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\title{Pointlike constituent quarks and scattering equivalences}

\author{Evan Sengbusch}
\altaffiliation[Also at ]{Department of Physics and Astronomy, The 
University of Iowa.}%Lines break automatically or can be forced with \\
\author{W. N. Polyzou}
%\footnote{This work supported 
%in part by the U.S. Department of Energy, under contract DE-FG02-86ER40286}}
\email{polyzou@uiowa.edu}
\affiliation{%
Department of Physics and Astronomy, \\The 
University of Iowa   
%This line break forced with \textbackslash\textbackslash
}%

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}

This paper illustrates how scattering equivalences can be used to
simplify current operators in relativistic constituent quark models.

\end{abstract}

\pacs{13.40Gp,12.39Ki,24.10Jv}% PACS, the Physics and Astronomy
                             % Classification Scheme.
\keywords{Constituent quarks, electron scattering,relativistic models}%

\maketitle
\section{Introduction}

Scattering equivalences are unitary transformations that leave the
scattering matrix unchanged\cite{wp}.  They parameterize the freedom to change
the representation of the wave function without changing experimental
observables.  In models with confining interactions, where scattering
is not relevant, scattering equivalences are distinguished because
they leave the constituent masses unchanged.  In this paper a
scattering equivalence is applied to a constituent quark model to
construct an equivalent model with a point-like quark current
operator.

The constituent quark model of Carlson, Kogut and Pandharipande
\cite{ckp} is used to illustrate the construction.  This model is
designed to fit the meson spectrum.  The Hamiltonian in \cite{ckp} is
interpreted as the mass operator of a relativistic constituent quark
model with a light-front kinematic subgroup.  The construction of a
consistent dynamical representation of the Poincar\'e group 
can be done following ref. \cite{fcwp1}\cite{bkwp}\cite{fc}.

It is possible \cite{wp} to find a conserved covariant quark current
operator that is consistent with this model and fits the data
\cite{amend}\cite{volmer} for the charge form factor of the pion.  
That this current cannot be a point-like
impulse current is seen by direct calculation.  The dashed curves in
figures 1 and 2 show the pion charge form factor that would be
predicted in this model if the quark current was a point-like current
in the light-front impulse approximation\cite{fcwp2}\cite{fcwp3}. 
Both the low- and high-momentum transfer predictions are inconsistent 
with the data.

A scattering equivalence is used to construct an equivalent model,
with the same meson mass spectrum, where the quark current operator is
transformed to an operator that can be accurately approximated by a
point-like light-front impulse current.

The mass operator in the Carlson, Kogut, and Pandharipande model 
has the form 
\beq
M= M_0+ V_c \qquad M_0 = \sqrt{m_q^2 + k^2} + \sqrt{m_{\bar{q}}^2 + k^2} 
\label{eq:CA}
\eeq
\beq
V_c (\lambda_1, \cdots , \lambda_4)  = 
\lambda_1 + {\lambda_2 \over r} + \lambda_3 r  + 
\delta \vec{s}_q \cdot \vec{s}_{\bar{q}} e^{-{r^2 \over 2\lambda_4^2} }  
\label{eq:CB}
\eeq
\beq
\delta = -  {\lambda_2 \over 3 m_q^2 \lambda_4^3 \sqrt{\pi}} .
\eeq
The parameters $\lambda_1, \cdots, \lambda_4$ of the original Carlson
Kogut and Pandharipande model 
are listed on the first line of table 1.
\begin{table}
\caption{\label{tab:table1}Interaction Parameters}
\begin{ruledtabular}
\begin{tabular}{cccccc}
Interaction & $m\, (GeV)$ & $\lambda_1 \, (GeV)$ & $\lambda_2$ & $\lambda_3 \,(GeV)^2$ & $\lambda_4\, (GeV)^{-1}$   \\
\hline					      		      
$V_c $ &$.360$ &$-.777$  & $-.5$ &$.197$ & $.66$ \\
$\bar{V}_c$ &$.360$ &$-.911$  & $-.14$ &$.049$ & $.35$ \\
\end{tabular}
\end{ruledtabular}
\end{table}

The eigenvectors and eigenvalues of $M$ are denoted by
\beq
M\vert m_n \rangle = m_n \vert m_n \rangle 
\label{eq:AO}
\eeq
where the $n=0$ state is the $\pi$ meson state and
$m_0= m_{\pi}$. 

The scattering equivalence is taken as a rank-one perturbation of the 
identity
\beq
A= I + \Delta 
\label{eq:AA}
\eeq
where $\Delta$ is proportional to a one-dimensional 
projection operator.  

To generate $\Delta$ let $\vert \bar{m_\pi}
\rangle$ be the lowest mass eigenstate of an operator $\bar{M}$
obtained from $M$ by adjusting parameters $\{\lambda_1, \cdots,
\lambda_4\}$ in the confining interaction: 
\beq
\bar{M} = M_0+ \bar{V}_c  \qquad
\bar{M} \vert \bar{m}_\pi \rangle =
\bar{m}_\pi \vert \bar{m}_\pi \rangle  
\label{eq:AP}
\eeq
\beq
\bar{V}_c = V_c(\bar{\lambda}_1, \cdots , \bar{\lambda}_4) .
\label{eq:AQ}
\eeq
until the
lowest mass eigenstate of $\bar{M}$ fits the pion charge form factor data
using a point-like quark current in the light-front impulse
approximation.  Denote this vector by $\vert \bar{m}_{\pi}
\rangle$.  The spectrum of $\bar{M}$ does not agree with the 
observed meson masses.  It should not be interpreted as a mass
operator; its role is to generate candidates for the transformed 
ground state wave function.  

In this model the parameters on the second line of table 1 lead to a
state $\vert \bar{m}_{\pi} \rangle$ that predicts the charge form factors
given by the solid lines in figures 1 and 2 in the light-front impulse
approximation.  These curves are in better agreement with the 
experimental results.

%The operator $\bar{M}$ has no physical interpretation in this
%construction; it is only used to generate $\vert p, \bar{m}_{\pi},
%\rangle$.
The operator $A$ is constructed to satisfy 
\beq
A \vert m_\pi \rangle = \vert \bar{m}_\pi \rangle 
\label{eq:AW}
\eeq
\beq
A\vert {m}_\perp \rangle = \vert \bar{m}_\perp \rangle
\label{eq:AX}
\eeq
and 
\beq
A \vert \psi \rangle = \vert \psi \rangle \qquad \mbox{for} \qquad 
\langle m_\pi \vert \psi \rangle = \langle m_\perp \vert \psi \rangle =0 .
\label{eq:AY}
\eeq

To construct the desired transformation define two orthogonal 
bases on the two-dimensional subspace spanned by the vectors 
$\vert m_\pi \rangle$ and $\vert \bar{m}_\pi \rangle$.  The orthonormal
basis functions are
\beq
\vert m_\pi \rangle,
\qquad
\vert  m_\perp \rangle := {\vert \bar{m}_\pi \rangle
-  \vert m_\pi \rangle \cos (\theta)  \over \sin (\theta)} 
\label{eq:AS}
\eeq
and 
\begin{figure}
\rotatebox{00}{\resizebox{3.0in}{!}{
\includegraphics{es1.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Pion form factor at low $Q^2$.}
\end{figure}
\beq
\vert \bar{m}_\pi \rangle, 
\qquad
\vert \bar{m}_\perp \rangle := {\vert {m}_\pi \rangle
-  \vert \bar{m}_\pi \rangle \cos (\theta)  \over \sin (\theta)} 
\label{eq:AT}
\eeq
respectively, where
\beq
\cos (\theta)= \cos (\theta)^* :=\langle m_\pi \vert \bar{m}_\pi \rangle
\qquad 
\sin (\theta ) > 0 .
\label{eq:AV}
\eeq
The overlap $\langle m_\pi \vert \bar{m}_\pi \rangle = \cos (\theta)$
can be chosen to be real as a consequence of the time-reversal
invariance of $V_c$ and $\bar{V}_c$. 
A unitary scattering equivalence $A$ satisfying 
(\ref{eq:AW}),(\ref{eq:AX}) and (\ref{eq:AY}) is given by: 
\[
A = I + \Delta =
\]
\beq
I-  \vert m_\pi \rangle \langle m_\pi \vert -
\vert m_\perp \rangle \langle m_\perp \vert  + 
\vert \bar{m}_\pi \rangle \langle m_\pi \vert +
\vert \bar{m}_\perp \rangle \langle m_\perp \vert .
\label{eq:AZ}
\eeq 
After some algebra $\Delta$ can be expressed 
as the rank-one operator:
\beq
\Delta =
- \rho  (  \vert m_\pi \rangle - \vert \bar{m}_\pi \rangle )
(\langle m_\pi \vert - \langle \bar{m}_\pi \vert )
\label{eq:BB}
\eeq
where
\beq
\rho = {\cos (\theta) +1 \over \sin^2 (\theta) }.
\label{eq:BC}
\eeq
In this example the overlap parameter is
\beq
\cos (\theta) := \langle m_{\pi} \vert \bar{m}_{\pi} \rangle = .731 .
\eeq

The scattering equivalence $A$ is used to define a transformed mass 
operator   
\beq
M' := A^{\dagger} (M_0 + V_c)A = M_0 + V'_c 
\eeq
which leads to a quark model with confining interaction 
\[
V_c' = V_c+ \Delta M + M \Delta^{\dagger} +  \Delta M \Delta^{\dagger}
=
\]
\[
V_c+\rho 
( \vert m_\pi \rangle - \vert \bar{m}_\pi \rangle )
\langle \bar{m}_\pi \vert (V_c- \bar{V}_c) 
\]
\[
+ \rho (V_c- \bar{V}_c)\vert \bar{m}_{\pi} \rangle (\langle m_{\pi} \vert
- \langle \bar{m}_{\pi} \vert )  
\]
\beq
+ \rho^2 
( \vert m_\pi \rangle - \vert \bar{m}_\pi \rangle )
\langle \bar{m}_{\pi} \vert(V_c- \bar{V}_c)\vert \bar{m}_{\pi} \rangle
(\langle m_{\pi} \vert - \langle \bar{m}_{\pi} \vert ) .  
\label{eq:BD}
\eeq 
The method of constructing $\vert \bar{m}_{\pi}\rangle$ leads to a
confining interaction with separable terms that have the same 
structure as the original confining interaction.
\begin{figure}
\rotatebox{00}{\resizebox{3.25in}{!}{
\includegraphics{es2.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Pion form factor at high $Q^2$.}
\end{figure}

By construction the mass operator
$M'$:
\begin{itemize}
\item[a.)] Has the same spectrum as $M$.
\item[b.)] Has the same pion wave function as $\bar{M}$ 
\item[c.)] Differs from $M$ by the short range modification (\ref{eq:BD}) 
to the confining interaction.
\end{itemize}

Because the operator $\Delta$ is kinematically invariant, the
operator $M'$ is also the mass operator for a unitary
representation of the Poincar\'e group with the light-front kinematic
subgroup.

Even though the scattering equivalence $A$ is constructed to transform the
ground state wave function, it also transforms the $n>0$ states to
preserve orthogonality.  The transformed states have the form
\beq  
\vert {m}'_n \rangle = A \vert m_n \rangle =
\vert m_n \rangle + \rho
\langle \bar{m}_{\pi} \vert m_n \rangle 
\left (\vert m_{\pi} \rangle 
-  \vert \bar{m}_\pi \rangle \right ).
\label{eq:BDD}
\eeq
The operator $M'$ has the same mass spectrum as the original Carlson,
Kogut, Pandharipande mass operator and has a pion eigenstate that can
be used with a point-quark impulse quark current to obtain the
measured pion charge form factor.  The pion 
charge form factor in
this model is given by the solid curves in figures 1 and 2. 

Figures 3 and 4 compare the coordinate (fig. 3) and momentum space
(fig. 4) wave functions of the pion for the mass operators $M$ (dotted
curve) and $M'$ (solid curve). 
The calculation of the wave functions are done using 
the method described in \cite{kp}.
The Carlson, Kogut, and Pandharipande
wave functions have a smaller size in configuration space and more
high-momentum components than the wave-functions of the transformed
mass operator. 

The transformation $A$ has the overall effect of softening the wave
functions of the original Carlson, Kogut, and Pandharipande model.
This is consistent with the calculations of Cardarelli
et. al. \cite{salme} who use a similar quark-antiquark model and
introduce single-quark form factors to obtain measured pion charge form
factors.

\begin{figure}
\rotatebox{00}{\resizebox{3.25in}{!}{
\includegraphics{psik-360.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Comparison of k-space wave functions.}
\end{figure}

The effect of the transformation $A$ is to add an additional short
range structure to the original confining interaction.  Compared to
the original Carlson, Kogut, and Pandharipande interaction, the
additional short-ranged part contains non-localities.  In a
relativistic quantum theory there is no preferred reason to favor a
local over a non-local interaction except for mathematical simplicity.
Local and non-local interactions have similar strengths and defects.  In
relativistic models with short-ranged interactions, both the local and
non-local interactions are consistent with two-body cluster
properties, but both fail to be consistent with microscopic locality.
Microscopic derivations of local interactions necessarily make
implicit assumptions that lead to local interaction, however these
assumptions are not based on physics principles. 
There is no reason to consider the transformed interaction to be any
more or less fundamental than the original interaction.

\begin{figure}
\rotatebox{00}{\resizebox{3.25in}{!}{
\includegraphics{psir-360.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Comparison of r-space wave functions.}
\end{figure}

\section{Conclusions} In this paper the freedom to change 
representation without changing the underlying physics is used to
construct a constituent quark model that fits the meson mass spectrum
and reproduces the pion form factor using only point-like constituent
quarks.

This is done using scattering equivalences, which are unitary
operators of the form $A=I+ \Delta$, where $\Delta$ is asymptotically
zero \cite{wp1}. 

In this example the desired scattering equivalence
is an easily constructed rank-one perturbation of the identity.
The only required input is the new state vector $\vert \bar{m}_{\pi}
\rangle$ and the overlap $\langle m_\pi \vert \bar{m}_{\pi} \rangle$.
The new representation does not require constituent quark/antiquark  
form factors.

While two-body currents are implicitly generated in the 
light-front impulse approximation \cite{fcwp2}, they are not 
explicitly needed to compute the form factor. 

The choice of $\Delta$ affects the predictions for elastic and
transition form factors involving other mesons.  Scattering
equivalences with higher rank $\Delta$s can be used to further
simplify the current in these models if it is warranted by the
physics.  

In general it may not be possible to always find a scattering
equivalence that completely removes the many-body contributions to a
current operator.  In this example, while figure 1 shows considerable
improvement in the agreement with experiment for the transformed
model, the agreement at lower values of momentum transfer is not
within the experimental error bars.  This is because it is difficult
to simultaneously fit both the high and low-momentum transfer data
without changing the constituent quark masses\cite{fcwp3}.  Scattering
equivalences are designed to leave the constituent particle masses
unchanged.  Wave functions that bring both the low and high momentum
transfer to within experimental uncertainties can be found for systems
of constituent quarks with smaller constituent masses\cite{fcwp3}.
These are associated with dynamical models that fit the meson spectrum
with smaller constituent quark masses \cite{fcwp3}.

This example started with a relativistic constituent quark model with a
light-front kinematic symmetry, and produced a new relativistic model
with the same light-front kinematic symmetry, the same mass spectrum,
where the pion charge form factor can be computed in the point-quark
impulse approximation.  This example illustrates that the constraints
imposed by the choice of kinematic subgroup and mass spectrum do not
determine the form factors.

Experiments can only determine a theory up to an overall scattering
equivalence.  The freedom to choose a particular scattering equivalent
representation for a dynamical computation in analogous to choosing a
convenient coordinate system.  This freedom may prove to be 
particularly important because scattering equivalences at the 
few-body level naturally lead to changes in the corresponding many-body 
theory.  

Scattering equivalences are also known to exist \cite{wp1} between
models with different kinematic subgroups.  These relationships can
also be exploited to construct equivalent current operators for models
with different forms of dynamics.  Existing calculations show that the
representation of the currents in different forms of dynamics can be
very different
\cite{fcwp2}\cite{fcwp3}\cite{salme}\cite{allen}\cite{fcdr}\cite{bertrand1}\cite{krutov}.
This same freedom also exists in local quantum field theory, where
scattering equivalent theories are partitioned into Borchers classes
\cite{sw}.  While the ambiguities in representations of the dynamics
are sometimes considered a liability, this paper shows that they lead
to a flexibility that can lead to a simplification of the dynamics.

\begin{acknowledgments}
This work supported 
in part by the U.S. Department of Energy, under contract DE-FG02-86ER40286
\end{acknowledgments}
\begin{thebibliography}{99}
\bibitem{ckp} J. Carlson, J. B. Kogut, V.R. Pandharipande, Phys. Rev. 
D{\bf 28},2807(1983). 
\bibitem{fcwp1} F. Coester and W. N. Polyzou, Phys. Rev. D{\bf 26}, 1348(1982).
\bibitem{bkwp} B.D. Keister and W.N. Polyzou, Adv. Nuc. Phys. {\bf 20}, 225(1991).
\bibitem{fc} F. Coester, Prog. Part. Nucl. Phys.{\bf 29},1(1992).
\bibitem{wp} W. N. Polyzou,  Phys. Rev. D{\bf 32},2216(1985).
\bibitem{amend} S. R. Amendolia et al. Nuc. Phys. B  {\bf 277}, 168 (1986).
\bibitem{volmer} J. Volmer et al. Phys. Rev. Lett. {\bf 86} , 1713  (2001).
\bibitem{fcwp2} P. L. Chung, F. Coester and W. N. Polyzou, 
Phys. Lett. B {\bf 205}, 545 (1988).
\bibitem{fcwp3} F. Coester and W. N. Polyzou,  nucl-th/0405082.
\bibitem{kp}B. D. Keister and W. N. Polyzou, J. Comp. Phys. {\bf 134}, 
231 (1997).
\bibitem{salme}F. Cardarelli et al. Phys. Lett. B {\bf 332},1(1994); 
{\bf 357} 267 (1995).
\bibitem{wp1} W. N. Polyzou, J. Math. Phys. {\bf 43},6024 (2002).
\bibitem{allen}T.W. Allen and W.H. Klink,  Phys. Rev. C{\bf 58},3670 (1998).
\bibitem{fcdr} B. Julia-Diaz, D. O. Riska, F. Coester, Phys. 
Rev. C{\bf 69},035212 (2004). 
\bibitem{bertrand1}B. Desplanques, nucl-th/0405060.
\bibitem{krutov} A. F. Krutov and V. E. Troitsky, 
Theor. Math. Phys.  {\bf 116}, 219 (1998).
\bibitem{sw} R. F. Streater and A. S. Wightman, PCT and Spin and Statistics, and all that, W. A. Benjamin, 1964. 
\end{thebibliography}

\end{document}

%\bibitem{bertrand} A. Amghar, B. Desplanques and L.Theussl, 
%Phys. Lett. B {\bf 574} , 201 (2003).

%If the inverse scattering problem has a solution in relativistic or
%non-relativistic quantum mechanics then it cannot be unique.  A large
%class of equivalent interactions that lead to the same $S$-matrix can
%be constructed using scattering equivalences \cite{wp}.  Different scattering
%equivalent models are equivalent representations of the dynamics.
%While there is no fundamental preference for a given representation
%among the class of equivalent representations, there are computational
%advantages to choosing a representation where one aspect of the dynamics
%is simple.
%In this paper we use a scattering equivalence 
%to simplify the structure of the quark current operator in a relativistic
%constituent quark model of mesons. 
% The general construction is
%discussed in section two.  The construction is applied to the model of
%\cite{ckp} in section three.
%
%\section{Construction} 


%The prominent role played by the choice of representation of the
%dynamics is a striking feature of relativistic direct-interaction
%quantum mechanics.  
%
%Models with different kinematic subgroups \cite{bkwp} (light-front,
%Euclidean, or Lorentz) that fit the experimental meson and baryon
%spectra require different representations of quark
%current\cite{salme}\cite{allen}\cite{fcdr}\cite{bertrand1}
%\cite{krutov} operators to fit the measured electromagnetic 
%properties of the pion and nucleon.
%
%The representation dependence of operators, like the current, is not
%surprising.  In local quantum field theory all fields in the same Borchers
%class \cite{sw} have the same scattering matrix.  In effective
%field theory, field redefinitions \cite{dick} lead to equivalent
%representations of dynamics on limited energy scales, and in quantum
%theories of a finite number of degrees of freedom the group of
%scattering equivalences \cite{fcwp1}\cite{wp} leads to different 
%representations of the dynamics with the same spectrum and 
%scattering matrix elements.


\vfill\eject 

\begin{figure*}
\rotatebox{00}{\resizebox{3.5in}{!}{
\includegraphics{es1.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Pion form factor at low $Q^2$.}
\end{figure*}

\vfill\eject 

\begin{figure*}
\rotatebox{00}{\resizebox{3.5in}{!}{
\includegraphics{es2.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Pion form factor at high $Q^2$.}
\end{figure*}

\vfill\eject 

\begin{figure*}
\rotatebox{00}{\resizebox{3.5in}{!}{
\includegraphics{psik-360.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Comparison of k-space wave functions.}
\end{figure*}

\vfill\eject 

\begin{figure*}
\rotatebox{00}{\resizebox{3.5in}{!}{
\includegraphics{psir-360.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Comparison of r-space wave functions.}
\end{figure*}

\end{document}

\begin{figure*}
\rotatebox{00}{\resizebox{7.5in}{!}{
\includegraphics{low_pi_star.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}$\pi^*$ form factor at low $Q^2$.}
\end{figure*}

\begin{figure*}
\rotatebox{00}{\resizebox{7.5in}{!}{
\includegraphics{high_pi_star.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}$\pi^*$ form factor at high $Q^2$.}
\end{figure*}



----


yAssume that it is possible to find a set of parameters 
$\{\bar{\lambda}_1, \cdots , \bar{\lambda}_n\}$ with the
property that experimental pion form factor can be expressed in terms
of a matrix element of a point-like impulse current, $J_p^{\mu} (0)$,
in the states $\vert p, \bar{m}_{\pi}, \rangle$, $\vert p', \bar{m}_{\pi}
\rangle$. 

The wave function $\langle k \vert \bar{m}_{\pi} \rangle$ is a
candidate for the transformed pion wave function.
The assumption means that the desired wave function can 
be found in the class of ground state solutions of mass operators with
different choices of the potential parameters.  This restricts the
possible candidates for the new wave function to a set of functions that
depend on a small number of parameters.  A larger class of wave
functions can be considered if there are no functions in the
above class that are able to reproduce the measured form factors.
The spectrum of $\bar{M}$ contains no physics, however,
without loss of generality one of the parameters,
$\bar{\lambda}_k$, can be taken to be an additive constant which 
can be adjusted so the eigenvalue $\bar{m}_{\pi} = m_{\pi}$.  This 
choice simplifies the notation, but has no effect on the form factor.

A general feature of relativistic quantum mechanical models is that
an impulse current cannot be compatible with current conservation
and current covariance.  The problem is that the left side of 
the relations
\beq
U(\Lambda ,a) J^{\mu} (x) U^{\dagger} (\Lambda,a) =
J^{\nu}(\Lambda x + a) \Lambda_{\nu} {}^{\mu}   
\eeq
and
\beq
[H,J^0 (x) ]_-= \sum_i [P^i , J^i (x) ]_- 
\eeq
involves the dynamics.  These dynamical constraints, when combined with 
space reflection and time reversal symmetries,  imply that the 
current matrix elements can be expressed in terms of a smaller set of
independent current matrix elements.  The number of independent
current matrix elements is precisely equal to the number of invariant
form factors.  For each kinematic subgroup there are natural choices
of independent current matrix elements.  It is possible to
consistently evaluate the independent current matrix elements, and all 
matrix elements related to them by kinematic transformations, using only
impulse currents.  In this paper, the impulse approximation means 
only that the independent matrix elements are calculated in the 
impulse approximation.   

The problem is to find a unitary operator $A=I+\Delta$ that maps
$\vert m_\pi \rangle$ to $\vert \bar{m}_\pi \rangle$.  Since $\vert
\bar{m}_\pi \rangle$ is not orthogonal to any of the states $\vert
m_n\rangle$, except where required by selection rules, the
transformation $A$ will effect all of the states.

A minimal solution uses a rank-one $\Delta$.  To construct the desired
transformation define two orthogonal bases on the two-dimensional
subspace spanned by the vectors $\vert m_\pi \rangle$ and $\vert
\bar{m}_\pi \rangle$.  If these vectors are identical then $A=I$, and there
is nothing to do.  Otherwise orthogonal bases on this 
two-dimensional subspace are
\beq
\vert m_\pi \rangle,
\qquad
\vert  m_\perp \rangle := {\vert \bar{m}_\pi \rangle
-  \vert m_\pi \rangle \cos (\theta)  \over \sin (\theta)} 
\label{eq:AS}
\eeq
and 
\beq
\vert \bar{m}_\pi \rangle, 
\qquad
\vert \bar{m}_\perp \rangle := {\vert {m}_\pi \rangle
-  \vert \bar{m}_\pi \rangle \cos (\theta)  \over \sin (\theta)} 
\label{eq:AT}
\eeq
where
\beq
\cos (\theta)= \cos (\theta)^* :=\langle m_\pi \vert \bar{m}_\pi \rangle
\label{eq:AU}
\eeq
\beq
\sin (\theta ) > 0 .
\label{eq:AV}
\eeq
The overlap $\langle m_\pi \vert \bar{m}_\pi \rangle = \cos (\theta)$ 
can be chosen to be real as a consequence of the time reversal
invariance of $V_c$ and $\bar{V}_c$.  The $\sin (\theta)$ is non zero,
otherwise $\vert m_{\pi} \rangle = \vert \bar{m}_{\pi} \rangle$.

By construction the states 
$\{ \vert m_\pi \rangle ,\vert \bar{m}_\pi \rangle\}$,
$\{ \vert m_\pi \rangle ,\vert m_\perp \rangle\}$, and 
$\{ \vert \bar{m}_\pi \rangle ,\vert \bar{m}_\perp \rangle\}$
all span the same two-dimensional subspace.  

The operator $A$ is chosen to satisfy 
\beq
A \vert m_\pi \rangle = \vert \bar{m}_\pi \rangle 
\label{eq:AW}
\eeq
\beq
A\vert {m}_\perp \rangle = \vert \bar{m}_\perp \rangle
\label{eq:AX}
\eeq
and 
\beq
A \vert \psi \rangle = \vert \psi \rangle \qquad \mbox{for} \qquad 
\langle m_\pi \vert \psi \rangle = \langle m_\perp \vert \psi \rangle =0 .
\label{eq:AY}
\eeq

A unitary $A$ satisfying (\ref{eq:AW}),(\ref{eq:AX}) and 
(\ref{eq:AY}) is given by: 
\[
A = I + \Delta =
\]
\beq
I-  \vert m_\pi \rangle \langle m_\pi \vert -
\vert m_\perp \rangle \langle m_\perp \vert  + 
\vert \bar{m}_\pi \rangle \langle m_\pi \vert +
\vert \bar{m}_\perp \rangle \langle m_\perp \vert .
\label{eq:AZ}
\eeq
The perturbation $\Delta$ is the matrix
\beq
\Delta = (\vert \bar{m}_\pi \rangle- \vert m_\pi \rangle) 
\langle m_\pi \vert
+
(\vert \bar{m}_\perp \rangle - \vert m_\perp \rangle )
\langle m_\perp \vert .
\label{eq:BA}
\eeq
It is useful to express $\Delta$ directly in terms of 
the non-orthogonal states $\vert m_\pi \rangle$ 
and $\vert \bar{m}_\pi \rangle$: 
\[
\Delta =
\]
\beq
- \rho  (  \vert m_\pi \rangle - \vert \bar{m}_\pi \rangle )
(\langle m_\pi \vert - \langle \bar{m}_\pi \vert )
\label{eq:BB}
\eeq
where
\beq
\rho = {\cos (\theta) +1 \over \sin^2 (\theta) }.
\label{eq:BC}
\eeq
In this representation $\Delta$ is easily seen to be a 
rank-one Hermetian operator.  Using $\Delta+ \Delta^{\dagger} + 
\Delta \Delta^{\dagger}=0$
and 
$\langle m_{\pi} \vert (V_c-\bar{V}_c) \vert \bar{m}_{\pi} \rangle =  
\langle \bar{m}_{\pi} \vert (V_c-\bar{V_c}) \vert m_{\pi} \rangle = 0
$ gives the transformed confining interaction:
\[
V_c' = V_c+ \Delta M + M \Delta^{\dagger} +  \Delta M \Delta^{\dagger}
=
\]
\[
V_c+\rho 
( \vert m_\pi \rangle - \vert \bar{m}_\pi \rangle )
\langle \bar{m}_\pi \vert (V_c- \bar{V}_c) 
\]
\[
+ \rho (V_c- \bar{V}_c)\vert \bar{m}_{\pi} \rangle (\langle m_{\pi} \vert
- \langle \bar{m}_{\pi} \vert )  
\]
\beq
+ \rho^2 
( \vert m_\pi \rangle - \vert \bar{m}_\pi \rangle )
\langle \bar{m}_{\pi} \vert(V_c- \bar{V}_c)\vert \bar{m}_{\pi} \rangle
(\langle m_{\pi} \vert - \langle \bar{m}_{\pi} \vert ) .  
\label{eq:BD}
\eeq 
By construction the mass operator
$M' = M_0 + V_c'$:
\begin{itemize}
\item[a.)] Has the same spectrum as $M$.
\item[b.)] Has the same pion wave function as $\bar{M}$ 
\item[c.)] Differs from $M$ by the short range modification (\ref{eq:BD}) 
to the confining interaction.
\end{itemize}

Even though the transformation $A$ was constructed to transform 
the ground state wave function, it must also 
transform the $n>0$ states to preserve orthogonality.  The 
transformed states have the form 
\beq  
\vert {m}'_n \rangle = A \vert m_n \rangle =
\vert m_n \rangle + \rho
\langle \bar{m}_{\pi} \vert m_n \rangle 
\left (\vert m_{\pi} \rangle 
-  \vert \bar{m}_\pi \rangle \right ).
\label{eq:BDD}
\eeq

If the current $J^{\prime \mu}(x)$ is identified with the point
quark-antiquark impulse current, $J^{\mu}_p (x)$, then the
transformation $A$ generates exchange current contributions to the
current in the original representation:
\[
\langle p', m'_n \vert J_p^{\mu} \vert  p, m'_{l}
\rangle =
\]
\beq
\langle p', m_n \vert [J_p^{\mu} + \Delta^{\dagger} J^{\mu}_{p} + 
J^{\mu}_{p} \Delta + 
\Delta^{\dagger} J^{\mu}_{p} \Delta] \vert  p, m_{l}
\rangle .
\label{eq:BE}
\eeq
Thus, form factors can be calculated using the current (\ref{eq:BE})
in the original representation or the point quark impulse 
current in the transformed representation (\ref{eq:BDD}).

This method provides an alternative means for describing 
electromagnetic properties of constituent quarks which replaces the  
constituent quark form factors by a different representation of 
the dynamics.

\section{Example} 

The simplicity of the construction is illustrated with an
example.  The constituent quark-antiquark model of Carlson,
Kogut and Pandharipande \cite{ckp} is taken as the constituent quark 
model that fits the meson spectrum.  The semi-relativistic Hamiltonian in
\cite{ckp} is interpreted as the mass operator in a fully 
relativistic model with the kinematic subgroup of the light front.
This relativistic model is described in the appendix.

The choice of kinematic subgroup does not effect the mass spectrum of 
the original model.  It does effect the value of the 
impulse current matrix elements.   

The mass operator in this model has the form
\beq
M = \sqrt{m_q^2 + k^2} + \sqrt{m_{\bar{q}}^2 + k^2} + V_c
\label{eq:CA}
\eeq
\beq
V_c (\lambda_1, \cdots , \lambda_4)  = 
\lambda_1 + {\lambda_2 \over r} + \lambda_3 r  + 
\delta \vec{s}_q \cdot \vec{s}_{\bar{q}} e^{-{r^2 \over 2\lambda_4^2} }  
\label{eq:CB}
\eeq
\beq
\delta = -  {\lambda_2 \over 3 m_q^2 \lambda_4^3 \sqrt{\pi}} .
\eeq
The model of \cite{ckp} also includes  spin-orbit and tensor interactions 
that vanish for the pion.  
The parameters $\lambda_1, \cdots, \lambda_4$ of the original Carlson
Kogut and Pandharipande model 
are listed on the first line of table 1.

\begin{table}
\caption{\label{tab:table1}Interaction Parameters}
\begin{ruledtabular}
\begin{tabular}{cccccc}
%Left\footnote{Note a.}&Centered\footnote{Note b.}&Right\\
%\hline
%\hline
Interaction & $m\, (GeV)$ & $\lambda_1 \, (GeV)$ & $\lambda_2$ & $\lambda_3 \,(GeV)^2$ & $\lambda_4\, (GeV)^{-1}$   \\
\hline					      		      
$V_c $ &$.360$ &$-.777$  & $-.5$ &$.197$ & $.66$ \\
$\bar{V}_c$ &$.360$ &$-.911$  & $-.14$ &$.049$ & $.35$ \\
%\hline
\end{tabular}
\end{ruledtabular}
\end{table}

%\begin{center}
%\begin{table} % [hbt]
%{\bf Table 1: Interaction Parameters } \\[1.0ex]
%\begin{tabular}{|l|l|l|l|l|l|}
%\hline
%Interaction & $m\, (GeV)$ & $\lambda_1 \, (GeV)$ & $\lambda_2$ & $\lambda_3 \,(GeV)^2$ & $\lambda_4\, (GeV)^{-1}$   \\
%\hline					      		      
%$V_c $ &$.360$ &$-.777$  & $-.5$ &$.197$ & $.66$ \\
%$\bar{V}_c'$ &$.360$ &$-.911$  & $-.14$ &$.049$ & $.35$ \\
%\hline
%\end{tabular}
% \label{tabspec}
%\end{table}
%\end{center}

The pion form factor can be expressed in terms of the
matrix elements of the $+$ component of the current in a frame where
the $+$ component of the momentum transfer vanishes\cite{fcwp2}
\cite{fcwp3}.  This matrix element is
evaluated with a point-like impulse current:
\beq
F_{\pi}(Q^2) =
\langle {q,\over 2} , m_\pi \vert \left ( J_{qp} ^{+}(0) + 
J_{\bar{q}p} ^{+} (0)\right  ) 
\vert -{q \over 2}, m_{\pi} \rangle
\label{eq:AR}
\eeq
where the states have the normalization
\beq
\langle p' , m_\pi  
\vert p, m_\pi \rangle = \delta (p^{+\prime} - p^+)
\delta^2 ( \vec{p}_{\perp}\,'- \vec{p}_{\perp}). 
\eeq
The calculation of the wave functions are done using 
the method described in \cite{kp}. If
the original Carlson, Kogut, Pandharipande wave pion function is used
in (\ref{eq:AR}), then the calculated form factor is larger than the
experimental form factor for both low \cite{amend} and high
\cite{volmer} momentum transfer.  The calculated form factor is given
by the dotted curves in figure 1 and figure 2.  

The parameters in the second line of table 1 define the interaction
$\bar{V}_c$.  The state $\vert \bar{m}_{\pi} \rangle$ is the ground
state of the mass operator $\bar{M}= M_0 + \bar{V}_c$.
The solid curves in figures 1 and 2 are the result of using $\vert
\bar{m}_\pi \rangle$ in (\ref{eq:AR}).  The calculated form factor
is consistent with the measured data, but the spectrum of 
the operator $\bar{M}$ no longer agrees with experiment. 

The method described in the previous section constructs a 
new mass operator, $M'$, that has the same spectrum as $M$ and 
the same pion wave function as $\bar{M}$. In this example the 
overlap parameter is
\beq
\cos (\theta) := \langle m_{\pi} \vert \bar{m}_{\pi} \rangle = .731 
\eeq
which gives $\rho = 3.71$.  The states $\vert
\bar{m}_{\pi} \rangle$ and $\vert m_\pi
\rangle$ and the overlap  $\langle \bar{m}_{\pi} \vert m_\pi
\rangle$ are used to define 
the unitary operator $A=I+\Delta$ using (\ref{eq:BB}).   
The transformed mass operator is $M' = M_0+
V_c'$ with $V_c'$ given by  (\ref{eq:BD}).  The operator $M'$ has the same 
mass spectrum as the original Carlson, Kogut, Pandharipande 
mass operator and has a pion eigenstate 
that can be used with the point impulse quark current to obtain 
the measured pion form factors.  By construction, the pion form 
factor in this model is given by the solid curves in
figures 1 and 2. 
Eigenstates of $M'$ for mesons 
other than the pion are obtained by applying $A$ to the corresponding 
eigenstates of $M$.

Figures 3 and 4 compare the coordinate (fig. 3) and momentum space
(fig. 4) wave functions of the pion for the mass operators $M$ (dotted
curve) and $M'$ (solid curve).  The Carlson, Kogut, and Pandharipande
wave functions have a smaller size in configuration space and more
high-momentum components than the wave-functions of the transformed
mass operator.  The form factor requires a more spread out wave
function in configuration space and a wave function with more low
momentum components in momentum space.  The transformation $A$ has the
overall effect of softening the wave functions of the original
Carlson, Kogut, and Pandharipande model.  This is consistent with the
calculations of Cardarelli et. al. \cite{salme}.  They use a similar
quark-antiquark model and need to introduce single quark form factors
to obtain measured pion form factors.  In \cite{salme} the quark form
factors also soften the wave function.

The pion decay constant, computed following \cite{fcwp3}, is $193$ MeV
in the light-front model with the original Carlson, Kogut,
Pandharipande parameters.  The transformation $A$ reduces this to 131
$MeV$, however it is still above the experimental value of 92 $MeV$.
This could be improved by considering more general transformations
$A$, or constituent quark models with smaller constituent quark masses
\cite{fcwp3}.

The overlap probability, $P=\vert \langle m_\pi \vert \bar{m}_\pi
\rangle \vert^2 = \cos^2 (\theta)$,  of the dotted and 
solid wave functions in fig. 3
and fig. 4 is .53.  This means that the overlap probability of
$\vert \bar{m}_{\pi} \rangle$ with the excited states of 
$M'$ is .47.  Figures 5 and 6 show the effect of the transformation 
$A$ on the point-like impulse form factors for the first radial 
excitation of the pion.  In these figures the dotted curve 
is the form factor in the original Carlson Kogut and Pandharipande 
model and the solid curve is the form factor in the transformed model.   
These plots indicate that the scattering equivalence also 
has significant effects on the excited mesons states with the 
same quantum numbers.   The form factors in the original and 
transformed models cross at about 1 $(GeV)^2$.  This is because 
the transformed excited state is not orthogonal to the 
original Carlson, Kogut, Pandharipande ground state, which has a 
significant amount of high momentum components.  The
operator $A$ constructed in this example is the simplest unitary 
operator that maps $\vert m_{\pi} \rangle$ to 
$\vert \bar{m}_{\pi} \rangle$.  Different choices of $A$ 
lead to different predictions for the form factors of 
the excited states. 
  
The effect of the transformation $A$ is to add an additional short
range structure to the original confining interaction.  Compared to
the original Carlson, Kogut, and Pandharipande interaction, the
additional short-ranged part contains non-localities.  In a
relativistic quantum theory there is no preferred reason to favor a
local over a non-local interaction except for mathematical simplicity.
Local and non-local interactions have similar strengths and defects.  In
relativistic models with short-ranged interactions, both the local and
non-local interactions are consistent with two-body cluster
properties, but both fail to be consistent with microscopic locality.
Microscopic derivations of local interactions necessarily make
implicit assumptions that lead to local interaction, however these
assumptions are not based on physics principles.  This shows that
there is no reason to consider the transformed interaction to be any
more or less fundamental than the original interaction.

\section{Conclusions} In this paper the freedom to change 
representation is used to construct a constituent quark model that fits
the meson mass spectrum and reproduces the pion form factor using only
point like constituent quarks.

This was done using scattering equivalences, which are unitary
operators of the form $A=I+ \Delta$, where $\Delta$ is asymptotically
zero \cite{wp}.  This paper utilized a limited class of these
operators where $\Delta$ is a finite rank operator on the internal
Hilbert space satisfying a domain restriction.  Desirable results were
achieved by considering only a small subset of rank-one $\Delta$'s.
Improvements to the fit of the pion form factor can be achieved by
considering a larger class of $\Delta$'s, such as compact $\Delta$'s.
Excellent results can be obtained with constituent quark models having
slighty lower constituent quark masses \cite{fcwp3}.  In this
application the quark masses cannot be used as parameters because they
appear in both the wave functions and the Clebsch-Gordon coefficients
of the Poincar\'e group.

The example in section three illustrates the simplicity and power of
the method. In this example the desired unitary scattering equivalence
is an easily constructed rank-one perturbation of the identity.
The only required input is the new state vector $\vert \bar{m}_{\pi}
\rangle$ and the overlap $\langle m_\pi \vert \bar{m}_{\pi} \rangle$.
The new representation does not require constituent quark/antiquark  
form factors.

The new model $M'$ is designed to be consistent with the meson spectrum
and the pion form factor.  For a given choice of $\Delta$ the model makes
predictions for elastic and transition form factors involving other
mesons which can (in principle) be tested against experiment. 
In addition, the transformed operators might be useful in 
formulating many-body constituent quark models of hadrons.   

The example started with relativistic constituent quark model with a
light-front kinematic symmetry, and produced a new relativistic model
with the same light-front kinematic symmetry, same mass spectrum,
where the pion form factor can be computed in point-quark impulse
approximation.  This example illustrates that constraints imposed by
the choice of kinematic subgroup and mass spectrum do not determine
the form factors.

Scattering equivalences are also known to exist \cite{wp} between
models with different kinematic subgroups.  These relationships can
also be exploited to construct equivalent current operators for models
with different forms of dynamics.  While the ambiguities in
representations of the dynamics are sometimes considered a liability,
this paper shows that they lead to a flexibility that can lead to a
simplification of the dynamics.

\begin{acknowledgments}
This work supported 
in part by the U.S. Department of Energy, under contract DE-FG02-86ER40286
\end{acknowledgments}

%\begin{verbatim}
\appendix
\section{}
%\section{}
%\end{verbatim}

%\section{Appendix} 
%\begin{appendix} 

The relativistic interpretation of the Carlson, Kogut, Pandharipande
semi-relativistic Hamiltonian is discussed in this appendix.

A relativistic quantum mechanical model is defined by 
a unitary representation of the Poincar\'e group acting 
on a model Hilbert space. The mass operator of this 
representation is necessarily a dynamical operator.

A representation of the quark-antiquark Hilbert space and a
representation of the Poincar\'e Lie algebra where the mass operator
is the semi-relativistic Carlson, Kogut, and Pandharipande Hamiltonian
is exhibited in this section.  The generators are chosen so the
generators of transformations that leave the light front $x^+=0$
invariant are kinematic.  This defines a relativistic light-front 
dynamics.

The Hilbert space in this model is the tensor product of the mass $m_q$ spin ${1
\over 2}$ irreducible representation of the Poincar\'e group
associated with the quark and the mass $m_{\bar{q}}$ spin ${1 \over
2}$ irreducible representation of the Poincar\'e group associated with
the antiquark.  The light-front components of the 
single particle momenta $p_i:= (p^+_i, p^1_i, p^2_i)
= (p^+_i, \vec{p}_{i \perp} )$ and the 3-component of the 
light front spin $j^3_i$ are a maximal set of commuting observables on the 
single quarks spaces.  The model Hilbert space is the tensor product of 
the space of square integrable functions of $(p_i, \mu_i)$;
\beq 
\psi_q (p_q, \mu_q) \otimes \psi_{\bar{q}} (p_{\bar{q}} , \mu_{\bar{q}})
\label{eq:DA}
\eeq

The tensor product of irreducible representations of the 
Poincar\'e group is reducible.  The Clebsch-Gordon 
coefficients of the Poincar\'e group are \cite{bkwp}
\[
\langle p_q, \mu_q, p_{\bar{q}}, \mu_{\bar{q}} 
\vert P, \mu (k,j; l,s ) \rangle
=
\]
\[
\delta (P-p_1 -p_2) {\delta (k -k(p_1,p_2)) \over \vert k^2 \vert} 
\sqrt{{ \omega_1 (k) \omega_2 (k) P^+ \over p_1^+ p_2^+ 
m(k)}} 
\]
\[ 
\times D^{1/2}_{\mu_1 \mu_1'} [B_f^{-1} (k_1) B_c (k_1)] 
D^{1/2}_{\mu_2 \mu_2'} [B_f^{-1} (k_2) B_c (k_2)] Y^l_{\mu_l} (\hat{k} )
\]
\beq
\times \langle {1 \over 2}, \mu_1,{1 \over 2} , \mu_2 \vert s, \mu_s \rangle
\langle l, \mu_l, s, \mu_s \vert j, \mu \rangle  
\label{eq:DB}
\eeq
where 
\beq
m=m(k)= \sqrt{k^2 +m_q^2} + \sqrt{k^2 +m_{\bar{q}}^2}
\label{eq:DC}
\eeq
is replaced by the kinematic relative momentum, $k$.
$B_f (p)$ and $B_c (p)$ are light-front and canonical 
Lorentz boosts respectively, and the momenta $k_i$ are defined by
\beq
k_i = B^{-1}_f (P) p_i 
%\]
%\beq
%\hat{z} \times \vec{k} = {p_2^+ \over P^+} \hat{z} \times \vec{p}_{1}  -  
%{p_1^+ \over P^+} \hat{z} \times \vec{p}_2
\label{eq:DD}
\eeq
The discrete indices $l,s$ label multiple copies of the same irreducible
representation that appear in the tensor product.  From the structure 
of the Clebsch-Gordon coefficient above it is obvious that $s\in {0,1}$ and
$\vert j-s \vert \leq l \leq j+s$.  

The Clebsch-Gordon coefficients lead to a representation of the quark
antiquark Hilbert space as the space of square integrable functions of
the variables $( P^+,\vec{P}_{\perp} , \mu , k,j, l,s )$.  Vectors 
in the Hilbert space are square integrable functions of theses variables
\beq
\psi (P,\mu,k,j,l,s) = \langle   P^+,\vec{P}_{\perp} , \mu , k,j, l,s
\vert \psi \rangle.
\label{eq:DE}
\eeq

The Carlson, Kogut and Pandharipande semi-relativistic Hamiltonian is 
embedded in this space as follows:
\[
\langle P, \mu (k,j;l,s) \vert M \vert P', \mu' (k',j';l',s')
\rangle =
\]
\beq 
\delta (P-P') \delta_{jj'} \delta_{\mu \mu'} 
\langle k,l,s \Vert M^j \Vert k',l',s' \rangle 
\label{eq:DF}
\eeq
This defines $M$ in terms of the kernel, 
$\langle k,l,s \Vert M^j \Vert k',l',s' \rangle $,
of the Carlson, Kogut and Pandharipande semi-relativistic Hamiltonian.   
This kernel is identified with the semi-relativistic Hamiltonian 
given by (\ref{eq:CA}) and (\ref{eq:CB}).

Given this mass operator the following operators \cite{bkwp}
\beq
\vec{E}= -i P^+ {\partial \over \partial \vec{P}_{\perp}}  
\qquad
K_3= -i P^+ {\partial \over \partial P^+}  
\label{eq:DH}
\eeq
\beq
J_3= j_3 - {1 \over P^+} \hat{z} \cdot ( \vec{P} \times \vec{E} )
\label{eq:DI}
\eeq
\beq
P_- = {\vec{P}_{\perp} \cdot \vec{P}_{\perp}  - M^2 \over P^+}
\label{eq:DJ}
\eeq
\[
\vec{J}_{\perp} =
\]
\beq
{1 \over P^+}\left ( {P^+ - P^-  \over 2} (\hat{z} \times \vec{E} )  
- (\hat{z} \times \vec{P} )K_3 + \vec{P}_{\perp}j_3 + 
M\vec{j}_{\perp}\right )
\label{eq:DK}
\eeq
along with  the multiplication operators 
$P^+$ and $\vec{P}_{\perp}$, define a set of self-adjoint operators 
on the space of square integrable functions of
$\{ P,\mu,k,j,l,s\}$ that satisfy the Poincar\'e Lie algebra
with the Carlson Kogut and Pandharipande mass operator (\ref{eq:DF}),
(\ref{eq:CA}) and (\ref{eq:CB}).

The interaction in the above expressions appears in both $P^-$ and
$\vec{J}_{\perp}$.  The remaining generators, which generate
transformations that leave the light front invariant, do not involve
$M$ thus are kinematic.  It is clear that the same properties hold if $M$
is replaced by $M'$.  Unlike Fock space motived models
\cite{bertrand}, 
the Poincar\'e symmetry is exact.

Note that to compute current matrix elements with an impulse current
it is necessary to use the Clebsch Gordon coefficients (\ref{eq:DB})
to transform the eigenstates of $M$ and $M'$ to single quark
variables.  Because the Clebsch Gordon coefficients depend on the
quark masses, (\ref{eq:DC}),  the quark masses were not allowed to
vary in determining $\bar{M}$.  

%\end{appendix}

%\begin{figure}
%\begin{center}
%\rotatebox{00}{\resizebox{4.0in}{!}{
%\includegraphics{es1.eps}}
%}
%\end{center}
%\caption{Pion form factor at low $Q^2$}
%\end{figure}
 
%\begin{figure} 
%\begin{center}
%\rotatebox{00}{\resizebox{4.0in}{!}{
%\includegraphics{es2.eps}}
%}
%\caption{Pion form factor at high $Q^2$}
%\end{center}
%\end{figure}
 
%\begin{figure} 
%\begin{center}
%\rotatebox{00}{\resizebox{4.0in}{!}{
%\includegraphics{psik-360.eps}}
%}
%\caption{Comparison of k-space wave functions}
%\end{center}
%\end{figure}
 
%\begin{figure} 
%\begin{center}
%\rotatebox{00}{\resizebox{4.0in}{!}{
%\includegraphics{psir-360.eps}}
%}
%\caption{Comparison of r-space wave functions}
%\end{center}
%\end{figure}
 
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D{\bf 28},2807(1983). 
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\bibitem{salme}F. Cardarelli et al. Phys. Lett. B {\bf 332},1(1994); 
{\bf 357} 267 (1995).
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Rev. C{\bf 69},035212 (2004). 
\bibitem{bertrand1}B. Desplanques, nucl-th/0405060.
\bibitem{krutov} A. F. Krutov and V. E. Troitsky, 
Theor. Math. Phys.  {\bf 116}, 219 (1998).
\bibitem{sw} R. F. Streater and A. S. Wightman, PCT and Spin and Statistics, and all that, W. A. Benjamin, 1964. 
\bibitem{dick} R. J. Furnstahl, H. W. Hammer, Negussie Tirfessa,
Nucl. Phys. A{\bf 689},846-868(2001). 
\bibitem{fcwp1} F. Coester and W. N. Polyzou, Phys. Rev. D{\bf 26}, 1348(1982). \bibitem{fcwp2} P. L. Chung, F. Coester and W. N. Polyzou, 
Phys. Lett. B {\bf 205}, 545 (1988).
\bibitem{fcwp3} F. Coester and W. N. Polyzou,  nucl-th/0405082.
\bibitem{kp}B. D. Keister and W. N. Polyzou, J. Comp. Phys. {\bf 134}, 
231 (1997).
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\bibitem{volmer} J. Volmer et al. Phys. Rev. Lett. {\bf 86} , 1713  (2001).
\bibitem{bertrand} A. Amghar, B. Desplanques and L.Theussl, 
Phys. Lett. B {\bf 574} , 201 (2003).
\end{thebibliography}

\vfill\eject 

\begin{figure*}
\rotatebox{00}{\resizebox{7.5in}{!}{
\includegraphics{es1.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Pion form factor at low $Q^2$.}
\end{figure*}

\vfill\eject 

\begin{figure*}
\rotatebox{00}{\resizebox{7.5in}{!}{
\includegraphics{es2.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Pion form factor at high $Q^2$.}
\end{figure*}

\vfill\eject 

\begin{figure*}
\rotatebox{00}{\resizebox{7.5in}{!}{
\includegraphics{psik-360.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Comparison of k-space wave functions.}
\end{figure*}

\vfill\eject 

\begin{figure*}
\rotatebox{00}{\resizebox{7.5in}{!}{
\includegraphics{psir-360.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}Comparison of r-space wave functions.}
\end{figure*}

\begin{figure*}
\rotatebox{00}{\resizebox{7.5in}{!}{
\includegraphics{low_pi_star.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}$\pi^*$ form factor at low $Q^2$.}
\end{figure*}

\begin{figure*}
\rotatebox{00}{\resizebox{7.5in}{!}{
\includegraphics{high_pi_star.eps}}}% Here is how to import EPS art
\caption{\label{fig:epsart}$\pi^*$ form factor at high $Q^2$.}
\end{figure*}

\end{document}

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Nucl. Phys. A{\bf 689},846-868(2001). 
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Phys. Lett. B {\bf 205}, 545 (1988).
\bibitem{fcwp3} F. Coester and W. N. Polyzou,  nucl-th/0405082.