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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, quasipotential equations, 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 edits that fix inconsistencies of notation and clarify
  definitions. 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.~IIA 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.~IIC, 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.~IIE which in general
  lead to different results. It is very important to provide stronger
  and more general motivations to discuss those models.  }

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 additional constraints noted above by the referee have been
addressed more directly in the paper.  There exists a limited range of
kinematic choices for each of the three $BT$ calculations (instant,
front, and point form).  The existence of these choices is a
manifestation of the fact that the $BT$ calculations do not exactly
agree with the $TP$ result, and thereby illustrate the quantitative
effect of the violation of cluster separability.  We have revised the
paper so that each of the three forms of dynamics has a similar choice
that relates three-vector components of the constituent-particle
momenta $b_3'$ and $b_3$ to the overall momentum transfer to the
three-particle system.  This corresponds to the instant- and
front-form cases along with point-form choice $A$.  One can make other
choices, such as point-form $B$, and also corresponding choices in the
instant and front form.  These lead to slightly different numerical
results, but with the same conclusions (effects small for typical
cases in nuclear physics, larger for quark models).  We have
eliminated point-form choice $B$ as an explicit topic, but have
inserted the more general comment about the freedom of choice in
kinematics as well as the observed numerical impact of those choices.

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

\end{enumerate}

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