% paper 2 by Z.Parsa, M.Zeitlin, A. Fedorova
% PAC97, Vancouver Canada 1997
%********************************************
\documentclass[a4paper]{pac97}
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\begin{document}
\title{ WAVELET APPROACH TO ACCELERATOR  PROBLEMS, II.\\
METAPLECTIC WAVELETS}
\author{A.~Fedorova and  M.~Zeitlin,
Institute of Problems of Mechanical Engineering,\\
 Russian Academy of Sciences, Russia, 199178, St.~Petersburg,\\
  V.O., Bolshoj pr., 61, e-mail: zeitlin@math.ipme.ru\\
Z.~Parsa, Dept. of Physics, Bldg.~901A, Brookhaven National Laboratory,\\
 Upton, NY 11973-5000, USA, e-mail: parsa@bnl.gov}
\maketitle
\begin{abstract}
This is the second part of a series of talks in which we 
present applications of wavelet 
analysis to polynomial approximations for a number of 
accelerator physics problems.
According to the orbit method and by using construction
from the geometric quantization theory we construct the
symplectic  and Poisson structures associated with generalized 
wavelets by using metaplectic structure and
 corresponding polarization. The key point is a consideration
of semidirect product of Heisenberg group and metaplectic group
as subgroup of automorphisms group of dual to symplectic space,
which consists of elements acting by affine transformations.
\end{abstract}
\section {Introduction}
In this paper we continue the application of powerful methods of wavelet 
analysis to polynomial approximations of nonlinear accelerator 
physics problems. In part 1 we considered our main example 
and general approach 
for constructing wavelet representation for orbital motion in
storage rings. But now we need take into account the Hamiltonian
or symplectic structure related with system (1) from part 1.
Therefore, we need to consider instead of compactly supported 
wavelet representation from part 1 the generalized wavelets,
which allow us to consider the corresponding symplectic structures. 
By using the orbit method and  constructions
from the geometric quantization theory we consider the
symplectic  and Poisson structures associated with Weyl--\-
 Heisenberg wavelets by using metaplectic structure and
the corresponding polarization. In part 3 we consider  applications
 to construction of Melnikov functions in the theory of
 homoclinic chaos in perturbed Hamiltonian systems.

In wavelet analysis the following three concepts are used now:
1).\ a square integrable representation $U$ of a group $G$, 
2).\ coherent states over G  
3).\ the wavelet transform associated to U.

We have three important particular cases:\\
a) the affine $(ax+b)$ group, which yields the usual wavelet
analysis
$$
[\pi(b,a)f](x)=\frac{1}{\sqrt{a}}f\left(\frac{x-b}{a}\right)
$$
b). the Weyl-Heisenberg  group which leads to the Gabor
functions, i.e. coherent states associated with windowed Fourier
transform.
$$
[\pi(q,p,\varphi)f](x)=\exp(i\mu(\varphi-p(x-q))f(x-q)
$$
In both cases time-frequency plane corresponds to the phase
space of group representation.\\
c). also, we have
 the case of bigger group, containing
both affine and We\-yl-\-Hei\-sen\-berg group, which interpolate between
affine wavelet analysis and windowed Fourier analysis: affine
Weyl--Heisenberg group [7]. But usual representation of it is not
square--integrable and must be modified: restriction of the
representation to a suitable quotient space of the group (the
associated phase space in that case) restores square --
integrability:
 $G_{aWH\longrightarrow}$ homogeneous space.
Also, we have more general approach which allows to consider wavelets 
corresponding to more general groups and representations [8], [9].
Our goal is applications of these results to problems of 
Hamiltonian dynamics and as consequence we need to take into account 
symplectic nature of our dynamical problem. 
 Also, the symplectic and wavelet structures
 must be consistent (this must
be resemble the symplectic or Lie-Poisson integrator theory).
 We use the
point of view of geometric quantization theory (orbit method)
instead of harmonic analysis. Because of this we can consider
(a) -- (c) analogously.
\section{METAPLECTIC GROUP AND RE\-PRE\-SEN\-TA\-TI\-ONS}
Let $Sp(n)$ be
symplectic group, $Mp(n)$ be its unique two- fold covering --
metaplectic group.   Let V be a symplectic vector space
with symplectic form ( , ), then $R\oplus V$ is nilpotent Lie
algebra - Heisenberg algebra:
$$[R,V]=0, \quad [v,w]=(v,w)\in
R,\quad [V,V]=R.$$
$Sp(V)$ is a group of automorphisms of
Heisenberg algebra.

 Let N be a group with Lie algebra $R\oplus
V$, i.e.  Heisenberg group.  By Stone-- von Neumann theorem
Heisenberg group has unique irreducible unitary representation
in which $1\mapsto i$. This representation is projective:
$U_{g_1}U_{g_2}=c(g_1,g_2)\cdot U_{g_1g_2}$, where c is a map:
$Sp(V)\times Sp(V)\rightarrow S^1$, i.e. c  is $S^1$-cocycle.

But this representation is unitary representation of universal
covering, i.e. metaplectic group $Mp(V)$. We give this
representation without Stone-von Neumann theorem.\
Consider a new group $F=N'\bowtie Mp(V),\quad \bowtie$ is semidirect
product (we consider instead of  $ N=R\oplus V$ the $
N'=S^1\times V, \quad S^1=(R/2\pi Z)$). Let $V^*$ be dual to V,
$G(V^*)$ be automorphism group of $V^*$.Then F is subgroup of $
G(V^*)$, which consists of elements, which acts on $V^*$ by affine
transformations. \\
This is the key point!

 Let $q_1,...,q_n;p_1,...,p_n$ be symplectic basis in V,
$\alpha=pdq=\sum p_{i}dq_i $  and $d\alpha$ be symplectic form on
$V^*$. Let M be fixed affine polarization, then for $a\in F$ the
map $a\mapsto \Theta_a$ gives unitary representation of G:
$ \Theta_a: H(M) \rightarrow H(M) $

Explicitly  we have for representation of N on H(M):
$$
(\Theta_qf)^*(x)=e^{-iqx}f(x),  \quad
 \Theta_{p}f(x)=f(x-p)
$$
The representation of N on H(M) is irreducible. Let $A_q,A_p$
be infinitesimal operators of this representation
$$
 A_q=\lim_{t\rightarrow 0} \frac{1}{t}[\Theta_{-tq}-I], \quad
 A_p=\lim_{t\rightarrow 0} \frac{1}{t}[\Theta_{-tp}-I],
$$
then
$$
A_q f(x)=i(qx)f(x),\quad A_p f(x)=\sum p_j\frac{\partial
f}{\partial x_j}(x)
$$
Now we give the representation of infinitesimal ba\-sic
elements. Lie algebra of the group F is the algebra of all
(non\-ho\-mo\-ge\-ne\-ous) quadratic po\-ly\-no\-mi\-als of (p,q) relatively
Poisson bracket (PB). The basis of this algebra consists of
elements
$1,q_1,...,q_n$,\  $p_1,...,p_n$,\ $ q_i q_j, q_i p_j$,\ $p_i p_j,
 \quad i,j=1,...,n,\quad i\leq j$,
 \begin{eqnarray*} & &PB \ is
\quad \{ f,g\}=\sum\frac{\partial f}{\partial p_j}
\frac{\partial g}{\partial q_i}-\frac{\partial f}{\partial q_i}
\frac{\partial g}{\ \partial p_i} \\
& &\mbox{and}  \quad
   \{1,g \}= 0 \quad for \mbox{ all} \ g,\\
& &\{ p_i,q_j\}= \delta_{ij},\\ & & \{p_i
q_j,q_k\}=\delta_{ik}q_j,
    \{p_i q_j,p_k\}=-\delta_{jk}p_i, \\ 
& & \{p_i p_j,q_k \}=
\delta_{ik}p_j+\delta_{jk}p_i,
 \{p_i p_j,p_k\}=0,\\ 
 & & \{ q_i q_j,q_k\}=0,
 \{q_i q_j,p_k\}=-\delta_{ik}q_j-\delta{jk}q_i
 \end{eqnarray*}
so, we have the representation of basic elements
\\
 $ f\mapsto A_f : 1\mapsto i, q_k\mapsto ix_k $,
\begin{eqnarray*}
& & p_l\mapsto\frac{\delta}{\delta x^l}, p_i q_j\mapsto
x^i\frac{\partial}{\partial x^j}+\frac{1}{2}\delta_{ij},\\
& &p_k p_l\mapsto \frac{1}{i}\frac{\partial^k}{\partial x^k\partial
x^l}, q_k q_l\mapsto ix^k x^l
\end{eqnarray*}
This gives  the structure of the Poisson mani\-folds to
representation of any (nilpotent) algebra or in other words to
continuous wavelet trans\-form.

\section{ THE SEGAL-BARG\-MAN
RE\-PRE\-SEN\-TA\-TI\-ON}
Let $$ z=\frac{1}{\sqrt{2}}(p-iq),\quad
\bar{z}=\frac{1}{\sqrt{2}}(p+iq),$$
$ p=(p_1,...,p_n)
,\quad
 F_n $ is the
space of holomorphic functions of n complex variables with
$(f,f)< \infty$, where $$ (f,g)=(2\pi)^{-n}\int
f(z)\overline{g(z)}e^{-|z|^2}dpdq $$
Consider a map  $U:
H\rightarrow F_n$ , where H is with real polarization,$F_n
$ is with complex polarization, then we have $$(U\Psi)(a)=\int
A(a,q)\Psi(q)dq,$$  where $$
A(a,q)=\displaystyle\pi^{-n/4}e^{-1/2(a^2+q^2)+\sqrt{2}aq}
$$
i.e. the Bargmann formula produce  wavelets.We also have
the representation of Heisenberg algebra on $F_n$ :
\begin{eqnarray*}
U\frac{\partial}{\partial q_j} U^{-1}&=&\frac{1}{\sqrt{2}}\left
(z_j- \frac{\partial}{\partial z_j}\right),\\
 Uq_j
U^{-1}&=&-\frac{i}{\sqrt{2}}\left(z_j+\frac{\partial }{\partial
 z_{j}} \right)
\end{eqnarray*}
and also : $ \omega=d\beta=dp\wedge dq,$
 where
$\beta=i\bar{z}dz $.
\section{ORBITAL THEORY FOR WAVELETS}
Let coadjoint action be
$$<g\cdot f,Y>=<f,Ad(g)^{-1}Y>,$$
 where $<,>$ is pairing
$ g\in G,\quad f\in g^*,\quad Y\in{\cal G}$ .

The orbit is
${\cal O}_f=G\cdot f\equiv G/G(f)$.

Also,let A=A(M) be algebra of functions,
V(M) is A-module of vector fields,
$A^p$  is A-module of p-forms.
Vector fields on orbit is
$$
\sigma({\cal O},X)_f(\phi)=\frac{d}{dt}(\phi(\exp tXf))\Big |_{t=0}
$$
where $\phi\in A({\cal O}),\quad f\in{\cal O}$. Then ${\cal O}_f$
are homogeneous symplectic manifolds with 2-form
$$
\Omega(\sigma({\cal O},X)_f,\sigma({\cal  O},Y)_f)=<f,[X,Y]>,
$$
and $d\Omega=0$. PB on ${\cal O}$ have the next form
$$
\{ \Psi_1,\Psi_2\}=p(\Psi_1)\Psi_2
$$
where p is $ A^1({\cal O})\rightarrow V({\cal O})$ with
definition
$\Omega (p(\alpha),X)$ $=$ $i(X)\alpha$. Here $\Psi_1,\Psi_2\in
A(\cal {O})$ and  $A({\cal O}) $ is Lie algebra with bracket
\{,\}.
\\ Now let N be a Heisenberg group. Consider adjoint and
coadjoint representations in some particular case.\\
 $N=(z,t)\in C\times R,
 z=p+iq$; compositions in N are $(z,t)\cdot(z',t')=
 (z+z',t+t'+B(z,z')) $, where $B(z,z')=pq'-qp'$. Inverse
 element is $(-t,-z)$. Lie algebra n of N is  $(\zeta,\tau)
 \in C\times R$ with bracket $[(\zeta,\tau),(\zeta',\tau')]=
 (0,B(\zeta,\zeta'))$. Centre is $\tilde{z}\in n $ and
generated by (0,1);
 Z is a subgroup $\exp\tilde{z}$.
Adjoint representation N on n is given by formula
  $$
Ad(z,t)(\zeta,\tau)=(\zeta,\tau+B(z,\zeta))
$$
Coadjoint:

 for $f\in n^*,\quad g=(z,t)$,
$$(g \cdot f)(\zeta,\zeta)=f(\zeta,\tau)-B(z,\zeta)f(0,1)$$ then
  orbits for which $f|_{\tilde z}\neq 0$ are plane in $n^*$
 given by equation $ f(0,1)=\mu$ . If $X=(\zeta,0),\quad
 Y=(\zeta ',0),\quad X,Y\in n$ then symplectic structure
  is
\begin{eqnarray*}
& &  \Omega (\sigma({\cal O},X)_f,\sigma({\cal
  O},Y)_f)=<f,[X,Y]>=\\
& &f(0,B(\zeta,\zeta'))\mu B(\zeta,\zeta')
\end{eqnarray*}
Also we have for orbit ${\cal O}_\mu=N/Z$ and ${\cal O}_\mu $ is
Hamiltonian G-space.

\section{ KI\-RIL\-L\-OV CHA\-RA\-C\-TER FOR\-MU\-LA  OR \\
 ANA\-LO\-GY OF GA\-BOR WAVELETS
}
Let U denote irreducible unitary representation of N with
condition $U(0,t)=\exp(it\ell)\cdot 1$, where $ \ell\neq
0 $,then U is equivalent to representation $T_\ell$ which acts in
$L^2(R)$ according to
$$
T_\ell(z,t)\phi(x)=\exp\left(i\ell(t+px)\right)\phi(x-q)
$$
If instead of N we consider E(2)/R we have $S^1$ case and we
  have Gabor functions on $S^1$.

\section{OSCILLATOR GROUP}
Let O be an oscillator group,i.e. semidirect product of R
and Heisenberg group N.

Let H,P,Q,I be standard basis in Lie algebra o of the group O
and $H^*,P^*,Q^*,I^*$ be dual basis in $o^*$. Let functional
f=(a,b,c,d)  be
$$
aI^*+bP^*+cQ^*+dH^* .
$$

Let us consider complex polarizations
\begin{eqnarray*}
h=( H,I,P+iQ ),     \quad 
\bar{h}=(I,H,P-iQ)
\end{eqnarray*}
Induced from h representation, corresponding to functional f
(for $a>0$), unitary equivalent to the representation
$$
W(t,n)f(y)=\exp (it(h-1/2)) \cdot U_{a} (n)V(t),
$$
where
\begin{eqnarray*}
V(t)&=&\exp[-it(P^2+Q^2)/2a] ,\\
 P&=&-{d}/{dx} ,\quad
 Q=iax,
\end{eqnarray*}
and $U_a(n)$ is irreducible representation of N, which have the
form $U_a(z)=exp(iaz)$ on the center of N.\\
Here we have:
U(n=(x,y,z)) is Schr\"{o}\-din\-ger representation, $U_t(n)=U(t(n))$
is the representation,which obtained from previous by
automorphism (time translation)
$n\longrightarrow t(n);\quad U_t(n)=U(t(n))$
is also unitary irreducible representation of N.
$$
V(t)=\exp(it(P^2+Q^2+h-1/2))
$$
is an operator, which according to Stone--von Neumann theorem
has the property
$$
U_t(n)=V(t)U(n)V(t)^{-1}.
$$

This is our last private case, but according to our approach we
can construct by using methods of geometric quantization theory
many "symplectic wavelet constructions" with corresponding
symplectic or Poisson structure on it.
Very useful particular spline--wavelet basis with uniform
exponential control on stratified and nilpotent Lie groups
was considered in [9].


Extended version and related results may be found in [1]-[6].

This research was supported in part under "New Ideas for Particle
Accelerators Program" NSF-Grant no.~PHY94-07194.
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