%\documentstyle[preprint,prc,eqsecnum,aps,epsf]{revtex}
%\documentclass[preprint,showpacs,eqsecnum]{revtex4}
\documentclass[showpacs,eqsecnum,aps]{revtex4}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{graphicx,color}
\pdfminorversion=5
%\usepackage{bm}
% \usepackage{graphics}
% \usepackage[dvips]{color}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\beqarray{\begin{eqnarray}}
\def\eeqarray{\end{eqnarray}}
\def\bsig{\mbox{\boldmath$\sigma$}}
\def\btheta{\mbox{\boldmath$\theta$}}
\def\bphi{\mbox{\boldmath$\phi$}}
\def\btau{\mbox{\boldmath$\tau$}}
\def\brho{\mbox{\boldmath$\rho$}}
\def\<{{\langle}}
\def\>{{\rangle}}
\def\q{{\mathbf q}}
\def\Q{{\mathbf Q}}
\def\p{{\mathbf p}}
\def\P{{\mathbf P}}
\def\v{{\mathbf v}}
\def\V{{\mathbf V}}
\def\B{{\mathbf B}}
\def\b{{\mathbf b}}
\def\d{{\mathbf d}}
\def\k{{\mathbf k}}
\def\x{{\mathbf x}}
\def\fq{{\tilde{\mathbf q}}}
\def\fQ{{\tilde{\mathbf Q}}}
\def\fp{{\tilde{\mathbf p}}}
\def\fP{{\tilde{\mathbf P}}}
\def\fB{{\tilde{\mathbf B}}}
\def\fb{{\tilde{\mathbf b}}}
\def\F{{\cal F}}
\def\half{{\frac{1}{2}}}
\usepackage{graphicx,color}

\begin{document}

\title{Model Tests of Cluster Separability In Relativistic Quantum Mechanics}

\author{B. D. Keister}
\affiliation{
Physics Division, National Science Foundation, Arlington, VA 22230
}

\author{W. N. Polyzou}
\affiliation{
Department of Physics and Astronomy, The University of Iowa, Iowa City, IA
52242}

\vspace{10mm}

\date{\today}

\begin{abstract}

  A relativistically invariant quantum theory first advanced by
  Bakamjian and Thomas has proven very useful in modeling few-body
  systems.  For three particles or more, this approach is known
  formally to fail the constraint of cluster separability, whereby
  symmetries and conservation laws that hold for a system of particles
  also hold for isolated subsystems.  Cluster separability can be
  restored by means of a recursive construction using unitary
  transformations, but implementation is difficult in practice, and
  the quantitative extent to which the Bakamjian-Thomas approach
  violates cluster separability has never been tested.  This paper
  provides such a test by means of a model of a current operator in a
  three-particle system for which (1) it is simple enough that there
  is a straightforward solution that satisfies Poincar\'e invariance
  and cluster separability, and (2) one can also apply the
  Bakamjian-Thomas approach.  The difference between these
  calculations provides a measure of the size of the corrections
  needed to restore cluster properties. Our estimates suggest that, in
  models based on nucleon degrees of freedom, the corrections that
  restore cluster properties are too small to effect calculations of
  observables.

\end{abstract}

\vspace{10mm}

\pacs{21.45+v}

\maketitle


\section{Introduction}

There are two distinct requirements for describing quantum mechanical
systems of particles under the requirements of special relativity.
The first requirement is Poincar\'e invariance: probabilities,
expectation values and ensemble averages for equivalent experiments
performed in different inertial frames are identical. A necessary and
sufficient condition for a quantum theory to be Poincar\'e invariant
is that the dynamics is described by a unitary  representation 
of the Poincar\'e group \cite{Wigner:1939cj}.  The
second requirement is cluster separability: isolated subsystems must
have the same observable properties as they would in a framework in
which the other ``spectator'' particles are absent entirely. 
This requirement justifies tests of special
relativity on isolated subsystems.  It applies both to systems of
particles interacting among themselves (e.g.~via the strong
interaction) and to the current operators that allow them to interact
with electroweak fields.

In local quantum field theory, both of these properties of the theory
follow as a consequence of the covariance, spectral properties and the
local commutation relations of the field operators.  Formal solutions
of field theories satisfy both requirements.  Truncations or
approximations may or may not be consistent with these
requirements, which must be verified explicitly.

These physical requirements also apply to Poincar\'e invariant quantum
mechanical models of systems with a finite number of degrees of
freedom.  The problem of constructing dynamical representations of the
Poincar\'e group was studied by Dirac~\cite{Dirac:1949cp} from an
algebraic perspective, using Poisson brackets.  He concluded that at
least three of the infinitesimal generators of the Poincar\'e group
must include interactions, although he did not solve the non-linear
problem of how to add interactions that preserve the Poincar\'e
commutation relations.  Bakamjian and Thomas~\cite{Bakamjian} provided
a solution to this problem for a system of two interacting particles.

Many realistic, Poincar\'e invariant, quantum mechanical models of
strongly interacting few-body systems are based on a generalization of
the Bakamjian-Thomas construction.  This construction can be used to
formulate models of systems of arbitrary numbers of particles and
systems that do not conserve particle number.  A representative sample
of few-body applications of Poincar\'e invariant quantum mechanics
based on the Bakamjian-Thomas construction include relativistic
constituent quark models
\cite{Melde:2008yr,Krutov:1999zz,Coester:2005cv,Sengbusch:2004sf,Boffi:2001zb},
relativistic few-nucleon models
\cite{Lin:2007kg,Witala:2004pv,Witala:2005nw,Witala:2008va,Witala:2011yq}
relativistic models involving electromagnetic probes
\cite{Huang:2008jd,Keister:1996ny,Keister:1988zz}, and relativistic
models with particle production \cite{Fuda:2009zz,Kamano:2008gr}.

The virtue of the Bakamjian-Thomas construction is that it provides a
means for constructing Poincar\'e generators for systems of
interacting particles.  The limitation of this construction is that
for systems of more than two particles, the dynamical representation of
the the Poincar\'e generators does not become a tensor product on
states representing asymptotically separated subsystems.  While this
limitation does not lead to observable consequences in the three-body
$S$ matrix or bound state observables, there are observable
consequences when the three-body system is embedded in the
four-particle Hilbert space, as it is in four-body problems or when an
electroweak probe interacts with a three-body system.

The most ambitious applications of the Bakamjian-Thomas construction
are relativistic Faddeev
calculations~\cite{Lin:2007kg,Witala:2004pv,Witala:2005nw,Witala:2008va,Witala:2011yq}
of three-nucleon scattering which have been performed using realistic
nucleon-nucleon forces and three-nucleon forces.  The next step in
developing a useful relativistic few-body theory would be to model
four-nucleon systems or electron scattering from a three-nucleon
system.  In both of these applications a three-nucleon subsystem is
embedded in a four-particle Hilbert space, and, for the first time,
there is the possibility of observable consequences of violations of
cluster properties.

The restoration of cluster properties can be achieved through an
inductive construction due to Sokolov
\cite{Sokolov:1977,Coester:1982vt}.  The Sokolov construction consists
of a hierarchy of unitary transformations, each of which preserves the
$S$ matrix for an associated subsystem.  The Bakamjian-Thomas
representation is retrieved by setting all of the unitary
transformations to the identity operator.  The Sokolov construction is
non-trivial and requires and is interaction dependent.  An isolated
three-body system is a special case where the cluster property can be
achieved by means of a single overall unitary transformation that
preserves the $S$ matrix.  For systems of more than three particles
there are observable differences between the two representations.  The
interested reader can find a complete discussion in
\cite{Coester:1982vt}.  

Actual construction of the Sokolov hierarchy of unitary transformation
is sufficiently complicated that the technique has never been used in
realistic calculations.  Before undertaking computationally intensive
four-body calculations, one would like to know the importance of the
corrections required by cluster properties.  While
Ref.~\cite{Coester:1982vt} argued that these corrections should be
small in nuclear physics applications, this was never quantified in
any model calculations.  An additional investigation is needed to
determine if these corrections can be ignored, can be treated
perturbatively, or must be treated exactly.  To address this question
we construct a model involving a current matrix element for a system
of three particles.  The model is simple enough that Poincar\'e
invariance and cluster separability are easily satisfied; this unusual
pair of features is due to the simplicity of the test model.  At the
same time, the model permits a Bakamjian-Thomas construction that
illustrates the quantitative impact of the breakdown of cluster
separability.  The conclusion of our preliminary analysis suggests
that the corrections required by cluster properties are too small to
be observable in nuclear physics applications.  The corrections are
more important for models based on sub-nucleon degrees of freedom.

\section{Test Model}

Poincar\'e invariance and the Dirac forms of dynamics for
Bakamian-Thomas ($BT$) constructions are discussed extensively in
Ref.~\cite{Keister:1991sb}.  The framework provided here makes use of
that discussion.  We approach the model in a heuristic way in order to
illustrate the issue of cluster separability combined with Poincar\'e
invariance with a minimum of formal development.

The simplest four-body model where we might observe the breakdown of
cluster separability consists of a three-body system interacting with
an electromagnetic probe.  In the one-photon exchange approximation
the relevant dynamical quantities are matrix elements of a current
density between eigenstates of the three-body four-momentum operator.

In our model we make the following simplifications.  First, we assume
that all of the particles are spinless. Second, we replace the
four-vector current density by a scalar current density.  We will see
that, even with these simplifications, there are differences in the
current matrix elements calculated using three-body eigenstates of a
$BT$ four-momentum operator compared to current matrix elements
calculated using three-body eigenstates of a four momentum operator
that clusters.  Our simplifying assumptions show that the violations
of cluster properties has nothing to do with spin or choices or the
orientation of the current.

The scalar current operator, $j(x)$, is assumed to be a one-body
operator.  The three-particle system consists of a bound pair (which
we label the 1-2 subsystem) and a third particle (3) that does not
interact with particles 1 or 2.  Only particle 3 has a ``charge'' with
respect to $j(x)$.  Therefore, the (12) subsystem acts as a spectator
with respect to the action of the current operator.  This can be
illustrated as a disconnected graph as shown in Fig.~\ref{fig:graph}.

\begin{figure*}[h]
\includegraphics[height=5cm]{current1+2.pdf}
    \caption{Graph of one-body current plus bound two-body spectator.}
    \label{fig:graph}
\end{figure*}

The three-body eigenstates that appear in the current matrix element
can be treated as eigenstates of the four-momentum of the tensor
product of a one-body and two-body Bakamjian-Thomas unitary
representation of the Poincar\'e group.  Alternatively they can be
treated as eigenstates of the four-momentum of a three-body
Bakamjian-Thomas unitary representation of the Poincar\'e group.
These representation are related by a unitary transformation of the
type that appears in the Sokolov construction.  Comparing the results
of the two calculations provides a measurement of the impact of the
unitary transformations that restore cluster properties.

\subsection{Tensor-Product ($TP$) Model:}
In this model the three-body eigenstates are tensor products of (12) and 3 eigenstates:
\begin{equation}
  \label{eq:3a}
  | \lambda_{12}, m ; \p_{12}, \p_3 \> = | \lambda_{12}; \p_{12} \>
\otimes| m,\p_3  \> .
\end{equation}
Since the current $j(x)$ acts only in the space of particle 3, 
the matrix element has the form
\begin{equation}
  \label{eq:m5}
  \< \lambda_{12} m; \p_{12}', \p_3' | j(0) | \lambda_{12} m; \p_{12} ,\p_3 \>
  = \delta(\p_{12}' - \p_{12}) f(\q^2),
\end{equation}
where
\begin{equation}
  \label{eq:m5a}
  \<  m;  \p_3' | j(0) |  m;  \p_3 \>
  = f(\q^2); \quad \q = \p_3' - \p_3,
\end{equation}
$\lambda_{12}$ is a bound-state eigenvalue of $M_{12}$, 
and $\p_{12}$ is its three-momentum.

If we specify initial momenta $\p_3$ and $\p_{12}$ and momentum
transfer $\q$, and integrate the matrix element over the final
spectator momentum $\p_{12}'$, the integral collapses due to the
spectator momentum delta function, and we have
\begin{equation}
  \label{eq:m6}
  \F_{TP} := \int d\p_{12}' \< \lambda_{12}, m; \p_{12}', \p_3' | j(0) |
  \lambda_{12}, m; \p_{12}, \p_3 \> = f(\q^2).
\end{equation}

Equation~\ref{eq:m6} is the Tensor-Product ($TP$) result.  It has no
dependence upon the momentum $\p_{12}$ of the bound-state spectator,
as expected from the physical requirement of cluster separability, and
it does not depend upon the specific values of $\p_3$ and $\p_3'$, as
long as $\p_3' - \p_3 = \q$.  This result also has the correct
properties of relativistic invariance.

\subsection{Discussion}

The model is a simple yet representative test because it has a
straightforward solution satisfying Poincar\'e invariance and cluster
separability (Eq.~\ref{eq:m6}), without the complications of the full
Sokolov construction.  Note that a full few-body system does not have
a simple solution that exhibits both of these properties.  At the same
time, the model is sufficiently non-trivial that it also permits an
analysis using the $BT$ scheme that provides a quantitative measure of
the violation of cluster separability that one would expect in more
complicated and realistic systems where a $BT$ approach would be
essential.  The violations of cluster properties are due to the size
of the difference between the unitary transform that relates the $2+1$
$BT$ representation to the $TP$ representation just
described.
 
In the following sections, we evaluate the same matrix element using
$BT$ representations of the three-particle system corresponding to
different forms of dynamics~\cite{Dirac:1949cp}.  The purpose of this
is to examine the quantitative sensitivity of the size of violations
of cluster properties in models based on each of Dirac's three forms
of dynamics.  These current matrix elements will each depend upon the
structure of the bound state, in violation of cluster separability,
and our goal is to examine the magnitude of that violation.  In all
cases, we specify the momentum transfer $\q$ and bound state momentum
$\p_{12}$ for a given calculation and then vary these momenta for
sensitivity tests.

\subsection{$BT$ Representation: Instant Form}

We now evaluate current matrix element in Eq.~\ref{eq:m5} using 
an instant-form $BT$ model.  Instant-form models have no
interactions in the generators of rotations and space translations. 

First, we change variables:
We replace the bound state and particle~3 momenta by the total
momentum of the system and the momentum of particle~3 Lorentz
transformed to the three-particle rest frame:
\begin{equation}
  \label{eq:b1}
  (\p_{12}, \p_3) \to (\P, \p),
\end{equation}
where 
\begin{equation}
  \label{eq:b2}
  \P = \p_{12} + \p_3;\quad \p = \p_3 + \Phi(\p_3,\P,M_\lambda)\P,
\end{equation}
and
\begin{equation}
  \label{eq:b3}
  \Phi(\p_3,\P,M_\lambda) =\frac {1}{ M_\lambda} 
  \left[ \frac {\P\cdot\p_3}{ E_\lambda + M_\lambda} - \omega_m(\p_1)\right];
  \quad 
M_\lambda =\sqrt{E_\lambda^2 - \P^2}; \quad
E_\lambda = \sqrt{m^2 + \p_3^2} + \sqrt{\lambda^2 + \p_{12}^2}.
\end{equation}
With this variable change the relation between the bound state 
and particle~3 in these bases is:
\begin{equation}
  \label{eq:b4}
  | \lambda , m; \p_{12}, \p_3 \> 
  = \left| \frac{\partial(\P ,\p)}{\partial(\p_{12},\p_3)}\right|^\half_{M_\lambda}
  | \lambda ,m; \P, \p \>.
\end{equation}

We also introduce single-particle momenta $\b_1, \b_2, \b_3 $.  
We change variables to the total momentum
of the two-particle pair (12) and the momentum of particle 1 Lorentz transformed to
the rest frame of the two-nucleon subsystem using:
\begin{equation}
  \label{eq:b6}
  \k = \b_1 + \Phi(\b_1,\b_{12},M^0_{12})\b_{12};\quad
  \b_{12} = \b_1 + \b_2;\quad M_{12} = 2\sqrt{m^2 + \k^2}.
\end{equation}
The three-nucleon invariant mass is
\begin{equation}
  \label{eq:b7}
  M_k := \sqrt{(M^0_{12})^2 + \p^2} + \sqrt{m^2 + \p^2}
\end{equation}
and the momentum of particle~3 Lorentz boosted to the rest frame
of the three-body system is:
\begin{equation}
\p = \p_3 + \Phi(\p_3,\P,M_k)\P,
\label{eq:b7a}
\end{equation}
where
\begin{equation}
  \label{eq:7b}
  \Phi(\p_3,\P,M_k)\P =\frac {1}{ M_k} 
  \left[ \frac {\P\cdot\p_3}{ E_k + M_k} - \omega_m(\p_1)\right];
  \quad 
%M_k =\sqrt{E_k^2 - \P^2}; \quad
E_k = \sqrt{m^2 + \p_3^2} + \sqrt{M_{12}^2 + \p_{12}^2}.
\end{equation}

In an instant-form Bakamjian Thomas dynamics, the connection between a
state with three free particles and that with particle~3 plus bound state is
\begin{equation}
  \< \lambda; \P', \p' \vert  \P, \p, \k\> =
  \delta (\P'-\P) \delta (\p' - \p) 
  \phi_\lambda (\k),
  \label{eq:7c}
\end{equation}
Note that $\p$ depends upon the free two-particle mass $M_{12}^{(0)}$
via Eq.~\ref{eq:b7a}, while $\p'$ depends upon the bound state mass
eigenvalue via $\lambda$ Eq.~\ref{eq:b2}.  The association of these
two three-momenta in the delta function of Eq.~\ref{eq:7c} is
therefore not fully consistent.  There is no observable consequence
for the two-body $S$ matrix or mass eigenvalues, but this association
will lead to a result that violates cluster separability. 
 
We now make use of Eq.~\ref{eq:7c} to compute the 
$BT$ counterpart to $\F_{TP}$ that was defined by Eq.~\ref{eq:m6}:
\begin{eqnarray}
  \label{eq:b11}
  \F_{BT} &:=& 
  \int   d\p_{12}'
  d\b_{12}' d\b_3'  d\k' 
  d\b_{12} d\b_3 d\k 
  \\ \nonumber
  &&\quad\times
  \left| \frac{\partial(\P', \p')}{\partial(\p_{12}',\p_3')}\right|^\half_{M_\lambda}
  \left| \frac{\partial(\P, \p)}{\partial(\p_{12},\p_3)}\right|^\half_{M_\lambda}
  \left| \frac{\partial(\b_{12}',\b_3')}{\partial(\P', \p')}\right|^\half_{M_k}
  \left| \frac{\partial(\b_{12},\b_3)}{\partial(\P, \p)}\right|^\half_{M_k}
  \\ \nonumber
  &&\quad\times
\phi_\lambda^*(\k')
  \< \b_{12}', \b_3', \k'  | j(0) | \b_{12}, \b_3, \k \>.
  \phi_\lambda(\k)
\end{eqnarray}
Since the current operator $j$ operates only in the space of particle~3, we have
\begin{equation}
  \label{eq:b12}
  \< \b_{12}', \b_3', \k'  | j(0) | \b_{12}, \b_3, \k \>
  = \delta(\k' - \k) \delta(\b_{12}' - \b_{12}) f[(\b_3' - \b_3)^2].
\end{equation}
The integral over $\p_{12}'$ can be converted 
to an integral over $\b_{12}'$ by means of Jacobians:
\begin{equation}
  \label{eq:b13}
  \int d\p_{12}' = \int d\b_{12}' 
  \left| \frac{\partial(\P', \p')}{\partial(\b_{12},'\b_3')}\right|_{M_k}
  \left| \frac{\partial(\p_{12}',\p_3')}{\partial(\P', \p')}\right|_{M_\lambda}.
\end{equation}
The final result is
\begin{equation}
  \label{eq:b14}
    \F_{BT}^{\rm instant} := 
    \int d\k  \left| \frac{\partial(\p_{12}',\p_3')}{\partial(\P', \p')}\right|^\half_{M_\lambda}
  \left| \frac{\partial(\P, \p)}{\partial(\p_{12},\p_3)}\right|^\half_{M_\lambda}
  \left| \frac{\partial(\P', \p')}{\partial(\b_{12}',\b_3')}\right|^\half_{M_k}
  \left| \frac{\partial(\b_{12},\b_3)}{\partial(\P, \p)}\right|^\half_{M_k}
  \left|\phi_\lambda(\k)\right|^2
  f[(\b_3' - \b_3)^2].
\end{equation}
The interesting observation is that this integral has a non-trivial
dependence on momentum, $\p_{12}$, of the bound state, 
in contrast to the $\p_{12}$ independence of
the tensor product result.  In this model the $BT$ and tensor product
representations are related one of Sokolov's unitary transformations,
$A_{(12)(3)}$, that preserve the three-body $S$-matrix:
\begin{equation}
| \lambda , m; \p_{12}, \p_3 \>_{TP} = A_{(12)(3)} | \lambda , m; \p_{12}, \p_3 \>_{BT}.  
\label{eq:b14.a}
\end{equation}
The scale of the $\p_{12}$ dependence in \ref{eq:b14} provides a
measure of the size of the violation of cluster properties that
results from ignoring $A_{(12)(3)}$ by replacing it with the unit operator.  
Since we know both states
in eq~\ref{eq:b14.a} we do not have to calculate 
$A_{(12)(3)}$ explicitly to determine its impact.

For this calculation, we vary the three-momentum transfer 
$\q$ and the spectator momentum $\p_{12}$.  The initial
momentum of particle 3 and the final momentum of the 
system are fixed in terms of these variables:
\begin{itemize}
\item $\p_3 = -\half\q$;
\item $\P' = \P + \q$.
\end{itemize}
Since in the instant form
\begin{equation}
  \label{eq:b15}
  \P = \p_{12} + \p_3 = \b_{12} + \b_3;\quad
  \P' = \p_{12}' + \p_3' = \b_{12}' + \b_3',
\end{equation}
and the current matrix element constrains $\b_{12}' = \b_{12}$, we
find that $\b_3' = \b_3 + \q$, in which case the argument of the form
factor is $\q^2$ as it is in the $TP$ case.  However, the final momenta
$\p_{12}'$ and $\p_3'$ are not constrained, and in general there are
non-vanishing contributions to this matrix element for $\p_3' \ne
\p_3 + \q$ and $\p_{12}' \ne \p_{12} + \q$.

In the nonrelativistic limit, where $\k$, $\q$ and $\p_{12}$ are all
small with respect to the relevant masses, the Jacobians are
approximately unity and can be factored out of the integral, leaving a
unit wave function normalization and a result identical to the $TP$
case.  The quantitative level of disagreement with the $TP$ result is
therefore linked to the extent to which the model goes beyond the
nonrelativistic limit.

\subsection{$BT$ Representation: Front Form}

Dirac's front-form dynamics is described in detail in 
Ref.~\cite{Keister:1991sb}.  In a front-form dynamics 
the generators of transformations that 
leave a plane tangent to the light cone invariant are free of interactions.
We provide a summary here.  

Basis states in the front form are described by light-front momenta 
\begin{equation}
  \label{eq:f1}
  \fp = (\p_\perp,p^+);\quad p^+ = p^0 + p^3.
\end{equation}
These generate translations in a hyperplane $x^+=0$ tangent to the light cone.

In the front form the Lorentz transformations used to define a
particle momentum in the rest frame of the three-particle system in
the instant form are replaced by boosts that leave the light front
invariant.  While these boosts are interaction independent, there is
an interaction dependence in the boost parameters.  The Bakamjian
Thomas model again identifies the momentum of particle~3 in the rest
frame of the three-particle system.  As in the instant-form case this
is an incorrect identification that has no consequences for the
two-body $S$ matrix.  With this modification the front-form result has
the same structure as the instant-form result:

\begin{equation}
  \label{eq:f2}
    \F_{BT}^{\rm front} := \int d\k
    \left| \frac{\partial(\fp_{12}',\fp_3')}{\partial(\fP', \fp')}\right|^\half_{M_\lambda}
    \left| \frac{\partial(\fP, \fp)}{\partial(\fp_{12},\fp_3)}\right|^\half_{M_\lambda}
        \left| \frac{\partial(\fP', \fp')}{\partial(\fb_{12}',\fb_3')}\right|^\half_{M_k}
    \left| \frac{\partial(\fb_{12},\fb_3)}{\partial(\fP, \fp)}\right|^\half_{M_k}
    \left|\phi_\lambda(\k)\right|^2
    f[(\fb_3' - \fb_3)^2].
\end{equation}
This quantity has an unphysical dependence on $\fb_{12}$ that does
not occur in the tensor product. 

For this calculation, we vary $\fq$ and $\fp_{12}$, with coordinate
axes chosen such that $q^+ = 0$ and $p_{12}^+ = 0$.  We define
$\p_\perp$ and $\fP'$ in terms of these quantities:
\begin{itemize}
\item $\p_\perp = -\half\q_\perp$;
\item $\fP' = \fP + \fq$.
\end{itemize}
Analogous to the discussion following Eq.~\ref{eq:b15}, we have that 
\begin{equation}
  \label{eq:f3}
  \fP = \fp_{12} + \fp_3 = \fb_{12} + \fb_3;\quad
  \P' = \fp_{12}' + \fp_3' = \fb_{12}' + \fb_3',
\end{equation}
The current matrix element constrains $\fb_{12}' = \fb_{12}$, and
therefore $\fb_3' = \fb_3 + \fq$, in which case the argument of the
form factor is $\q^2$ as it is in the $TP$ case.  However, the final
momenta $\fp_{12}'$ and $\fp_3'$ are not constrained, and in general
the integral has non-zero contributions from
$\fp_3' \ne \fp_3 + \fq$ and $\fp_{12}' \ne \fp_{12} + \fq$.

\subsection{$BT$ Representation: Point Form}

Dirac's point-form dynamics are also described in detail in
Ref.~\cite{Keister:1991sb}.  In this case the Lorentz group
is non-interacting.  We provide a summary here.

Basis states in the point form are described by four-velocity vectors
$\v$.  Momenta are obtained by multiplying the four-velocities by
(interacting or non-interacting) masses.  Thus, we seek to evaluate
matrix elements of the current operator $j(x)$ between states of
particle~3 and the (12) bound state with initial four-velocity $\V$
and final four-velocity $\V'$.

The derivations proceed in a fashion similar to the instant form
except that we must take care to use velocities in cases where the
associated masses can acquire interactions.

With these conventions, the point-form 
$\F_{BT}$ has the structure
\begin{equation}
  \label{eq:p1}
    \F_{BT}^{\rm point} := \int d\k
    \left| \frac{\partial(\p_{12}',\p_3')}{\partial(\V' ,\p')}\right|^\half_{M_\lambda}
    \left| \frac{\partial(\V, \p)}{\partial(\p_{12},\p_3)}\right|^\half_{M_\lambda} 
    \left| \frac{\partial(\V', \p')}{\partial(\b_{12}',\b_3')}\right|^\half_{M_k}
    \left| \frac{\partial(\b_{12},\b_3)}{\partial(\V, \p)}\right|^\half_{M_k}
    \left|\phi_\lambda(\k)\right|^2
    f[(\b_3' - \b_3)^2].
\end{equation}

For the instant- and front-form calculations, we chose to fix three
three-momenta in a way that kept the physical momentum transfer in the
particle-3 current matrix element.  For the point form, we provide two
different constraint choices, each analogous to the previous
calculations, but leading to different results.

\begin{description}
\item {A.} Constrain the momentum transfer to particle 3:
\begin{itemize}
\item $\p_3 = -\half\q$;
\item $\p_3' = \p_3 + \q$.
\end{itemize}
Note in this case that the velocities $\V$ and $\V'$ are the same for
the kinematics using the interacting mass $\lambda$ or the free mass
$M_{12}$.  This implies that
\begin{equation}
  \label{eq:p2}
  \b_3 = \p_3';\quad \b_3' = \p_3',
\end{equation}
and therefore that $\b_3' -\b_3 = \q$.  One consequence is that in general, $M_\lambda(\V'-\V) \ne \q$.

\item {B.} Constrain the final velocity to match momentum transfer $\q$.
\begin{itemize}
\item $\p_3 = -\half\q$;
\item $\V' = \V_{\rm calc}$.
\end{itemize}
To obtain $\V_{\rm calc}$, we use temporary quantities
\begin{equation}
  \label{eq:p3}
  \P'_{\rm temp} = \P + \q
\end{equation}
and
\begin{equation}
  \label{eq:p4}
  P'{}^0_{\rm temp} = P^0.
\end{equation}
These are the results that one expects in the tensor-product representation.
From these relations we obtain 
\begin{equation}
  \label{eq:p5}
  M'_{\rm temp} = \sqrt{(P'{}^0_{\rm temp})^2 - \P'_{\rm temp}{}^2},
\end{equation}
and then define
\begin{equation}
  \label{eq:p6}
  \V_{\rm calc} := {\P'_{\rm temp}\over M'_{\rm temp}}.
\end{equation}
Substituting, we get
\begin{equation}
  \label{eq:p7}
  \V_{\rm calc} = {\P + \q \over 
    \sqrt{(P^0)^2 - (\P + \q)^2}}.
\end{equation}
Thus, the three-vector $\V'$ is constrained, but the actual values of
$M'$ and $\P'$ may be different than the temporary values used to
obtain $\V_{\rm calc}$.  In this form,
  \begin{eqnarray}
    \label{eq:p8}
    \V_{\rm int} &=& \V_{\rm ext}; \nonumber \\
    \V'_{\rm int} &=& \V'_{\rm ext},
  \end{eqnarray}
which also implies that
\begin{eqnarray}
  \label{eq:p9}
    \b_3 &=& \p_3; \nonumber \\
    \b_3' &=& \p_3',
\end{eqnarray}
\end{description}

\section{Results and Discussion}
In this section we discuss results of calculations of the form factor
$\F_{BT}$ for instant, front and two point-form kinematic choices.
Initially we consider scales that are relevant for systems of 
nucleons interacting with two-body interactions.  Then we turn to
examples more appropriate to hadron models.

In all figures we show the figure of merit:
\begin{equation}
  \label{eq:p10}
  \frac{(\F_{BT}-\F_{TP})}{\F_{TP}},
\end{equation}
which represents the relative error induced by ignoring the unitary
transformation eq.~\ref{eq:b14.a} that restores cluster
separability.

\subsection{Malfliet-Tjon Deuteron Wave Function}
To model realistic conditions for nuclear physics, we use a
bound-state spectator with deuteron properties constructed from
Malfliet-Tjon~\cite{Malfliet:1968tj} potential IV, which contains both
attractive and repulsive forces.  Ref.~\cite{Keister:1991sb} discusses
how to make a phase-equivalent relativistic model that has the same
wave functions as the non-relativistic model

We first consider the figure of merit as a function of the 
momentum transfer and momentum of the bound state in each of Dirac's forms
of dynamics and for the bound state momentum perpendicular and parallel
to the momentum transfer.  The  results are shown in the six figures below.
In all cases the expected results are given by the flat planes.

The fractional deviation of this $BT$ model calculation from the $TP$
result that satisfies cluster separability is very small for the
instant- and front-form cases, typically of order $10^{-3}$ or smaller
at the highest values of the three-momenta $\q$ and $\p_{12}$.  The
deviations for the point-form Option A are somewhat larger, and larger
still for Option B.  

\begin{figure*}[h]
  \begin{minipage}{0.45\linewidth} \centering
    \includegraphics[width=\linewidth]{instant_3d_parallel.pdf}
    \caption{Model differences for instant-form $BT$ calculation,
$\q\|\p_{12}$.}
    \label{fig:instant-par}
  \end{minipage} \hspace{0.05\linewidth}
  \begin{minipage}{0.45\linewidth} \centering
    \includegraphics[width=\linewidth]{instant_3d_perp.pdf}
    \caption{Model differences for instant-form $BT$ calculation,
$\q\perp\p_{12}$.}
    \label{fig:instant-prp}
  \end{minipage}
\end{figure*}

\begin{figure*}[h]
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{ff_3d_parallel.pdf}
    \caption{Model differences for front-form $BT$ calculation, $\q\|\p_{12}$.}
    \label{fig:front-par}
  \end{minipage}
  \hspace{0.05\linewidth}
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{ff_3d_perp.pdf}
    \caption{Model differences for front-form $BT$ calculation, $\q\perp\p_{12}$.}
    \label{fig:front-prp}
  \end{minipage}
\end{figure*}

\begin{figure*}[h]
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{pf_wp_3d_parallel.pdf}
    \caption{Model differences for point-form $BT$ calculation A, $\q\|\p_{12}$.}
    \label{fig:point-A-par}
  \end{minipage}
  \hspace{0.05\linewidth}
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{pf_wp_3d_perp.pdf}
    \caption{Model differences for point-form $BT$ calculation A, $\q\perp\p_{12}$.}
    \label{fig:point-A-prp}
  \end{minipage}
\end{figure*}

\begin{figure*}[h]
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{pf_brad_3d_parallel.pdf}
    \caption{Model differences for point-form $BT$ calculation B, $\q\|\p_{12}$.}
    \label{fig:point-B-par}
  \end{minipage}
  \hspace{0.05\linewidth}
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{pf_brad_3d_perp.pdf}
    \caption{Model differences for point-form $BT$ calculation B, $\q\perp\p_{12}$.}
    \label{fig:point-B-prp}
  \end{minipage}
\end{figure*}

\subsection{Binding Energy Variation}

The calculations above assumed a bound state with a wave function
having a standard dependence on the relative momentum of the
constituent nucleons.  The next set of curves illustrates the figure
of merit for fixed values of $\q$ and $\p_{12}$ as we vary the binding
energy and momentum scale of the wave function.  The variations that
we consider are still scales that are relevant to nuclear systems.
Figures~\ref{fig:MT-BE-par} and~\ref{fig:MT-BE-perp} show the
results of calculations vary the binding energy with the Malfliet-Tjon
wave function.

As with the earlier cases that employed the bound state binding energy,
the fractional deviation of the $BT$ results from the $TP$ benchmark is
quite small, of order $10^{-3}$ or less, for all cases except
point-form Option B.

\begin{figure*}[h]
  \begin{minipage}{0.45\linewidth}
    \centering
%    \includegraphics[width=\linewidth]{all_3d_parallel_slice_1.pdf}
   \includegraphics[width=\linewidth]{all_be_parallel.pdf}
    \caption{Model differences for $BT$ calculations as a function of
      two-body spectator binding energy, $\q=10~{\rm fm}^{-1}\|\p_{12}$.}
    \label{fig:MT-BE-par}
  \end{minipage}
  \hspace{0.05\linewidth}
  \begin{minipage}{0.45\linewidth}
    \centering
%    \includegraphics[width=\linewidth]{all_3d_perp_slice_1.pdf}
    \includegraphics[width=\linewidth]{all_be_perp.pdf}
    \caption{Model differences for $BT$ calculations as a function of
      two-body spectator binding energy, $\q=10~{\rm fm}^{-1}\perp\p_{12}$.}
    \label{fig:MT-BE-perp}
  \end{minipage}
\end{figure*}

\subsection{Wave Function Scale Variation}

We also examined sensitivity to the scale of the wave function by
replacing the Malfliet-Tjon wave function with a Gaussian form:
\begin{equation}
  \label{eq:g1}
  \phi(\k) = {1\over\sqrt{N}} e^{-(k/k_0)^2}.
\end{equation}
Figures~\ref{fig:Gaussian-par} and~\ref{fig:Gaussian-perp} show
the results for the spectator momentum perpendicular and parallel of
the momentum transfer in all three forms of dynamics.

These results mirror those discussed above: the fractional deviation
of the $BT$ results from the $TP$ benchmark is quite small, of order
$10^{-3}$ or less, for all cases except point-form Option B.

\begin{figure*}[h]
  \begin{minipage}{0.45\linewidth}
    \centering
%    \includegraphics[width=\linewidth]{all_3d_parallel_slice_1.pdf}
    \includegraphics[width=\linewidth]{all_k0_parallel.pdf}
    \caption{Model differences for $BT$ calculations as a function of
      two-body Gaussian wave function scale, $\q=10~{\rm fm}^{-1}\|\p_{12}$.}
    \label{fig:Gaussian-par}
  \end{minipage}
  \hspace{0.05\linewidth}
  \begin{minipage}{0.45\linewidth}
    \centering
%    \includegraphics[width=\linewidth]{all_3d_perp_slice_1.pdf}
    \includegraphics[width=\linewidth]{all_k0_perp.pdf}
    \caption{Model differences for $BT$ calculations as a function of
      two-body Gaussian wave function scale, $\q=10~{\rm fm}^{-1}\perp\p_{12}$.}
    \label{fig:Gaussian-perp}
  \end{minipage}
\end{figure*}

\subsection{Implications for Nuclear Theory}

Our results indicate that Bakamjian-Thomas models, which explicitly
satisfy the requirements of Poincar\'e invariance, can be utilized in
typical nuclear physics problems with minimal quantitative error due
to the lack of cluster separability using Dirac's instant- or
front-form dynamics.  For the point form, particularly Option B, one
must exercise care, since the effects can be large enough to play a
role when comparing results to experimental data.  

We believe that there is a physical basis behind the distinctive
results for the point form, which center on the use of velocities,
whereas a current matrix element depends upon momentum transfer.  In
all of the calculations discussed above, the $BT$ approach links
``external variables'' such as $\p_3$ and $\p_3'$ that describe the
observable kinematics to ``internal variables'' such as $\b_3$ and
$\b_3'$ that are used to calculate such quantities as the particle-3
current matrix element.  The momentum transfer $\q$ plays an explicit
role in the calculation of the ``inner'' quantities for the instant
and front forms, and even point-form Option A.  However, with Option
B, the effect of $\q$ is buried in a calculated vector $\V$ rather
than a momentum that enters the internal kinematics directly.  This
option thus has a weaker connection to the physical kinematics
dictated by the $TP$ benchmark.


\subsection{Implications for Hadron Models}

The final set of figures show the results of calculations with scales
that are more appropriate models of hadrons based on sub-nucleonic
degrees of freedom.  

We note here that QCD confinement precludes separating arbitrary
subsystems by large distance scales, so the general requirement of
cluster separability is irrelevant for models of hadrons.  The issue
may be relevant, however, for systems of hadrons described by
subnucleonic degrees of freedom.

In these cases, in order to understand the relevant scales, we replace
particle~3 masses in the above calculations by constituent quark
masses of 220 MeV, and consider two-body (``diquark'') masses ranging
from 200 to 600 MeV for Gaussian wave functions with a 1~fm${}^{-1}$
scale, and Gaussian wave functions with scale ranging from
0.5~fm${}^{-1}$ to 10~fm${}^{-1}$, with a diquark mass of 600 MeV.

The results are similar to those appropriate to nuclear physics
discussed above, except that the scale of deviation from the $TP$
benchmark is somewhat larger, as high as $1\%$ for instant and front
forms, and of order unity for the point form options.  This not
surprising given that the mass/momentum scale variation for these
calculations is much higher than for typical cases in nuclear physics
with nucleons.

\begin{figure*}[h]
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{all_be_parallel_quark.pdf}
    \caption{Model differences for $BT$ calculations as a function of
      two-body spectator mass, $\q=10~{\rm fm}^{-1}\|\p_{12}$.}
    \label{fig:quark-BE-all-par}
  \end{minipage}
  \hspace{0.05\linewidth}
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{all_be_perp_quark.pdf}
    \caption{Model differences for $BT$ calculations as a function of
      two-body spectator mass, $\q=10~{\rm fm}^{-1}\perp\p_{12}$.}
    \label{fig:quark-BE-all-prp}
  \end{minipage}
\end{figure*}

\begin{figure*}[h]
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{all_k0_par_quark.pdf}
    \caption{Model differences for $BT$ calculations as a function of
      Gaussian wave function scale, $\q=10~{\rm fm}^{-1}\|\p_{12}$.}
    \label{fig:quark-k0-all-par}
  \end{minipage}
  \hspace{0.05\linewidth}
  \begin{minipage}{0.45\linewidth}
    \centering
    \includegraphics[width=\linewidth]{all_k0_perp_quark.pdf}
    \caption{Model differences for $BT$ calculations as a function of
      Gaussian wave function scale, $\q=10~{\rm fm}^{-1}\perp\p_{12}$.}
    \label{fig:quark-k0-all-prp}
  \end{minipage}
\end{figure*}

\section{Summary}

Bakamjian-Thomas formulations, which explicitly satisfy the
requirements of Poincar\'e invariance, do not satisfy cluster
separability above the three-particle level, i.e.\ in systems that
involve three-body subsystems whose total momentum must vary.  The
cluster properties can be restored via a hierarchy of unitary
transformations.  These transformations depend upon the full solution
of a three-body problem, and are difficult to implement in practice.

Rather than attempt to calculate directly the size of these unitary
transformations (e.g.\ the difference of matrix elements from those of
the unit operator), we have developed a simple model in which the
exact result consistent with cluster separability is known, and then
compare to it the results of Bakamjian-Thomas calculations.

We conclude from these model studies that Bakamjian-Thomas models,
which explicitly satisfy the requirements of Poincare\'e invariance,
can be utilized in typical nuclear physics problems with minimal
quantitative error due to the lack of cluster separability using
Dirac's instant- or front-form dynamics.  For the point form, one must
exercise care, since the effects can be large enough to play a role
when comparing results to experimental data.

We also examined models utilizing mass/momentum scales appropriate for
quark models.  QCD confinement precludes separating
arbitrary subsystems by large distance scales, so the general
requirement of cluster separability is irrelevant for models of
hadrons.  The issue may be relevant, however, for systems of hadrons
described by subnucleonic degrees of freedom.  In such cases, the
deviations from the model benchmark are larger than those for typical
nuclear physics calculations with nucleons, though they are still
manageable for the instant and front forms.

\bibliography{packmodel}

\end{document}


\section{TODO Items}

\begin{itemize}
\item Figs.~\ref{fig:MT-BE-par} and \ref{fig:MT-BE-perp} should have
  something like ``BE'' or ``binding energy'' for the $x$-axis label,
  instead of $P_{12}$.
\item I couldn't find figures for $k_0$ (Gaussian) sweeps, so at this
  point figures labeled \ref{fig:Gaussian-par} and
  \ref{fig:Gaussian-perp} are copies of \ref{fig:MT-BE-par} and
  \ref{fig:MT-BE-perp}.
\item $Q$ should probably be replaced with $q$ in the figure labels
  and captions, since we now use $\q$ or $q$ in the text.
\item I notice that you separate momenta by commas in the Jacobians.
  I haven't gone back through to make this consistent.
\item I'm figuring we might submit this to Phys.\ Rev.  I know in the
  past they did not allow macros, i.e.\ we had to expand them back
  out.  Also I don't know whether they accept other packages like AMSTeX.
\end{itemize}
