sandro Graffi for suggesting the problem
addressed in this paper and for many stimulating discussions. The first author thanks the
department of mathematics of the University of Bologna, where part of this work was performed
for its hospitality and the C.N.R. for partial financial support.}\endthanks
\date{(16 December 1994)}
\enddate
\abstract{ We present a unified framework for the quantization of a
family of discrete dynamical systems of varying degrees of ``chaoticity". The
systems to be quantized are piecewise affine maps on the two-torus, viewed as
phase space, and include the automorphisms, translations and skew translations. We
then treat some discontinuous transformations such as the Baker map and
the sawtooth-like maps.
Our approach extends some ideas from geometric
quantization and it is both conceptually and calculationally
simple.}
\endabstract
\endtopmatter
\def\tende#1#2{\matrix \phantom{x}\\ \longrightarrow\\ {#1\rightarrow #2
\atop\phantom{x}} \endmatrix}
\define\Lh{L^2 (\R^2,{dqdp \over 2\pi\hbar})}
\define\LT{L^2(\T, {dqdp \over 2\pi\hbar})}
\define\T{{\Bbb T}^{2}}
%ensembles de nombres
\define\pR{\pbb }
\define\pC{\pbb C}
%\font\bb=msym10
\define\R{\Bbb R}
\define\Q{\Bbb Q}
\define\C{\Bbb C}
\define\N{\Bbb N}
\define\Z{\Bbb Z}
\define\I{\Bbb I}
\define\Qg{\widehat g}
\define\Qf{\widehat f}
\define\Qfh{{\widehat f}^{\hbar}}
\define\Qgh{{\widehat g}^{\hbar}}
\define\Qq{\widehat{q}}
\define\Qp{\widehat{p}}
\define\QB{\widehat{B}}
\define\Di{{\Cal S}'(\R^2)}
\define\Df{{\Cal S}_{\hbar}'(\theta)}
\define\Dfp{{\Cal S}_{\hbar}'(\theta')}
\define\nav{\nabla_v}
\define\eh{\frac{i}{2\hbar}}
\define\hw{{\Cal H}_w}
\define\hz{{\Cal H}_z}
\define\aw{{\Cal A}_w}
\define\dw{{\Cal D}_w}
\define\dz{{\Cal D}_z}
\define\dwt{{\Cal D}_w(\theta,N)}
\define\dzt{{\Cal D}_z(\theta,N)}
\define\hwt{{\Cal H}_w(\theta,N)}
\define\hzt{{\Cal H}_z(\theta,N)}
\define\hvt{{\Cal H}_v (\theta,N)}
\define\dwtp{{\Cal D}_w(\theta',N)}
\define\hwtp{{\Cal H}_w(\theta',N)}
\define\hztp{{\Cal H}_z(\theta',N)}
\define\thetap{\theta '}
\define\K{{\Cal K}_w}
\define\wx{w^tx}
\define\vx{v^tx}
\define\qwx{\widehat{w^tx}}
\define\qvx{\widehat{v^tx}}
\define\Lp{{\Cal L}_{\theta}}
\define\a1{\alpha_1}
\define\bt{\beta_{\theta}}
\define\qkw{q_k(w,\theta)}
\define\qlw{q_\ell(w,\theta)}
\define\qlz{q_\ell(z,\theta)}
\define\qkz{q_k(z,\theta)}
\define\wl{q_\ell(w,\theta)}
\define\wlp{q_\ell(w,\theta')}
\define\zk{q_k(z,\theta)}
\define\g1{\gamma_1}
\define\ga2{\gamma_2}
\define\mt{\widetilde m}
\define\nt{\widetilde n}
\define\xkl{x_{k\ell}}
\define\ykl{y_{k\ell}}
\define\teti{\tilde{\theta}}
\magnification = 1200
\baselineskip=16 pt
\TagsOnRight
\nologo
\subheading{1. Introduction}\par
\bigskip
Interest in the quantization of discrete dynamical systems on compact phase
spaces comes from the desire to understand the
possible signature of classical chaotic dynamics in quantum mechanics.
Recall for example that it is expected and in some cases proved that the asymptotic
properties ($\hbar\to 0$) of the eigenfunctions of quantized systems depend on
the degree of ``chaoticity'' of the corresponding classical ones (see, for
instance, \cite{Sar} and references therein). The torus forms an excellent
testing ground for these ideas. Indeed, the simplest ergodic
systems are the irrational translations on the torus, whereas the simplest
hyperbolic dynamical systems are certain area-preserving maps \cite{AA,CFS}.
Among
these, the best known are the toral automorphisms, the Baker transformation
and some discontinuous maps such as the sawtooth map considered in \cite{Ch,LW,V,Li}. It has
been shown there that their singularities do not destroy the ergodicity and mixing properties
one expects for hyperbolic maps.
One way in which the classical singularities will show up at the quantum level is as follows.
For the linear automorphisms the classical and the quantum evolution are identical, as in the
harmonic oscillator. The singularities will destroy this property, so that, to control the
semiclassical behaviour of the eigenfunctions a non trivial Egorov theorem will be needed.
Similarly, the statistics of the eigenvalues of the quantum propagator should be more
generic than in the linear case, where they are determined by purely arithmetic properties.
Clearly, before being able to address this kind of problems, one needs to develop a
quantization for the systems considered.
Since none of the above examples is obtained by evaluating a smooth Hamiltonian flow
on the torus at discrete times, the usual quantization schemes all fail and a direct
attack is needed.
In this paper we will show how to extend the most elementary part of geometric quantization
\cite{Bl,GuSt, Ko, Sn, Wo} beyond its natural context in order to construct a unified and
simple framework for the quantization of all of the above systems. Some of them
had not been quantized before, such as the translations and certain piecewise affine hyperbolic
maps. It will turn out that the unitary matrices describing the quantum
evolution of each of those systems can be computed straightforwardly and with relatively
little effort in this way.
The toral automorphisms and the Baker transformation were quantized
respectively in \cite{HB,DE, DGI} and in \cite{BV} and they have been studied
intensely ever since, both numerically and analytically
\cite{ Ke1, Ke2, Ke3, DGI, Eck, Sa}. The methods of quantization used in these
papers look very different from each other. Our approach reproduces the same results
in those cases.
In order to get a more precise flavour of the ideas to be developed,
recall that in classical mechanics the dynamics of a system is obtained
by integrating a Hamiltonian vector field $X_H$ on a symplectic manifold
$(M,\omega)$. Here $H\in C^{\infty}(M)$ and $X_H$ is defined by
$X_H \rfloor\omega=dH.$ In quantum mechanics, the dynamics is given by a
unitary flow $U_t$ on a Hilbert space
${\Cal H}_{\hbar}$. A {\it quantization} is a set of rules allowing
to associate to $(M,\omega)$ a Hilbert space ${\Cal H}_{\hbar}$ and to
each function $f$ on $M$ in a suitable class ${\Cal C}$, a self-adjoint
operator $\widehat f$ on ${\Cal H}_{\hbar}$. One then says that
$U_t = \exp[\,(-i/\hbar)\widehat H t\,]$ is the quantization of the
classical flow of $X_H$. Typical requirements \cite{Be} are that the map
$f\mapsto \widehat f$ is linear, injective, unital, {\it i.e.} that it satisfies
$\widehat 1 = Id_{{}\,{\Cal H}_{\hbar}}\,$, and that it is compatible with the natural
involutions, $(\,\widehat f\,)^* = \widehat {\bar f\,}\,$.
Moreover, one requires the classical limit condition
$
(1/i\hbar)\,[\,\widehat f,\widehat g\,]\,\,\tende \hbar 0
\,\,\widehat {\{f,g\}}\,.
$
When the classical evolution is not a flow, but a discrete map, this scheme is
clearly not sufficient.
We extend here some of the simplest ideas
of geometric quantization beyond their natural range of applicability
to obtain a unified framework for the quantization of
a reasonably large class of area preserving maps on the torus. We will show
that, in spite of its reputation, the essence of geometric quantization is
intuitive, simple and well suited for such generalizations. For that purpose,
we first present in Section 2 a revisited version of the
geometric quantization on $T^*\R$, just to
demonstrate how it permits to {\it
reformulate} quantum mechanics for systems having $T^*\R$ as phase space
and to quantize linear flows. At
several points, we shall use physical or intuitive arguments to motivate parts
of the construction that are usually justified in terms of very general
geometrical objects.
We then apply this approach to the quantization of toral automorphisms
in Section 3: the resulting quantum propagators are identical to the ones
obtained elsewhere by other methods \cite{HB,DE}.
In the final Section 4 we shall obtain the
quantization of translations, skew-translations as well as of
a class of {\it piecewise linear} hyperbolic maps such as the Baker
transformation and other maps studied, for instance, in \cite{Ch,LW,Li,V}. Those
maps do not preserve the natural geometric structures associated with the torus,
and therefore geometric quantization as such does not apply to them.
The proposed extension, however, will provide a definite answer.
\bigskip
\bigskip
\subheading{2. Geometric quantization on $T^*\R$}\par
\bigskip
As usual we call $(q,p)$ the coordinates of $T^*\R\cong\R^2$ and choose the
standard symplectic form $\omega = dq\wedge dp$ that gives the canonical
Poisson bracket
$\lbrace q,p \rbrace = 1\,$. Our goal is to realize the space of the quantum
states ${\Cal H}_{\hbar}$ as a subspace of $\Di$, equipped with a suitable
Hilbert space structure, and to establish a correspondence between classical
and quantum observables, so as to be able to describe the physical properties
of the quantum system. To this purpose we recall a first result, the validity
of which is easily checked by a direct computation: there exists a map
$f\in C^{\infty}(\R^2)\to\Qf\in L(\Di,\Di)\,,$ which is linear, unital
and satisfies the classical limit condition.
This map is explicitly given by
$$
\Qf=-i\hbar\nabla_{X_f} +f,\tag{2.1}
$$
where $X_f = (\partial_p f)\,\partial_q - (\partial_q f)\,\partial_p$
is the Hamiltonian vector field associated to $f$ and
$\nabla_X = X - (i/\hbar)\,X\rfloor \theta $
denotes the covariant derivative with respect to the connection form
$\theta = {1\over 2}(pdq - qdp)\,$.
Note that the use of $\nabla_X$ guarantees the local gauge invariance of
the construction (see \cite{Wo, Sn} for details).
It is moreover worth remarking that, if $\hat f$ in (2.1) is replaced
by $-i\hbar X_f$, then the unital property fails to hold, thereby
violating the uncertainty principle. In particular we have
$\widehat q = i\hbar\,\partial_p + q/2\,$ and
$\widehat p = -i\hbar \,\partial_q + p/2\,,$
so that, indeed, the canonical commutation relation
$\lbrack \widehat q, \widehat p \rbrack = i \hbar$ is satisfied.
The correspondence between $f$ and $\widehat f$ given in (2.1) is referred to
as {\it prequantization} \cite{Ko}.
We now need some conditions to choose the subspace ${\Cal H}_{\hbar}$ of $\Di$
and the Hilbert space structure it has to carry for it to correspond to the quantum
Hilbert space of states. Note first that the equation $i\hbar\partial_t\psi_t=\Qf\psi_t$
is easily solved on $\Di$. Writing $\psi_t=\exp{[\,-(i/\hbar)\widehat f t\,]}\,\psi$,
one has
$$\align {}&\bigl(\exp[\,{\frac{i}{\hbar}\,\widehat f t}\,]\,\psi\bigr)(q,p)=\\
{}&{}\quad\quad\exp{[\,-\frac{i}{\hbar}\int_0^t ds\,
\bigl(\frac{1}{2}(p(s)\dot q(s)-q(s)\dot p(s))-f(q(s),p(s))\bigr)\,]}\,
\psi(q(t),p(t))\,.
\endalign
$$
where $(q(s),p(s))$ is the solution of the Hamilton equations
$\dot q=\partial_p f$, $\dot p=-\partial_q f$, with initial conditions $(q,p)$. Note that
the prequantized flow $\exp{[\,(i/\hbar)\widehat f t\,]}$ makes sense also when
$\psi\in\Di$.
The idea is then to try to pick ${\Cal H}_{\hbar}$ in such a way that
$\exp{[\,(i/\hbar)\widehat f t\,]}$ is a unitary one-parameter group for
a suitable large class ${\Cal C}$ of functions $f$.
This allows then for the interpretation of $\Qf$ as the quantized observable.
An a priori obvious choice would be $\Lh$.
It is nevertheless not suitable as the quantum Hilbert space.
Indeed it is easily seen that the spectra of $\Qq$ and $\Qp$ are not simple:
actually, the generalized eigenspaces are infinite dimensional, which is
in contradiction with standard quantum mechanics on
$L^2(\R)$. Otherwise stated, $\Qq$ (or $\Qp$) is not a complete set of
commuting observables on $\Lh$, or, equivalently, $\Qq$ and
$\Qp$ do not generate an irreducible algebra.
To put this more precisely, recall that the Heisenberg group is the group
$H=\R^3$ (as a set) equipped with the group law
$(a,b,\phi)(a', b', \phi') = (a+a', b+b', \phi +\phi'+{1\over 2}(ab'-a'b)).$
$H$ acts on $\R^2$ by $(a,b,\phi)(q,p)=(q+a, p+b)$. The
prequantized operators $\hat{q}$, $\hat{p}$ generate a unitary representation of
$H$ on $\Lh$ given explicitly by
$$[U(a,b,\phi)\,\psi](q,p)
=\exp[\,{ - {i \over \hbar}\phi}\,]\,
\,\exp[\,{ -{i \over 2\hbar}(ap-bq)}\,]\,\,\,\psi(q-a,p-b).\tag{2.2}$$
This representation is not irreducible on $\Lh$.
There is a second problem with (2.1) which is worthwhile mentioning.
It is easy to see that, if $H(q,p) = p^2/2 + V(q)$, then
$\widehat H \not = \widehat p^{\,2}/2 + V(\widehat q)\,.$
It is then clear that the correspondence (2.1) is far from reproducing the
Schr\"odinger equation.
Some conditions have to be imposed on
the quantum Hilbert space ${\Cal H}_{\hbar}\subset \Di$ in order to
avoid the previous difficulties.
For the irreducibility of the algebra generated by $\hat q$ and $\hat p$ we
should require
$\phantom{ii}(i)$ ${} U(a,b,\phi)$ restricts to a unitary irreducible
representation of $H$ on ${\Cal H}_{\hbar}$.
\noindent To reproduce the Schr\"odinger equation we should impose:
$\phantom{i}(ii)$ ${}\quad\exists n_0\in\N^*$, and a dense subspace $D$ of
${\Cal H}_{\hbar}$ so that $\widehat p$, ${\Qp}^{\,2}$, and ${\Qq}^{\,n}\,$
($1\leq n\leq n_0$) are essentially self-adjoint on $D$ and ${\Qp}^{\,2}=
\widehat{p^2\,}$,
${\Qq}^{\,n}=\widehat{q^n}$ on $D$.
\noindent Note that this is equivalent to requiring the correct form of
the Schr\"odinger equation for all polynomial potentials of order at most
$n_0$. We are however already asking too much if we take $n_0\geq 2$, as we now show.
\proclaim{Proposition 2.1} If $\psi\in\Di$ satisfies ${\Qp}^{\,2}\psi={\widehat p}^2\psi$
and ${\Qq}^{\,2}\psi=\widehat{q^2}\psi$, then $\psi=0$.
\endproclaim
The proof of this proposition is a simple calculation that we omit.
In conclusion, the requirements $(i-ii)$ can not be satisfied on any non trivial subspace
of $\Di$. Hence we can not even quantize in the proposed manner Hamiltonians with quadratic,
let alone general polynomial potentials. The best we can still hope to do is to impose
$(i)$ and a weakened version of $(ii)$, as we now explain.
Given $w\in\R^2$, with $w=(w_1,w_2)$, let $v\in\R^2$ such that $\omega(w,v)=1$,
we define the subspace
$${\Cal D}_w=\{\psi\in\Di \,\vert\, \nabla_w\psi=0\,\},\tag{2.3}$$
where $X_{h_w}$ is the Hamiltonian vector field associated to
$h_w(x)={}^T\!w\,x=w_1q+w_2p$ and $\nabla_w:\,=\nabla_{X_{h_w}}$. Here and
in the following $x\equiv (q,p)$. We then have
\proclaim{Lemma 2.1}
Let $w\in\R^2$ and $v\in\R^2$ such that $\omega(w,v)=1$. Then $\psi\in{\Cal D}_w$ if and only if
there exists a tempered distribution $f_v$ on the line such that
$$
\psi(x)=f_v(h_w(x))\, \exp[\,{-\frac{i}{2\hbar}\,h_w(x)h_v(x)\,]}.\tag{2.4}
$$
\endproclaim
\demo{Proof} We have
$\nabla_{w}=(w_2{\partial}_{q} -w_1{\partial}_{p})-(i/2\hbar)h_w(x)$.
Consider the map $(q,p)\mapsto (y_1,y_2)=(h_w(x),h_v(x))$ which is
linear and with determinant equal to unity. $\nabla_w\psi=0$ becomes
$\partial_{y_2}\,\eta(y_1,y_2)=-(i/2\hbar)\,y_1\,\eta(y_1,y_2)\,,$
with $\eta(y_1,y_2)=\psi(q,p)$. Its general solution is
$\eta(y_1,y_2)=f_v(y_1)\,\exp[\,{-(i/2\hbar)\,y_1y_2}\,]\,,$ thus
proving the lemma.
\qed\enddemo
\remark{Remark} If $v'\in\R^2$ satisfies $\omega(w,v')=1$, then $v'=v+rw$,
with $r\in\R$. It is easy to see that
$f_{v'}(y)=\exp[\,(i/2\hbar)ry^2\,]\,f_v(y)$.
We will therefore omit the indication of the dependence of $f$ on $v$.
We then have the following Lemma.
\endremark
\proclaim{Lemma 2.2} Let $\psi\in\Di$ and $w\in\R^2$. Then the following
are equivalent:
\roster
\item
$\widehat h{}_w^{\,2}\psi=\widehat{h_w^2}\psi$;
\item
$\widehat h{}_w^{\,n}\psi=\widehat{h_w^n}\psi$, for all $n\in\N$;
\item
$\psi\in\aw : =\{\eta\in\Di\,\vert\,(\nabla_{X_w})^2\eta=0\}$;
\item
Let $v\in\R^2$ such that $\omega(w,v)=1$.
Then there exist $\varphi_0,\varphi_1\in{\Cal S}'(\R)$ such that
$$
\psi(x)=\bigl(h_v(x)\,\varphi_0(h_w(x)) +\varphi_1(h_w(x))\bigr)\,
\exp[\,{-\eh\,h_w(x)h_v(x)}\,]\,.
$$
\endroster \noindent
Moreover, if $u\in\R^2$, then $\widehat h_u\aw\subset \aw.$
\endproclaim
\demo{Proof}
A direct calculation shows
$
\widehat {h_w^n}=-i\hbar n\,h_w^{n-1}\nabla_{w} + h_w^n\,.
$ Using $[\nabla_w,h_w(x)]=0$ to compare
$\widehat {h_w^n}$ to $(\widehat{h_w})^n$, and the previous lemma, the result follows
easily.\qed
\enddemo
The lemma suggests to weaken $(ii)$ by imposing, $(\widehat{h_w})^n=\widehat {h_w^n}$, for
some choice of $w$. This would imply $D\subset \aw$.
Now it is
not hard to see that the eigenvalues of $\hat q$ and $\hat p$ on $\aw$ are
doubly degenerate. In order to satisfy $(i)$ it would be natural to pick
$D$ in a subspace of $\aw$ on which this degeneracy is lifted. It is
easy to describe all subspaces of $\aw$ that are, like $\aw$ itself,
invariant under all $\widehat h_u$, and on which the eigenvalues of all
$\widehat h_u$ are non-degenerate. Although there seems to be no physical
criteria permitting to select one such subspace, $\dw$ (see (2.3)) satisfies the above
requirements and it is customary in geometric quantization to construct $\hw$ as a subspace
of $\dw$ because of its geometric appeal.
The condition $\nabla_w\psi=0$ is called a
{\it polarization condition} in this context. Note that we can identify
$\dw$ with
${\Cal S}'(\R)$ and that
$\widehat h_w$ then acts as a multiplication operator while $\widehat h_v$ as a
derivative operator. A calculation as in the proof of Proposition $2.1$ shows that
if $u\in\R^2$ is not a multiple of $w$, then $\widehat{h_u^2}\,\dw\cap\dw=\{0\}$, thus
excluding a priori the quantization of quadratic Hamiltonians, as already pointed out.
Let us now briefly show how one can nevertheless correctly describe the
quantization of quadratic Hamiltonians within the framework of geometric
quantization (see [GuSt, Wo] for details).
Recall that for a quadratic polynomial
$f(q,p) = (\lambda/2)q^2 + \mu qp + (\nu/2)p^2\,$ the flow of
$X_f$ is linear and can be written as
${}^T\!(q(t),p(t))= A(t)\,{}^T\!(q,p)\,,$ with $A(t)\in SL(2,\R)$
(${}^T$ denotes the transpose). The prequantized flow then reads
$$\bigl(\exp[\,{\frac{i}{\hbar}\,\widehat ft}\,]\,\psi\bigr)(q,p)
=\psi(A(t)\binom{q}{p})\,:\,= (U(A^{-1}(t))\psi)(q,p)$$
and the map $A\mapsto U(A)$ gives a unitary representation of $SL(2,\R)$
on $\Lh$. We now observe that $U(A)$ satisfies
$U(A)\dw={\Cal D_{{}\,{}^T\!\!A^{-1}w}}\,.$
We will explain below that it is possible to equip a suitable subspace $\hw$ of $\dw$ with a
Hilbert space structure and then to identify the Hilbert spaces for different
values of $w$ by means of unitary maps $P_{zw}:\hw\rightarrow{\Cal H}_z$. This
is a particular case of a general construction which allows to compare Hilbert
spaces corresponding to different real or complex polarizations (BKS kernels
\cite{Wo,GuSt, Sn}). The quantized linear transformation $V(A)$ is then
defined by $V(A)=D_\hbar P_{w,{}^T\!\!A^{-1}w}\circ U(A):\hw\to\hw$ (see (2.8)).
We start by showing how to equip suitable subspaces $\hw$ of the $\dw$ with a
Hilbert space structure. Note first that (2.4) implies that if
$\psi_1,\psi_2\in\dw$, then $\bar{\psi_1}\psi_2$ is a function of $h_w(x)$.
Moreover
$$
[\,\overline{U(a,b,\phi)\psi_1}\,U(a,b,\phi)\psi_2\,](q,p)=
\bar{f_1}f_2(h_w(x)-aw_1-bw_2).
$$
This suggests defining a Hilbert subspace $\hw$ of $\dw$ by
$$
\hw=\{\psi\in\dw\vert\,\int\vert\psi\vert^2(y)\,dy\,<\,\infty\},\tag{2.5}
$$
equipped with the obvious scalar product
$\langle\psi_1,\psi_2\rangle_{w}\,:\,=\int \bar{\psi_1}\psi_2(y)\,dy\,.$
The choice of the Lebesgue measure in (2.5) is dictated by the requirement
that $U(a, b,\phi)$ be unitary on $\hw$.
Let $w=(w_1,w_2)$ and $z=(z_1,z_2)$ be linearly independent and consider the two
corresponding Hilbert spaces ${\Cal H}_{w}$ and ${\Cal H}_{z}$.
We denote by $v=(v_1,v_2)$ and $u=(u_1,u_2)$ two fixed vectors such that
$\omega(w,v)=\omega(z,u)=1$.
Consider $\psi\in{\Cal H_{w}}$ and $\varphi\in{\Cal H_{z}}$.
It is then easy to see that $\bar{\varphi}\psi$ belongs to $L^1(\R^2,dq\,dp)$.
The following proposition then follows from a straightforward calculation that we omit
\cite{GuSt}.
\proclaim{Proposition 2.2}
Let $w,z\in\R^2$ be linearly independent. Let $\Delta=\omega(w,z)$. Then there exists a
unique continuous linear map $P_{zw}: {\Cal H_{w}}\to {\Cal H_{z}}$
such that, $\forall \psi\in {\Cal H}_w$ and $\forall \varphi\in {\Cal H}_z$
$$
\langle\varphi,P_{zw}\psi\rangle_{z}
=\int\bar{\varphi}\psi\,\frac{dq\,dp}{2\pi\hbar}\,. \tag{2.6}
$$
Moreover, if $D_\hbar\in\C$, with $\vert D_{\hbar}\vert=\sqrt{2\pi\hbar
\,\vert\Delta\vert\,}$\,, then $D_{\hbar}P_{zw}$ is unitary.
\endproclaim
The proof of the proposition provides an explicit expression for $P_{zw}$:
$$
(P_{zw}\,\psi)(x)=\frac{1}{2\pi\hbar\Delta}\,
\exp{[\,-\frac{i}{2\hbar}\,h_z(x)h_u(x)\,]}\,
\int f(y)\,\exp{[\,-\frac{i}{\hbar}S_{zw}(y,h_z(x))\,]}\, dy\,,
$$
where $S_{zw}$ is the quadratic form
$$
S_{zw}(y_1, y_2)=\frac{1}{2\Delta}\,\Bigl[\,(y_1,y_2)\left
(\matrix \omega(v,z) & 1\\ 1 & \omega(u,w)
\endmatrix\right)\binom{y_1}{y_2}\,\Bigr]\,.\tag{2.7}
$$
Note that $P_{zw}$ extends to $\dw$ (see \cite{Fo}).
The previous result allows to associate to any linear map $A\in SL(2,\R)$ and
to any given $z\in\R^2$ a well defined unitary operator, unique up to a phase,
in the following manner. Given $A\in SL(2,\R)$ and $z\in\R^2$, it follows
immediately that $\forall \psi\in\hz$ of the form (2.4), we have
$$
(U(A)\,\psi)(x)=f(h_{{}\,{}^T\!\!A^{-1}z}(x))\,
\exp{[\,-\eh\,h_{{}\,{}^T\!\!A^{-1}z}(x)\,h_{{}\,^T\!\!A^{-1}u}(x)\,]},
$$
where $U(A)$ is the previously defined prequantum action.
Hence $U(A)\hz={\Cal H}_{{}\,^T\!\!A^{-1}z}$ and we can define
$V(A):\hz\to \hz$ by
$$
V(A)=D_\hbar P_{z,{}^T\!\!A^{-1}z}\circ U(A).\tag{2.8}
$$
$V(A)$ is an unitary integral operator representing the quantum propagator
associated to the classical symplectic transformation $A$. Indeed, to see that
it agrees with Schr\"odinger quantum mechanics (up the the choice of a phase),
note that in the case where $z=(1,0)$ and $A=\left(\matrix a & b\\ c
& d \endmatrix\right)$, with $b\neq 0$, we recover the well known formula
($u=(0,1)$, $w=(d,-b)$, $v=(-c,a)$, $\Delta=b$) for the integral kernel
of $V(A)$, {\it i.e.}
$$
V(A)(y_1,y_2)=\frac{1}{\sqrt{2\pi\hbar b}}\,
\exp{[\frac{i}{2\hbar b}(ay_1^2-2y_1y_2+dy_2^2)\,]}\,.
$$
The correct phase for $V(A)$ is not obtained by the very simple approach
we have presented. This can be done with a considerable amount of
additional work \cite{Fo, GuSt}: this problem is however of no concern
in the present framework, since a global phase does not change the quantum
dynamics of a single transformation.
\bigskip
\bigskip
\subheading{3. Quantization of toral automorphisms}\par
\bigskip
We shall now apply our previous construction to quantization on the torus
$\T =\R^2/\Z^2$, with canonical symplectic structure $\omega_{\T}$, such that
$d\pi^*\,\omega_{\T}=dq\wedge dp$, where $\pi:\R^2\to\T$ is the usual covering
map. In the first place, we need to identify the quantum Hilbert space.
The periodicity of the system in $q$ and in $p$ will be taken into account
along the same lines well known in solid state physics, namely by considering
distributions on $\R^2$ with quasiperiodic boundary conditions both in $q$ and
$p$. This approach is calculationally convenient and we shall show its
equivalence with the geometric quantization procedure. It has the advantage of
being readily extendable to the geometrically non-natural situations of Section 4.
Let us introduce $\,u_1=U(1,0,0)\,$ and $\,u_2=U(0,1,0)\,$ as in (2.2).
Given $\hbar\in \R^+$ and $\theta\in\T$, we denote by $\Df$ the space of all the
tempered distributions $\psi$ on the plane satisfying the following conditions:
$$
u_1 \,\psi(q,p)=\exp[\,{2\pi i\,\theta_1}\,]\,\psi(q,p)\,,\quad\quad\quad
u_2 \,\psi(q,p)=\exp[\,{2\pi i\,\theta_2}\,]\,\psi(q,p)\,.\tag{3.1}
$$
Computing $(u_1u_2-u_2u_1)\psi$ using (2.2) and (3.1) one can easily see that
this space is non trivial if and only if
$
2\pi\hbar N=1
$
for some $N\in\N$. We shall refer to this as the {\it prequantum condition} and,
from now on, we shall assume it to be satisfied. In this case,
$\forall n=(n_1,n_2)\in\Z^2$ and $\psi\in\Df\,$,
$$
\psi(x+n)=\exp{[\,-2\pi i\,(\theta_1 n_1+\theta_2 n_2)\,]}\,
\exp{[\,\frac{i}{2\hbar}(qn_2-pn_1)\,]}
\exp{[\,-\frac{i}{2\hbar}n_1n_2\,]}\, \psi(x)\,,\tag{3.2}
$$
where, as in Section 2, $x=(q,p)$.
Given now $\psi(q,p)\in\Df$ and $w\in\R^2$, one checks readily that
$\nabla_w \psi\in\Df$. We then define, in analogy with (2.3), the corresponding
space of linearly polarized sections $
\dwt=\Df\cap\dw=\{\psi\in\Df\vert\, \nabla_w\psi=0\}\,.
$
We will only consider polarizations of the torus for which $w_2/w_1\in\Q$.
This is equivalent to requiring that the flow lines of $X_w$ are circles.
In this case, up to rescaling $w$ by a constant multiple, we can
assume $w=(w_1,w_2)\in\Z^2$ with $g.c.d.\,(w_2,w_1)=1$.
\proclaim{Theorem 3.1}
Let $w=(w_1,w_2)\in\Z^2$ as above, where $g.c.d.(w_1,w_2)=1\,.$
Then $\dwt$ is a complex vector space of dimension $N$. Choosing $v\in\Z^2$
with $\omega(w,v)=1$, any $\psi\in\dwt$ can be written uniquely as
$$
\psi(x)=\sum_{k\in\Z}c_k\,\exp{[\,-i\pi N\, h_w(x)h_v(x)\,]}\,
\delta(h_w(x)-q_k(w,\theta))\,,\tag{3.3}
$$
where
$$
\qkw=k/N -(1/2)\,w_1w_2+(1/N)\,\omega(w,\theta)\,,\tag{3.4}
$$
and $\forall k\in\Z$, the $c_k\in\C$ satisfy
$$
c_{k+N}=\exp{[\,2\pi i\,\alpha_{\theta}(N,w)\,]}\,c_k\,,\tag{3.5}
$$
with
$$
\alpha_{\theta}(N,w)=(N/2)\,v_1v_2+\omega(v,\theta)\,.\tag{3.6}
$$
Conversely, any $\psi\in\Di$ of the form $(3.3)$ with the $c_k$'s satisfying
$(3.5-6)$ belongs to $\dwt$.
\endproclaim
\demo{Proof} Let $\psi\in\dwt$, then Lemma 2.1. implies that it is of the form
$$
\psi(x)=f(h_w(x))\,\exp{[\,-i\pi N\,h_w(x)h_v(x)\,]},\tag{3.7}
$$
where $v\in\R^2$ is chosen such that $\omega(w,v)=1$.
It will be convenient to take $v\in\Z^2$. Note that, since
$g.c.d.\,(w_1,w_2)=1\,$, such $v$ always exists.
Using (3.2) and making the simple observation that
$$h_w(a)h_v(b)-h_v(a)h_w(b)=\omega(a,b)\,\quad\forall a,b\in\R^2\,,\tag{3.8}$$
one obtains, for any $n\in\Z^2$ and $t\in\R\,$, that $f$ must satisfy
$$
\align
f(t+h_w(n))=&\exp{[\,i\pi N\,(2th_v(n)+h_w(n)h_v(n))\,]}\\
{}&{}\quad\quad\exp{[\,-i\pi N\,n_1n_2\,]}\,
\exp{[\,-2\pi i\,h_n(\theta)\,]}\, f(t)\,.\tag{3.9}\\
\endalign
$$
Choosing $n=m$, where $m=(-w_2,w_1)$, and noting that
$h_w(m)=0\,$, $h_v(m)=\omega(w,v)=1\,$, $h_m(\theta)
=\omega(w,\theta)\,$,
one concludes that $f$ is of the form ($c_k\in\C$)
$$
f(t)=\sum_{k\in\Z} c_k\,\delta(t-q_k(w,\theta))\,,\tag{3.10}
$$
where the $q_k(w,\theta)$ are given in (3.4). Therefore (3.9) yields
$$
\sum_{k\in\Z} c_k\,\delta(t-q_k+h_w(n))= \sum_{k\in\Z}\,c_k\,\exp{[\,2\pi i\,
\bt (k,n,N)\,]}\,\delta(t-q_k)\,,
$$
where
$$
\bt(k,n,N)=Nq_k h_v(n)+\frac{N}{2}\,h_w(n)h_v(n)-\frac{N}{2}\,n_1n_2
-h_n(\theta)\,.\tag{3.11}
$$
Note that $\bt(k,n,N)$ depends on $k$ only through a term
$kh_v(n) =0\,\mod 1$.
Clearly we can drop this term and replace $\bt(k,n,N)$ by $\bt(n,N)$ defined by
$$
\bt(n,N)=-\frac{N}{2}w_1w_2h_v(n)+\omega(w,\theta)h_v(n)+
\frac{N}{2}\bigl(h_w(n)h_v(n)-n_1n_2\bigr)-h_n(\theta).\tag{3.12}
$$
Observing that $q_k -h_w(n)=q_{k-Nh_w(n)}$ (see (3.4)), we find the following
condition on the $c_k$, $\forall n\in\Z^2$ and $\forall k\in\Z$:
$$
c_{k+Nh_w(n)}=\exp{[\,2\pi i\,\bt (n,N)\,]}\,c_k.\tag{3.13}
$$
We will show that the solution space of (3.13) is exactly $N$-dimensional.
First note that, for (3.13) to have any non-trivial solution at all, it is necessary that
$$
\bt(n,N)=\bt(\nt,N)\,\mod 1\,,\tag{3.14}
$$
whenever $h_w(n)=h_w(\nt)$, {\it i.e.} whenever $\exists r\in\Z$
so that $\nt=n+rm$, ($m=(-w_2,w_1)$).
To prove (3.14), remark first that $\forall n,n'\in\Z^2$
$$
\bt(n+n',N)=\bt(n,N)+\bt(n',N)\,\mod 1\,.\tag{3.15}
$$
This follows immediately from (3.11) upon using (3.8).
Moreover, one has
$$
\bt(m,N)=N\,[\,-\frac{1}{2}\,w_1w_2+\frac{1}{N}\,\omega(w,\theta)\,]
+\frac{N}{2}\,w_1w_2-\omega(w,\theta)=0\,\mod 1\,.
$$
This, together with (3.15) implies (3.14). We can then choose $c_0,c_1,\cdots,c_{N-1}$ freely and
define $c_k$ for all other $k$ using (3.13).
To assure that the resulting solutions satisfy (3.13) $\forall k\in\Z$, and not only for
$k=0,1,\cdots,N-1$, condition (3.15) is necessary and sufficient.
Finally, to compute $\alpha_{\theta}(N)$, note that
$\alpha_{\theta}(N)=\bt(n,N)$ for any $n\in\Z^2$ such that $h_w(n)=1$.
If we take $n=(v_2,-v_1)$, then $h_v(n)=0$, and (3.12) yields (3.6).\qed
\enddemo
\remark{Remark}
In particular, if $w=(1,0)$, it easy to see that the corresponding space
of polarized sections contains all distributions of the form
$f(q)=\sum_{k} c_k\,\delta(q-k/N-\theta_2/N)\,,$ where
$c_{k+N}=\exp{[\,-2\pi i\,\theta_1\,]}\,c_k$.
\endremark
Given now $w,v\in\Z^2$ and $\theta\in\T$ as before, the previous proposition
allows us to identify the space of sections $\dwt$ with $\C^N$, as follows:
$$
(c_0,\cdots,c_{N-1})\in\C^N \mapsto \psi(q,p)\in\dwt,\tag{3.16a}
$$
where
$$
\psi(q,p)=\sum_{k\in\Z} c_k\,\exp{[\,-i\pi N\, h_w(x)h_v(x)\,]}\,
\delta (h_w(x)-\qkw).\tag{3.16b}
$$
Here, for $k\notin \{0,\cdots,N-1\}$, the $c_k$'s are defined by (3.5).
In analogy with the results of Section 2, we give $\dwt$ a Hilbert space
structure. Here also, the choice of the inner product will be dictated by
the requirement that the Heisenberg group acts unitarily. We shall denote by
${\Cal H}_w(\theta,N)$ the quantum Hilbert space thus obtained.
Setting $m=(-w_2,w_1)$ and $\mt=(v_2,-v_1)$ it is easy to see that $m$ and
$\mt$ form a basis of $\R^2$ and, in addition, that $\forall n\in\Z^2$, there
exist unique $\alpha,\beta\in\Z$ such that $n=\alpha m+\beta\mt$.
Moreover, by using (2.2), one computes, for all
$\psi\in\dwt$ and for any $\alpha,\beta\in\R$ as
in (3.5),
$$
\align
\bigl(U(\alpha m)\psi\bigr)(x)&=\sum_{k\in\Z}[U(\alpha m)c]_k\,
\exp{[\,-i\pi N\, h_w(x)h_v(x)\,]}\,\,\delta(h_w(x)-q_k)\,,\\
\bigl(U(\beta\mt)\psi\bigr)(x)&=\sum_{k\in\Z}c_k\,
\exp{[\,-i\pi N\,h_w(x)h_v(x)\,]}\,\,\delta(h_w(x)-(q_k+\beta))\,,\\
\endalign
$$
where
$$
\bigl(U(\alpha m)c\bigr)_k=\exp{[\,2\pi iN\, q_k\alpha\,]}\,c_k\,.
$$
>From these results and Theorem 3.1, we see that
$U(a,b) \dwt \subset \dwt$ if and only if $N(a,b) \in\ \Z^2$ and then
$$
\bigl(U(\frac{\ell}{N}\,m)\,c\bigr)_k=\exp[\,{2\pi i\, q_k\ell}\,]\,c_k\,,
\quad\quad\quad \bigl(U(\frac{\ell}{N}\,\mt)\,c\bigr)_k= c_{k-\ell}\,.\tag{3.17}
$$
The natural Hilbert structure making $U(m)$ and $U(\mt)$ unitary is given by
$$
\langle\psi_2,\psi_1\rangle_w=
\sum_{k=0}^{N-1}{\bar{d}}_k c_k.\tag{3.18}
$$
where $\psi_1\cong (c_0,\cdots c_{N-1})$, $\psi_2\cong (d_0,\cdots d_{N-1})$.
As in section 2, we can construct a natural identification (or pairing) between
$\hwt$ and $\hzt$ when $w$ and $z$ are linearly independent.
We first introduce the equivalent of the right hand side of (2.6).
If $\psi_1\in\hwt$ and $\psi_2\in\hzt$, then $\bar{\psi_2}\psi_1$
can be interpreted as a distribution on the plane. Indeed, although the product of
distributions is not defined in general, it makes sense in this case because of the
particular form of $\psi_1$ and $\psi_2$: $\delta(h_w(x)-q_l(w, \theta))$ and
$\delta(h_z(x)-q_k(z, \theta))$ are supported on transversal lines, so that we have
no trouble defining their product. Clearly, $\bar{\psi_2}\psi_1$ is $\Z^2$-periodic
and, as such, passes to a distribution on $\T$. Hence $\int_{\T}
\bar{\psi_2}\psi_1\,\frac{dq\,dp}{2\pi\hbar}$ makes sense as the value of the
distribution
$\bar{\psi_2}\psi_1$ on the function $(2\pi\hbar)^{-1}$ on $\T$.
We then have, in analogy with Proposition 2.2:
\proclaim{Proposition 3.1}
Given $w,z\in\Z^2$ as above with $\Delta=\omega(w,z)>0$ and $\theta \in\T$, there
exist a unique vector space homomorphism
$P_{zw}(\theta, N): \hwt\to \hzt$,
such that $\forall \psi\in\hwt, \forall\varphi\in\hzt$
$$
\langle\varphi,P_{zw}(\theta, N)\psi\rangle_{\hzt}=\int_{\T}\bar
{\varphi}\psi\,
\frac{dq\,dp}{2\pi\hbar}.\tag{3.19}
$$
Moreover, using the identifications defined in $(3.16)$, the matrix
representation of $P_{zw}(\theta, N)$ is
$$P_{zw}(\theta, N)_{kr}=\frac{N}{\Delta}\,\sum_{p=0}^{\Delta-1}
\exp{[2\pi i\,\alpha_\theta (N,w)p]}\,
\exp{[-2\pi iN\,S_{zw}
(q_r(w,\theta)+p,q_k(z,\theta))]}.\tag{3.20}$$
\endproclaim
\demo{Proof} That $P_{zw}$ is defined as a vector space homomorphism
by (3.19) is clear. To prove the rest of the proposition, we compute the right hand side
of (3.19). Recall that this can be done by "integrating"
$\bar{\varphi}\psi$ over {\it any} fundamental domain of the torus. We start by
describing a suitable choice. Let $J=\left(\matrix 0 & 1\\ -1&0\endmatrix\right)$ and
define
$g_1=(1/\Delta)\, Jz,\quad g_2=-(1/\Delta)\, Jw$.
Then $g_1,g_2$ is a basis of $\R^2$ dual to $w,z$ since
$
h_w(g_1)=h_z(g_2)=1\,,\,\, h_w(g_2)=h_z(g_1)=0.
$
The unit cell of the dual lattice has volume $\Delta^{-1}$. Taking $L=(L_1,L_2)\in\R^2$,
define
$$
T(L)=\{x\in\R^2\,\vert x=\alpha g_1+\beta g_2,L_1\leq\alpha1$,
$g\in\Z$) then:
$$
V(A,\theta, N)_{\ell,k}=\frac{1}{\sqrt{N}}
\exp{[\frac{2\pi i}{N}\,(g\ell^2-\ell k+gk^2)\,]}\,.
$$
\endproclaim
\demo{Proof of Proposition $3.2$}
1) Let $v\in\Z^2$ with $\omega(w,v)=1$ and set $m=-Jw$, $\mt=Jv\,.$
We define, for $\teti\in\lbrack 0,1\lbrack\,\times\,\lbrack 0,1\lbrack$ ,
$$
S(\teti)=\sum_{\alpha,\beta\in\Z} (-1)^{N\alpha\beta}\, \exp{\lbrack\,-2\pi
i(\alpha\teti_1+\beta\teti_2)\,\rbrack}\, U(\alpha m+\beta\mt)\,.
$$
It is then easy to see that $S(\teti)$ is a continuous operator from ${\Cal S}(\R^2)$
to $\Di$ which extends uniquely to a map from $\Di$ to $\Di$. Moreover, a calculation
shows
$$
\align S(\teti)&\,=\,
\Bigl(\sum_{\alpha\in\Z} \exp[\,-2\pi i\teti_1\alpha\,]\,U(\alpha m)\Bigr)\,
\Bigl(\sum_{\beta\in\Z} \exp[\,-2\pi i\teti_2\beta\,]\,U(\beta\mt)\Bigr)\\
&{}\quad\quad\,=\,
\Bigl(\sum_{\beta\in\Z} \exp[\,-2\pi i\teti_2\beta\,]\,U(\beta\mt)\Bigr)\,
\Bigl(\sum_{\alpha\in\Z} \exp[\,-2\pi i\teti_1\alpha\,]\,U(\alpha m)\Bigr)
\endalign
$$
and
$$
U(\alpha' m+\beta '\mt)\,S(\teti)\,=\,(-1)^{N\alpha'\beta'}\,
\exp[\,2\pi i\,(\teti_1\alpha'+\teti_2\beta')\,]\,S(\teti)\,.
$$
Since
$$
\binom{\alpha'}{\beta '}=\left (\matrix v_1 & v_2\\ w_1 & w_2\endmatrix\right)
\binom{n_1}{n_2}
$$
for $\alpha' m+\beta '\mt=n$, we have
$$ u_1\,S(\teti)\,=\,(-1)^{Nv_1w_1}\,
\exp[\,2\pi i\,(v_1\teti_1+w_1\teti_2)\,]\,S(\teti)\,,
\tag{3.22}
$$
$$ u_2\,S(\teti)\,=\,(-1)^{Nv_2w_2}\,
\exp[\,2\pi i\,(v_2\teti_1+w_2\teti_2)\,]\,S(\teti)\,.
$$
As a result $S(\teti)\,\Di\subset \Df$, with
$$
\theta_1=(N/2) v_1w_1+\lbrack v_1\teti_1+w_1\teti_2\rbrack
\quad\quad\quad
\theta_2=(N/2) v_2w_2+\lbrack v_2\teti_1+w_2\teti_2\rbrack
\tag{3.23}
$$
For $\psi \in {\Cal D}_w$ as in (2.4), a simple computation using the
Poisson formula yields
$$
\lbrack \,S(\teti)\psi\,\rbrack(x)=\exp[\,-i\pi N\, h_w(x)h_v(x)\,]
\,\sum_{k\in\Z} d_k(\theta)\,
\delta[\,h_w(x)-(k+\teti_1)/N\,]
\tag{3.24}
$$
where
$$ d_k(\theta)=\frac{1}{N}\,\sum_{\beta\in\Z} \exp[\,-2\pi i\,\beta\teti_2\,]\,
f((k+\teti_1)/N-\beta)\,.\tag{3.25a}
$$
Note that
$$ d_{k+N}(\theta)=\exp[\,-2\pi i\,\teti_2\,]\, d_k\,.\tag{3.25b}$$
Using (3.23), one establishes
$$
(k +\teti_1)/N=\qkw-(1/2)\,w_1w_2\,[\,v_2 - v_1-1\,],\tag{3.26}
$$
with $\qkw$ as in (3.4), and
$$
\teti_2=-\alpha_{\theta}(N,w)-(N/2)\, v_1v_2\,[\,w_1 - w_2-1\,],
\tag{3.27}
$$
with $\alpha_{\theta}(N,w)$ defined in (3.6). Note that the relation
$w_1v_2-w_2v_1=1$ implies that the last term in (3.26) and in (3.27) is an
integer. Hence (3.24) becomes
$$
\lbrack \,S(\teti)\psi\,\rbrack(x)=\exp[\,-i\pi N\, h_w(x)h_v(x)\,]\,
\sum_{k\in\Z} c_k(\theta)\,\delta[h_w(x)-\qkw]
\tag{3.28}
$$
with $ c_k(\theta)=d_{k+\frac{N}{2}w_1w_2[v_2-v_1-1]}(\theta)$ satisfying
(3.5), thanks to (3.25) and (3.27). Hence (3.28) is written in the form (3.3)
which shows
$ S(\teti)\psi\in\dwt\,.
$
Recall now from (3.16) and (3.18) that $\hwt\cong \C^N$. As a result
$$
\align
N^2\,\int_0^1\int_0^1d^2\theta\,\hwt&\cong N^2\,\int_0^1\int_0^1
\C^N\,d^2\theta\\
&\cong \,L^2(\,[0,1[\,\times\,
[0,1[\,;\,\C^N,\,N^2 d^2\theta).
\endalign
$$
On the other hand, if $f\in {\Cal S}(\R)\subset L^2(\R,dy)$, then, using (3.25), and
performing a change of variables in the integral, using (3.23), yields
$$ \align
N^2\,\int_0^1 d\theta_1\int_0^1 d\theta_2&\,\sum_{r=0}^{N-1}
\vert c_r(\theta)\vert^2=\\
&N^2\,\int_0^1 d\teti_1\int_0^1 d\teti_2\,\sum_{r=0}^{N-1}
\vert c_r(\theta)\vert^2=\int_{\R} \vert f(y)\vert^2\, dy.
\endalign
$$
Hence the map
$$
f\mapsto (c_0(\theta),\cdots,c_{N-1}(\theta))
\in L^2(\,[0,1[\,\times \,[0,1[\,; \C^N,\,
N^2 d^2\theta)
$$
extends to a natural isometry on all of $L^2(\R,dy)$. It is easily seen to be onto
and hence unitary.
Since $L^2(\R,dy)\cong\hw$, this proves (1).
\newline 2) $\forall \psi_1\in\hw$, $\psi_2\in\hz$, we have
$$
\align \langle\psi_2, P_{zw}U(a,b)\psi_1\rangle_z
& =\int\frac{dq\,dp}{2\pi\hbar}\,\,
\overline{\psi_2}\,U(a,b)\psi_1 \\
& = \int\frac{dq\,dp}{2\pi\hbar}\,\,
\overline{U(a,b)^*\psi_2}\,\psi_1 =\,\langle U(a,b)^*\psi_2, P_{zw}\psi_1
\rangle_z.
\endalign
$$
3) We know from the remark after
Proposition 2.2 that $P_{zw}$ extends from $\hw$ to a map from $\dw$ to $\dz$.
Moreover, in view of (2), $P_{zw} \dwt\subset\dzt$. Hence, defining
${\tilde P}_{zw}(\theta, N)$ to be the restriction of $P_{zw}$ to $\dwt$, it follows that
${\tilde P}_{zw}(\theta, N):\hwt\to\hzt$ and consequently that
$P_{zw}= N^2\,\int d^2\theta\,
{\tilde P}_{zw}(\theta, N)$. By computing the
explicit formula for ${\tilde P}_{zw}(\theta,N)$, we will show
$N\,{\tilde P}_{zw}(\theta,N)= P_{zw}(\theta, N)$, establishing the proposition.
Taking $\psi\in\dwt$ in the form (3.3), and using
the definition of $P_{zw}$ in Proposition 2.2, we have
$$
\align [\,&{\tilde P}_{zw}(\theta,N)\,\psi\,](x_2)=\\
&(1/2\pi\hbar\Delta)\,\exp[\,-i\pi N\,h_z(x_2)h_u(x_2)\,]\,
\sum_{k\in\Z}c_k \,\exp[\,-2\pi iN\,S_{zw}(q_k(w,\theta),h_z(x_2))\,]\,,
\tag{3.29}
\endalign
$$
where $S_{zw}$ is given in (2.7).
Letting $k=\ell\Delta N+r$, $\ell\in\Z$ and $r\in\{0,\cdots,\Delta N-1\}\,,$
the {\it r.h.s.} of (3.29) reads
$$
\align
\exp[\,-i\pi N\,&h_z(x_2)h_u(x_2)\,]\,\cdot\\
\frac{N}{\Delta}\,&\sum_{\ell\in\Z}\sum_{r=0}^{N\Delta-1} c_r\,
\exp[\,2\pi i\,\ell\Delta\alpha_{\theta}(N,w)\,]\,\exp[\,-2\pi i
N\,S_{zw}(q_r+\ell\Delta,h_z(x_2))\,]\,.
\endalign
$$
Since $(1/2)\,\omega(v,z)\,\ell^2\Delta=(1/2)\,\omega(v,z)\,\ell\Delta\,\,
\mod 1\,,$ we have
$$
S_{zw}(q_r+\ell\Delta,h_z(x_2))
=S_{zw}(q_r,h_z(x_2))+\ell\,[q_r\omega(v,z)+h_z(x_2)]+(1/2)\omega(v,z)
\ell^2\Delta\,,
$$
so that
$$
\align
\exp[\,&-2\pi iN\,S_{zw}(q_r+\ell\Delta,h_z(x_2))\,]=\\
{}&\exp[\,-2\pi i N\,S_{zw}(q_r,h_z(x_2))\,]\,
\exp[\,-2\pi i N\,(q_r\omega(v,z)+h_z(x_2)+\omega(v,z)\Delta/2)\ell\,]\,.
\endalign
$$
This yields:
$$
\align
[\,&{\tilde P}_{zw}(\theta,N)\psi\,](x_2)=
\exp[\,-i\pi N\,h_z(x_2)h_u(x_2)\,]\,\cdot\\
{} & \Bigl(\frac{N}{\Delta}\,\sum_{r=0}^{N\Delta-1}c_r\,\exp[\,-2\pi i
N\,S_{zw}(q_r,h_z(x_2))\,]\Bigr)\,\Bigl(\sum_{\ell\in\Z}
\exp[\,-2\pi iN\,(h_z(x_2)-A)\ell\,]\Bigr)\,,
\endalign
$$
where
$$ A=-q_r(w,\theta)\,\omega(v,z)-(1/2)\,\omega(v,z)\Delta+(\Delta/N)\,
\alpha_{\theta}(N,w),
$$ and where $q_r(w,\theta)$ and $\alpha_{\theta}(N,w)$ are given by (3.4)
and (3.6) respectively. Note that we can replace $A$ by anything else mod
$1$, a freedom we will use in order to get a convenient form for
${\tilde P}_{zw}(\theta,N)$. In particular one has
$$
\align
A&=-(r/N)\,\omega(v,z)+(1/2)\,z_1z_2-(1/N)\,\omega(\theta,z)\quad\mod 1\\
&=q_{[-r\omega(v,z)]}(z,\theta)\quad\mod 1\,,
\endalign
$$
so that
$$
\align [&{\tilde P}_{zw}(\theta,N)\,\psi](x_2)=
\exp[\,-i\pi N\,h_z(x_2)h_u(x_2)\,]\cdot\\
&{}\quad\quad\Bigl(\frac{N}{\Delta}\,\sum_{r=0}^{N\Delta-1}c_r\,
\exp[\,-2\pi iN\,S_{zw}(q_r,h_z(x_2))\,]\Bigr)\,
\Bigl((1/N)\,\sum_{k\in\Z}\delta
[y_2-q_{k-r\omega(v,z)}]\Bigr)\\
&=\exp[\,-i\pi N\,h_z(x_2)h_u(x_2)\,]\\
&{}\frac{1}{\Delta}\quad\sum_{r=0}^{N\Delta-1}c_r\,\sum_{k\in\Z}
\exp[\,-2\pi iN\,S_{zw}(q_r(w,\theta),
q_{k+r\omega(z,v)}(z,\theta))\,]
\,\delta[\,y_2-q_{k+r\omega(z,v)}(z,\theta)\,]\\
&=\exp[\,-i\pi N\,h_z(x_2)h_u(x_2)\,]\\
&{}\frac{1}{\Delta}\quad\sum_{r=0}^{N\Delta-1}c_r\,\sum_{\ell\in\Z}
\exp[\,-2\pi iN\,S_{zw}(q_r(w,\theta),
q_{\ell}(z,\theta))\,]
\,\delta[\,h_z(x_2)-q_\ell(z,\theta)\,]\,.
\endalign
$$
Using (3.5) and comparing to (3.20), one sees that
$\tilde P_{zw}(\theta, N)= (1/N)\,
P_{zw}(\theta, N)$. Hence $P_{zw}= N\int d^2\theta\, P_{zw}(\theta, N)$.\qed
\enddemo
To summarize, by applying the ideas of geometric quantization in their simplest
form, one can easily quantize linear transformations on $\R^2$ as well as on
$\T$. We stress again that the construction is simple and calculationally
very convenient. Indeed, although the proofs of Propositions 3.1 and 3.2 are
somewhat involved in the general case, they reduce to trivialities when
$\Delta =1$, as in Corollary 3.1 and in the following sections. In that case (3.20)
does not involve a sum and the unitarity of $P_{zw}$ is then immediate.
We shall now show that the reformulation of geometric quantization we have just
presented allows for an immediate generalization to a class of piecewise linear or
affine linear transformations of the torus.
\bigskip
\bigskip
\subheading{4. Quantization of piecewise linear and affine transformations}\par
\bigskip
\noindent {\sl $($A$)$ Translations and skew translations.}
\medskip
The simplest transformations on the torus are undoubtedly the translations
$x=(q, p)\mapsto(q+a, p+b)\,\mod 1\,$. If $a={r_1}/{s_1}$ and $b={r_2}/{s_2}$
(with $g.c.d.$ $(r_i, s_i)=1$, $i=1,2\,$), then we can write
$(a,b)=(r/s)\,(-w_2,w_1)$ for integer $r,s$ with $g.c.d.$ $(r, s)=1$,
$w\in\Z^2$, and $g.c.d.$ $(w_1, w_2)=1$. Here $s$ is the least common
multiple of $s_1$ and $s_2$,
which is also the common period of all orbits under this translation.
Taking $k\in\N^*$, $N=sk$, we saw in Section 3 (see (3.17)) how to quantize
this translation. The expression of the
quantum translation $U(a,b)$ (i.e. (3.17) with $\ell=rk$) shows that its eigenfunctions
are concentrated on the circles
$$\omega(x,(a,b))=(r/s)\,q_i\qquad i=0,\dots, ks-1$$ and that they are $k$-fold
degenerate. The quantum propagator is easily seen to have the same
period as the classical translation since
$$U^s(a,b) = \exp [\,2\pi i\,(-\frac{w_1w_2}{2}\,\ell s
+ r\omega(w,\theta))\,]\,\, id_{\hwt}.$$
It follows that, as in the multidimensional harmonic oscillator with
commensurate frequencies \cite{DBIH}, these degeneracies can be used to
construct eigenfunctions of $U(a,b)$ that, in the
classical limit ($k\to\infty$), concentrate on any given classical orbit.
The approach of Section 3 does not a priori permit the quantization of
translations of the form $(a,b)=\alpha\,({r_1}/{s_1},{r_2}/{s_2}),\,\,
\alpha\notin\Q$, much less of ergodic translations, for
which $a/b\notin\Q$. The reason is that the corresponding prequantized translations
do not preserve the spaces
$\hwt$.
Since the ergodic translations are undoubtedly the simplest ergodic
dynamical systems, it would be interesting to circumvent this difficulty and
to nevertheless construct a quantum analog for them. We will see that
this can be done very naturally within the framework of Section 3.
The situation is actually very similar to the one encountered when quantizing
linear flows. Indeed, there we saw that
$U(A)\hwt={\Cal H}_{{}\,^T\!\!A^{-1}w}(\theta, N)$
for a suitable choice of $\theta$ and then we used the natural pairing between
${\Cal H}_{{}\,^T\!\!A^{-1}w}(\theta, N)$ and $\hwt$ to construct $V(A)$. Here
we will see that $U(a,b)\hwt=\hwtp$ with $\theta'$ given in Lemma 4.1 below. Although
in this case
we can never choose $\theta$ so that $\theta'=\theta$, we will construct an
identification
$D_{\hbar} P_{vw}(\theta,\theta')$ between $\hwtp$ and $\hvt$ in analogy with (3.19).
Since there is also a natural identification
$D_{\hbar}P_{wv} (\theta)$ between $\hvt$ and $\hwt$ (Proposition 3.1), we define
the unitary quantum translation $Q_w(a,b)$ by $$
Q_w(a,b)=D_{\hbar}^2 \,P_{wv} (\theta,\theta)\circ P_{vw}(\theta,\theta')\circ
U(a,b)\,:\,\hwt\to\hwt\,.\tag{4.1}
$$
Note that this reduces to (3.17) when the translation has the required form, and that
the $Q_w(a,b)$ depend continuously on $(a,b)$. On the other hand, the construction is
$w$-dependent and it is clear that the $Q_w(a,b)$ can not generate a unitary
representation of the full Weyl-Heisenberg group.
\proclaim{Lemma 4.1}
\roster
\item $U(a,b)\,\Df=\Dfp\,$, with
$
(\theta_1',\theta_2')=(\theta_1 -Nb,\theta_2 +Na)\quad \mod 1\,.
$
\item $U(a,b)\nabla_w \psi=\nabla_w U(a,b)\psi\,$
for any $w\in\R^2$, $(a,b)\in\R^2$, $\psi\in\Di\,$.
\item $U(a,b):\hwt\to\hwtp$ is unitary.
\endroster
\endproclaim
\demo{Proof}
Both (1) and (2) follow from a simple computation. That $U(a,b)$ maps
$\hwt$ isomorphically onto $\hwtp$ is an immediate consequence of (1) and (2).
To check the unitarity, let $\psi\in\hwt$ with
$$
\psi(q,p)=\sum_{k\in\Z} c_k\,\exp[\,{-i\pi N\, h_w(x) h_v(x)}\,]\,
\delta\bigl(h_w(x)-q_k(\theta,w)\bigr)\,.
\tag{4.2}
$$
For convenience, we write $\tau=(\tau_1,\tau_2)=(a,b)$. Now we introduce
$\ell=(\ell_1,\ell_2)\in\Z^2\,$, $I_{1/N}=\,]-{1\over N},{1\over N}\,[\,$
and $\beta=(\beta_1,\beta_2)\in\,I_{1/N}^2\,$, uniquely determined by
$$
\align
\tau_i&={\ell_i}/{N}+\beta_i\\
\theta'_i&=\theta_i+(-)^i\,N\,\beta_{3-i}\,\in\,[\,0,1\,[\,,
\endalign
$$
with $i=1,2$. A direct calculation shows that
$$
[\,U(a,b)\psi\,](q,p)=\sum_{k\in\Z} d_k\,\exp[\,{-i\pi N\,h_w(x) h_v(x)}\,]
\,\delta\bigl(h_w(x)-q_k(\theta',w)\bigr)\,,
$$
where
$$
d_k=c_{k-h_w(\ell)}\,
\exp[\,{i\pi N\,\bigl(2 q_k(\theta',w)h_v(\tau)-h_w(\tau)h_v(\tau)\bigr)}\,]\,
\tag{4.3}
$$
Recalling the identification $\psi\cong (c_0,\cdots,c_{N-1})$ and
$[U(a,b)\psi]\cong (d_0,\cdots,d_{N-1})$, the unitarity of $U(a,b)$ is
now immediate from (4.2).\qed
\enddemo
Given now $U(a,b)\psi\cong (d_0,\cdots,d_{N-1})\in {\Cal H}_w(\theta ',N)$, we can
proceed in the spirit of Proposition 3.1 to define $P_{vw}(\theta,\theta'):
{\Cal H}_w(\theta', N)\to\hvt$ as follows:
$$\langle\psi_2, P_{vw}(\theta, \theta')\psi_1\rangle_{\hvt}=
\int_{[0,1)\times [0,1)}\,{\overline\psi_2}\psi_1\,
{dq\,dp\over2\pi\hbar}.\tag{4.4}$$
A simple calculation then yields
$$[P_{vw}(\theta,\theta')\psi](q,p)=N\,\sum_{\ell}[P_{vw}(\theta,\theta')\psi]_{\ell}\,
\exp{[i\pi Nh_w(x)h_v(x)]}\,\delta(h_v(x)-q_{\ell}(v,\theta))
$$
where
$$
\align
[P_{vw}(\theta,\theta')\psi]_{\ell}&=N\, \sum_{k=0}^{N-1} d_k\,\exp{[-2i\pi N S_{vw}
(q_k(w,\theta'),q_{\ell}(v,\theta))]}\\
&=N\, \sum_{k=0}^{N-1}d_k\,\exp{[-2i\pi N q_k(w,\theta')q_{\ell}(v,\theta)]}
\endalign
$$
It is easy to see that $\parallel D_{\hbar}P_{vw}(\theta,\theta')
U(a,b)\psi\parallel_{{\Cal H}_v (\theta,N)}^2=\parallel \psi\parallel_{{\Cal H}_w(\theta
',N)}^2$, where $\vert D_{\hbar}\vert =N^{-3/2}$.
When $(a,b)$ is ergodic, the eigenfunctions of the $Q_w(a,b)$ can on general
grounds be expected to be equidistributed on the torus in the classical limit,
in sharp contrast to what happens in the periodic case.
Note that it is now easy to quantize skew translations of the form
$(q,p)\mapsto(q+a, p+kq)$ which are
ergodic if $a$ is irrational and $k$ a non-zero integer \cite{CFS}. They are just the
composition of a linear transformation and a translation.
\bigskip
\noindent {\sl $($B$)$ Piecewise affine transformations.}
\medskip
A first class of piecewise affine maps studied in \cite{Ch} is the following.
Take $A=\left(\matrix a & b\\ c & d \endmatrix\right)\in SL(2,\Z)$, apply it to
$[ 0,1)\times [ 0,1)$, then cut the resulting parallelogram into
strips along the direction $(a,c)$ and shift the strips around with translations
parallel to $(a,c)$. Combining Section 3 and Section 4A, one can easily obtain a
quantization for this class of transformations.
%The maps
%$A=\left(\matrix 1 & b\\ b & 1+b^2 \endmatrix\right)$, $b\in\R$.}
\medskip
Let us now turn to another class of discontinuous maps described in \cite{Ch,LW,V}.
Consider the map $A_1=\left(\matrix 1 & b\\ 0 & 1\endmatrix\right)$, $b\in\R$
restricted to the strip $0\leq p\leq 1$ and taken modulo $1$ in $q$. This defines a map $A_1$
on the torus, discontinuous on the circle $\{p\in\Z\}$ if $b\notin\Z$.
Similarly, construct a map $A_2$ on the torus by restricting $A_2=\left(\matrix 1 & 0\\ b &
1\endmatrix\right)$, $b\in\R$ to the strip $0\leq q\leq 1$ and taking $p$ modulo $1$.
This map will be discontinuous on the circle $\{q\in\Z\}$ if $b\notin\Z$.
The map $A=A_2A_1$, which is a discontinuous
hyperbolic area preserving map on the torus, is ergodic and exponentially
mixing, \cite{Ch,Li,LW,V}.
We now propose a quantization of $A_i$, $i=1,2$ in the spirit of Section 3.
Calling $V_i$ the quantization of $A_i$, we will define the quantum propagator $V$ of
$A$ by $
V=V_2V_1$.
We saw in Section 2 that $U(A_1)\dw={\Cal D}_{(A_1^{-1})^T w}$. If, however, $a\notin\Z$
then $U(A_1)\Df\not\subset\Dfp$ for any choice of $\theta$ and $\theta '$.
This situation is similar to, but slightly more complicated than, the one of the
previous paragraph, where $U(a,b)\dw=\dw$, but $U(a,b)\Df=\Dfp$.
So there is again no geometrically natural definition of the quantum propagator
associated to $A_1$.
This reflects the fact that $A_1$ is not a continuous automorphism of the torus.
The approach of Section 3 nevertheless suggests an obvious way to quantize $A_1$.
For that purpose, note that the image of $\lbrack 0,1)\times\lbrack 0,1)$ under
$A_1$ is
$$
F_1=\{ (q,p)\in\R^2\,\vert\,0\leq p<1,\,\,bp\leq qFrom equation ($4.6$) one sees that its eigenfunctions are indeed concentrated on the
invariant circles.
Finally, the construction of $V_2$ is completely analogous, with the roles of $w$
and $v$ interchanged.
The resulting quantum propagator $V=V_2V_1$ on $\hwt$ is readily seen to be
$$
V=D_2\circ{\Cal F}_N^{-1}\circ D_1\circ {\Cal F}_N.\tag{4.7}
$$
Here $D_2$ is the diagonal matrix with entries $\exp{[i\pi N b q_{\ell}^2]}$.
The non trivial structure of $V$ comes from the fact that it is the product
of two non commuting matrices $V_1, V_2$.
\bigskip
\noindent {\sl $($C$)$ The Baker transformation.}
\medskip
Given the matrix $A=\left(\matrix 2 & 0\\ 0 & 1/2\endmatrix \right)$, we consider the
piecewise affine map $B$ defined on the unit square ($0\leq q < 1, 0\leq p\leq 1)$ by
$$
B(q,p)=\cases
Ax\,,\phantom{T(-1,1/2)\circ A}\quad 0\leq q< 1/2\,,\\
T(-1,1/2)\circ A\,,\phantom{Ax}\quad 1/2\leq q<1\,, \endcases
$$
where $T(a,b)x=(q+a,p+b)$.
This map is called {\it the Baker transformation},
and its dynamical properties have been studied in detail (see \cite{AA,LW}).
Note that it has the same structure as the piecewise affine maps described
above. First one applies a linear map, then one slices the resulting rectangle
and shift the parts around. There is one major difference, however, leading to some
additional complications for the quantization. The linear part of the Baker transformation
does not send $\lbrack 0,1)\times\lbrack 0,1)$ into another fundamental domain of $\T$.
Even though the Baker transformation is not continuous on the torus, the tools we developed
in the previous section
can again be used to associate a corresponding quantum
operator to this map, as we now show. In particular, as in \cite{BV, Sa},
we take the point of view that the correct quantum Hilbert spaces for this problem are
still the ones constructed in Section 3 (see below).
It then suffices to mimic the approach of the previous section, with some
minor changes to account for the discontinuities of the map.
The resulting quantum operator is identical to the one obtained in \cite{BV,Sa} by
completely different arguments.
We shall first define a prequantized version ${\QB}$ of $B$ on distributions on $\R^2$
with support in the left or right half of the unit square.
Suppose $\psi$ is a distribution supported on $0\leq q<{1\over 2}\,$,
${}\,\,0\leq p\leq 1\,$. Then we define
$$
(\QB\psi)(q,p)=U(A)\,\psi(q,p)\,.
$$
Note that the support of $\QB\psi$ is contained in $0\leq q<1\,$,
${}\,\,0\leq p\leq {1\over 2}\,$. If, on the other hand, $\psi$ is supported in
${1\over 2}\leq q<1\,$, ${}\,0\leq p\leq 1$, then
$$
(\QB\psi)(q,p)=[\,U(-1,1/2)\circ U(A)\,]\psi(q,p)
$$
and its support is now contained in $0\leq q<1\,$, ${}\,\,{1\over 2}\leq p\leq 1$.
Given $N\in \N$, and $w=(1,0)$, recall that $\dwt$ is the space of distributions $\psi$
of the form:
$$
\psi(q,p)=\sum_{k\in\Z}c_k\,\exp[\,{-i\pi N\,pq}\,]\,\delta(q-q_k)\,,
$$
where $q_k=k/N +\theta_2/N $ and, in addition,
$c_{k+N}=e^{-2\pi i\,\theta_1}\,c_k$ for any $k\in\Z\,.$ Therefore, because of
the latter relations, no information is lost if we restrict $\psi$ to the
unit square, namely
$$
\psi(q,p)=\sum_{k=0}^{N-1} c_k\, \exp[\,{-i\pi N\,pq}\,]\,
\delta(q-k/N-\theta_2/N) \,\,\chi_{[0,1]}(p)\,,\tag{4.8}
$$
where $\chi_{[0,1]}$ is the characteristic function of the unit interval. We shall
write $H_1(\theta)$ for the space of distributions of the form (4.8), equipped with the
inner product (3.18). This is the quantum Hilbert space for the Baker map in the {\sl
position representation}, which is realized as a space of distributions on the phase
space of the problem. Similarly, we introduce $H_2(\theta)$, which is the space of
distributions ${\Cal D}_v(\theta, N)$ with $v=(0,1)$, restricted to the unit square, i.e.,
$\psi\in
H_2(\theta)$ iff
$$\psi = \sum_{\ell=0}^{N-1} d_l\exp{[i\pi Npq]}\,\delta(p-p_l)\,\chi_{[0,1]}(q),$$
where $p_\ell =\ell/N + \theta_1/N$. $H_2(\theta)$ is the quantum
Hilbert space of the Baker transformation in the {\sl momentum} representation. We have
a natural identification between $H_1(\theta)$ and $H_2(\theta)$, given by the
pairing of section 3, which in this case is just the finite Fourier transform
(see the remark after Proposition 3.1).
We now observe that we have a natural decomposition $H_1(\theta)=H_L(\theta)
\bigoplus H_R(\theta)$.
Indeed, each $\psi\in H_1(\theta)$ can
be uniquely written as $\psi=\psi_L + \psi_R$, where
$$
\align
\psi_L&=\sum_{0\leq q_k < {1\over 2}} c_k\,\exp{[\,-i\pi N\,pq\,]}\,\delta(q-q_k) \,\,
\chi_{[0,1]}(p)\,,\\
\psi_R&=\sum_{{1\over 2}\leq q_k<1} c_k\,\exp{[\,-i\pi N\,pq\,]}\,\delta(q-q_k) \,\,
\chi_{[0,1]}(p)\,,\\
\endalign
$$
have their respective supports in $0\leq q<{1\over 2}$, and
${1\over 2}\leq q<1$. We can now compute
$$
(\QB\psi)(q,p)=(\QB\psi_L)(q,p)+(\QB\psi_R)(q,p)\,.
$$
This gives
$$
\align
(\QB\psi_L)(q,p)&=2\sum_{0\leq q_k <{1\over 2}} c_k\,\exp[\,{-2\pi iN\,q_kp}\,]\,
\delta(q - 2q_k)\,\chi_{[0,1]}(2p)\,,\\
(\QB\psi_R)(q,p)&=2\, \exp[\,{2\pi i\, (\theta_2-N/4)\,]}\,
\sum_{{1\over 2}\leq q_k < 1}\,c_k\,\exp[\,{-2\pi iN\,(q_k-1/2)\,p}\,]\,\cdot\\
{}&\phantom{2\, \exp[\,{2\pi i\, (\theta_2-\frac{N}{4})\,]}\,
\sum_{k=N/2}^{N-1}\,c_k\,\quad\quad} \delta(q+1-2q_k)\,\,\chi_{[0,1]}(2p-1)\,.
\endalign
$$
Note that the support of $\QB\psi_L$ is contained in $0\leq q<1,\,
0\leq p\leq {1 \over 2}$, whereas the support of $\QB\psi_R$ is contained in
$0\leq q<1,\,{1\over 2}\leq p\leq 1$. It is clear that $\QB\psi$ obtained in this way is
not an element of $H_1(\theta)$ (for
any $\theta$), nor of
any $\dzt$. Hence, we have no hope of applying the general results on pairing of
the previous section directly to define the quantum propagator. It will nevertheless
turn out that we can again define, in the spirit of (3.19), a natural projection $P\hat
B\psi$ of the
distribution $\hat B\psi$ onto $H_2(\theta)$.
\proclaim{Proposition 4.1} If $N$ is even and $0<\theta_1,\,\theta_2<1$, then
there exists a
unitary map $2^{-1/2}N^{-3/2}\,P\QB:H_1(\theta)
\to H_2(\theta)$, uniquely defined by:
$$\langle\psi_2,P\hat B\psi_1\rangle_{H_2(\theta)}=\int_{[0,1)\times [0,1)}
\overline\psi_2 \hat B\psi_1 \,
{dq\,dp\over 2\pi\hbar}.\tag{4.9}$$
Specifically,
$$
P(\QB\psi)(q,p)=\sum_{\ell=0}^{N-1} \bigl(P(\QB\psi)\bigr)_\ell\,
\exp[\,{i\pi N\, qp}\,]\,\delta(p-p_\ell)\,\chi_{[0,1]}(q),
$$
with, for $\ell