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\centerline{\bf Response to Referee Report}

We have made several changes to the manuscript that we believe address the referee's very helpful comments.

\begin{enumerate}

\item {\it I suggest that the authors go through their paper from the
  standpoint of a general PRC reader, who is very interested in
  relativistic few-body physics, but not only in the approach of
  Ref.[3]. For example, some readers may be interested in approaches
  based on relativistic field theory, like the Bethe-Salpeter
  equation, or other covariant approaches based on Green functions and
  Feynman diagrams.  The present paper should be brought into a form
  which is interesting and useful also for those more general readers.
}

We have added a brief discussion in the introduction regarding the
choices that one can make when implementing the constraints of Lorentz
invariance in quantum theories, including field theory and
Bakamjian-Thomas constructions, and their respective connections in
practice to the requirement of cluster separability.  As noted in the
introduction, implementations of field theory that depend upon
truncations (such as ladder approximations to the Bethe-Salpeter
equation or other integral equations driven by subsets of Feynman
diagrams) do not automatically satisfy cluster separability and would
need to be studied on a case-by-case basis.  Such studies go beyond
the scope of this paper.

\item {\it If one works through the paper from this perspective, first
  of all one still finds problems with the notations: For example, the
  2-body mass eigenvalue is called $\lambda_{12}$ in Eqs.(2.1) -
  (2.4), but in (2.7) and below it is denoted as $\lambda$. More
  importantly, it is not clear why it is necessary to introduce new
  symbols $(b_1, b_2, b_3)$ for the single-particle momenta in
  Eq.(2.9). The momenta $p_{12}=p_1 + p_2$ and $p_3$ used in
  (2.5)-(2.8) do not refer to a particular frame, so it would be
  helpful if the authors could simply continue to use $(p_1, p_2,
  p_3)$ instead of $(b_1, b_2, b_3)$. In order to clarify the second
  change of variables in Eq.(2.9), it would be helpful to indicate
  this change of variables like it was done in Eq.(2.5) for the first
  change if variables. There are several other places where new
  notations can be avoided, for example the symbol $M^0_{12}$ in
  Eq.(2.9), which should be simply $2m$, because the authors consider
  3 particles with equal mass m.  }

We have made several thorough edits that fix several inconsistencies
of notation.  Two specific points:
\begin{itemize}

\item The momenta $(b_1, b_2, b_3)$ are distinct from $(p_1, p_2,
  p_3)$, and are calculated in different ways.  This difference is now
  explicitly identified in the text preceding Eq. (3.12) of the new
  version of the manuscript.

\item We have replaced the notation $M^0_{12}$ by $M_k$ in Eq. (3.16).
  The non-interacting mass of the two-body is not $2m$ (the
  non-relativistic version) but rather the sums of the one-body
  energies in a frame where the total momentum of the two-body system
  vanishes.
\end{itemize}

\item {\it Second, the generally interested reader is left alone with
  conceptual problems: For example, the term [Tensor-Product (TP)
    Model] used in Sect. II A is unclear. The properties (2.2) and
  (2.3) seem so general and reasonable that one should not consider
  them as model assumptions.  In an approach based on relativistic
  field theory and Green functions, these properties would follow
  almost trivially from the relevant Feynman diagrams. Is it really
  necessary to use BT Representations in Sect. II C, which fail to
  satisfy those properties? It seems that there are many ambiguities
  in those representations, because it is necessary to introduce
  additional constraints like A and B in Sect. II E which in general
  lead to different results. It is very important to provide stronger
  and more general motivations to discuss those models.  }

We agree that Tensor Product (TP) should refer to a representation
rather than a model, and have corrected the text accordingly.  We have
also strengthened the point that, for more realistic systems than the
simple four-body model that we developed, one cannot simply write down
a TP result that is both Poincar\'e invariant and cluster separable.
If that we true, then indeed there would be no point in using the BT
construction.  It is only because the model is so simple that we can
write down a TP result and then compare the BT solutions to it.  This
comparison has never been made in a quantitative way for a specific
case until now.

As noted above, a full field theory involving products of propagators manifestly satisfies cluster separability.  However, approximate schemes such as those involving three-dimensional reductions of a Bethe-Salpeter equation, no longer involve products of propagators with completely independent four-momenta, and cluster separability is no longer manifestly satisfied.  Such schemes must demonstrate separability - either exactly, or to the extent that it is satisfied approximately.  In short, the separability requirement is easy to write down, and in certain cases (exact field theories) is trivially satisfied, but for many realistic calculations involving strong interactions, the requirement is non-trivial.

The ambiguities to which the referee refers have to do with the comparison between a BT calculation and a TP result.  One manifestation of the cluster separability problem for BT constructions without the Sokolov restoration is that the full set of momenta do not quite match between the calculations.  Thus one must choose (constrain) a set of momenta for comparison purposes.  We present what we believe are natural choices, and they are spelled out for the specific cases of instant, front and point forms.  The point form is somewhat distinct from the instant and point forms because states are labeled with velocities rather than momenta, and we found two possible choices from which one could make a BT vs TP comparison.  We note here that for the instant- and front-form calculations, as well as one of the point-form calculations, the BT-TP difference is of order 10$^{-3}$.  The main conclusion of the paper is that the {\it impact} of the violation of cluster separability is quantititatively small.

We hope that with these revisions the paper will be suitable for publication.

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