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\begin{center}{\Large{\bf New representations of the Poincar\'{e} group
describing two interacting bosons}} \vspace{8mm}
E. Frochaux \\ D\'{e}partement de Math\'{e}matiques, Ecole
Polytechnique F\'{e}d\'{e}rale de Lausanne, CH-1015 Lausanne, Switzerland \\
e-mail frochaux@masg1.epfl.ch
\end{center} \vspace{12mm}
{\large {\bf Abstract}}\vspace{5mm}
New representations of the Poincar\'{e} group are given, which describe two
bosons with interaction in four space-time dimensions. The quantum frame is
the Schr\"{o}dinger picture in momentum space. More precisely we start from
the relativistic free model with Hilbert space $L^2(I\!\!R^3 \times I\!\!R^3,
\sigma_2)$, where $\sigma_2$ is the Lorentz invariant measure.
We add to the free Hamiltonian and the free Lorentz generators new interaction
terms, without changing the Poincar\'{e} algebra commutation rules, and such
that the algebra representation can be integrated to a unitary representation
of the group on $L^2(I\!\!R^6,\sigma_2)$. The physics of these models can be
investigated through the bound state equation (a {\em relativistic
Schr\"{o}dinger equation}) and through the scattering matrix. Asymptotic
completeness is obtained in some cases. Finally we give an example for which
a bound state exists and for which the scattering matrix can be written down
explicitely. This example assures that an interaction between the particles
can effectively occur in these models.
The present paper is an extended and improved version of a previous one
entitled "Relativistic quantum models for two bosons with interaction in
the Schr\"{o}dinger picture", available at mp-arc 96-545.
\vspace{12mm}
{\large {\bf 1. Introduction}}\vspace{5mm}
A new family of unitary representations of the Poincar\'{e} group in four
space-time dimensions is given, which describe two boson systems with
interaction. The quantum frame is the Schr\"{o}dinger picture of Quantum
Mechanics in momentum space. The physics of these models can be obtained
from the scattering operator and the bound state equation, an eigenvalue
equation of the kind of the Schr\"{o}dinger equation. The existence of a
bound state and the non triviality of the scattering matrix confirm that
the interaction is effective. To our knowledge, this is the first example
of relativistic quantum models for a finite number of interacting particles.
Moreover they admit a rather simple mathematical construction.
The quantum frame of the Schr\"{o}dinger picture means that the Hilbert
state space is simply made of functions of the momenta. In particular the
time variable does not appear explicitly. However, these models satisfy
the relativistic principles, because they consist of unitary representations
of the Poincar\'{e} group. An elementar example of such a model is given
by the free model for two bosons ('free' means without interaction). Our
models are obtained by perturbations of the Hamiltonian and the Lorentz
generators of the free model, in such a way that the commutation rules of
the Poincar\'{e} algebra remain satisfied. The exisence of solutions to
this problem is rather surprising. Indeed, such a perturbation of the
Hamiltonian and the Lorentz generators has already been proposed by Dirac
in the late 40's [1], in the framework of classical mechanics. Dirac thought
that the classical theory should be established first, and that the quantum
version would be obtained afterwards by applying the canonical quantization.
Therefore his paper concerns only classical physics, and it does not really
obtain an interesting conclusion. In the early 60's, Currie continued this
approach, still in the classical frame. He obtained in [2] that, under
additional natural assumptions, such modifications cannot generate an
interaction (this is known as the `non-interaction theorem'). The present
paper shows that by going directly to the quantum frame, this difficulty can
be overcome.
The result given here is the generalization of similar results in 2-d (two
space-time dimensions) [3] and in 3-d [4]. Although the family of models in
[3] is very general (the kernel of the interaction part of the Hamiltonian
can be chosen almost arbitrarily in the centre-of-mass frame), in 3-d and
4-d we have only managed to construct models with special interaction terms
(unfortunately, local interactions by a potential are not included). Moreover,
the generalization to more than two particles [5] and to bosons with different
masses [6], which is easy in 2-d, seems to be very difficult in higher
space-time dimensions.
Originally, the result in 2-d was suggested by an new approch of the bound
state problem in 2-d Quantum Field Theory (QFT). Let us briefly describe
this new approach.
The mathematical construction of bosonic 2-d QFT models (the so-called
'weakly coupled $P(\varphi)_2$ models') by Glimm, Jaffe, Spencer [7], has
made possible a rigorous approach of the relativistic bound state problem.
In this framework the Bethe-Salpeter method has been rigorously established
and extensively studied [8]. This method provides a clear and precise
theory for relativistic quantum particles with interaction at low energy,
which is satisfactory both from the physical and the mathematical point
of view. However, there are some drawbacks. First, the numerical calculation
of the bound state mass from the Bethe-Salpeter equation is extremely
difficult. Second, the connection to the corresponding problem in Quantum
Mechanics (QM) is not completely made: only the non-relativistic limit is
established, but not its relativistic corrections. In particular, it is not
known how to get the relativistic corrections to the Schr\"{o}dinger
equation. This problem could be related to the famous 'inverse problem'
in QM, because the Bethe-Salpeter method consists in the search of the
bound states by analysing the scattering amplitudes, while the corrections
to the Schr\"{o}dinger equation refers to a 'potential approach', which
has never been obtained clearly. Finally, the mathematical rigour of
this method depends on the mathematical construction of the QFT model
in question, which is missing at four dimensional space-time and for
gauge interactions.
Before trying to study the last mentioned difficulties, it should be
natural to improve first the bound state description of the $P(\varphi)_2$
models, in order to develop intui\-tion and new ideas. To treat the bound
state problem from an new point of vue, one can try to construct the
eigenspace of a bound state. This leads indeed to an original approach,
because the standard strategy of QFT is precisely to avoid working with
states of interacting particles (because the notion of 'interacting particle'
itself is not defined, and therefore only the vacuum state and the
asymptotic states are used in QFT). Now, in a relativistic theory, the mass
of a bound state is an eigenvalue of the mass operator $M$, and thus its
states are eigenvectors of $M$, which is an unambiguous notion. A new
method can be developed, adapted to QFT models in which the existence
of bound states is already known (from the Bethe-Salpeter method). This
has been done by combining the variational and the perturbation methods
([9], [10]). Let us explain in a few words how this works. The aim is
to minimize the Rayleigh quotient $R(\psi)=(\psi,M \psi) \ \| \psi
\|^{-2}$ over the largest set of vectors $\psi \in D(M)$, $\psi \neq 0$,
orthogonal to the vacuum state and to the one-particle states. $R(\psi)$
is calculated by perturbation with respect to the coupling constant. More
precisely, it is calculated explicitely at first orders, and the
remainder is estimated, thanks to the mathematical construction of
these models. In doing this, it appears that it is sufficient to restrict
ourselves to a subspace made of vectors called 'zero-time vectors', which
do not involve time variables. Then the minimization of the explicitly
calculated terms of $R(\psi)$ is performed, in varying $\psi$ in a subset
of zero-time vectors for which the remainder is uniformly bounded. The
vectors $\psi_0$ exhibited by the minimization can be considered as
perturbation of eigenvectors. This gives a method to construct
approximations of the eigenspace, once the existence of a bound
state is known.
The interest of this methods is that it brings several surprises. First,
it leads to an eigenvalue equation for the bound state mass, which is
similar to the Schr\"{o}dinger equation (made of a 'kinetic' plus an
'interaction' terms) and which can be considered as an approximation of a
`relativistic Schr\"{o}dinger equation' ([9], [11]). Second, a subspace is
found which is stable under the Poincar\'{e} transformation (at first
perturbation orders) and nevertheless carries interaction (sect. 3 of [7],
[11]). Such representations of the Poincar\'{e} group can be studied for
themselves, without reference to QFT. The third surprise is that such
representations can be constructed non-perturbatively [3]. They provide
a set of models for two particles with effective interaction in the
Schr\"{o}dinger picture (the time variable is not introduced, due to the
restriction to zero-time vectors). Moreover, the complete set of such
representations can be given, in the weak coupling regime. They are
all constructed from an interaction kernel which, in the centre-of-mass
frame, can be chosen arbitrarily (apart from some analytic conditions) [3].
The last surprise is that generalizations are possible, to $N > 2$ particle
systems [5] and to higher dimensional space-time. The case of two particles
in three space-time dimensions has been obtained for special interactions
(which involve the projection on the s-waves) [4]. With such an interaction,
the direct connection with QFT fails. The present paper concerns the
generalization to four space-time dimensions, for two particle systems.
The difficulty in passing from 3-d to 4-d comes from the highter dimension
of the Poincar\'{e} group (passing from 7 to 10 dimensions) and from the
non-integrability of some kernels. This has the consequence that, in order
to follow the same general exposition as for the 3-d case [4], all the
proofs have had to be completely rewritten. However, provided the
interaction satisfies again the special property given above, unitary
representations of the Poincar\'{e} group can be constructed. Moreover,
the study of the physical content of these models (bound state equation,
scattering matrix) confirms that they describe two particles systems with
interaction in four space-time dimensions. Note that these are probably
the first relativistic quantum models for two interacting particles in
four space-time dimensions, and that they admit a rigorous
mathematical construction which is remarkably simple.
The paper is organized as follows. Section 2 presents the models, given by
a family of operators which (formally) satisfy the commutation rules of
the Poincar\'{e} algebra, provided some conditions on the 'interaction
kernel' $h$ (the kernel of the interaction term of the Hamiltonian)
hold. In particular, if $h$ satisfies the special property given above,
then these conditions reduce to a single `fundamental equation' for $h$.
In section 3 the complete set of solutions to this equation is given, in
the weak coupling regime. Then we focus on a particular subset of
solutions which satisfy all the analytic properties we need for the
following proofs. Section 4 shows that the formal representations of
the Poincar\'{e} algebra, we have just obtained, are mathematically
well defined and can be integrated to give unitary representations of
the Poincar\'{e} group. Section 5 gives the eigenvalue equation for
the bound states (the `relativistic Schr\"{o}dinger equation') and
section 6 establishes the existence of two-particle asymptotic states
and of the scattering operator. Asymptotic completness is obtained for
special $h$. Finally, section 7 gives a simple example (one-dimensional
perturbation) for which the bound state equation admits a solution and
for which the scattering matrix can be written explicitly. Its non
triviality can easily be seen, assuring the non triviality of this set
of models.
\vspace{12mm}
{\large {\bf 2. The models}}\vspace{5mm}
The Lie algebra $\cal G$ of the Poincar\'{e} group ${\cal P}_{+}^{\uparrow}$
in four space-time dimensions is generated by the ten operators $H$
(Hamiltonian), ${\vec P} = (P_1,P_2,P_3)$ (momentum), ${\vec J} =
(J_1,J_2,J_3)$ (angular momentum) and ${\vec L} = (L_1,L_2,L_3)$
(generators of the Lorentz transformations), satisfying the 45 commutation
rules
\begin{eqnarray}
& [P_j,P_k] = 0 & \hspace{10mm} [P_j,J_k] = i \varepsilon_{j,k,\ell}
P_{\ell} \hspace{10mm} [J_j,J_k] = i \varepsilon_{j,k,\ell} J_{\ell} \\
& [P_j,H] = 0 & \hspace{10mm} {[H,J_j]} = 0 \\
& [P_j,L_k] = i\delta_{j,k}H & \hspace{10mm} {[J_j,L_k]} = i
\varepsilon_{j,k,\ell} L_{\ell} \\
& [H,L_j] = iP_j & \hspace{10mm} [L_j,L_k] = - i \varepsilon_{j,k,\ell}
J_{\ell}
\end{eqnarray}
for all $1 \leq j,k \leq 3$ where $\delta_{i,j}$ is the Kronecker tensor
and $\varepsilon_{j,k,\ell} \in \{0,1\}$ is the totally antisymetric
tensor with $\varepsilon_{1,2,3}=1$ (we have used the Einstein summation
convention). This algebra admits two Casimir operators, the mass operator
$M$, the square of which is given by
\begin{eqnarray}
M^2 \ := \ H^2 \ - \vec{P}^2
\end{eqnarray}
and the Pauli-Lubanski operator
\begin{eqnarray}
W^2 \ := \ \left( \vec{P} \cdot \vec{J} \ \right)^2 \ - \ \left( -H
\vec{J} + \vec{P} \wedge \vec{L} \ \right)^2
\end{eqnarray}
where the dot ``$\cdot$'' and the wedge ``$\wedge$'' denote the ordinary
scalar and vector pro\-ducts respectively.
We start from the representation describing one spinless particle of mass
$m>0$, in the Schr\"{o}dinger picture, given by the following choice of
operators
\begin{eqnarray}
\vec{P} \phi({\vec p}) & = &\ \vec{p} \ \phi({\vec p}) \nonumber \\
H \phi({\vec p}) & = & \omega ({\vec p}) \phi({\vec p}) \nonumber \\
\vec{J} \phi({\vec p}) & = & -i \ \vec{p} \wedge \vec{\nabla}
\phi({\vec p}) \\
\vec{L} \phi({\vec p}) & = & -i\omega({\vec p}) \ \vec{\nabla}
\phi({\vec p}) \nonumber
\end{eqnarray}
for almost all ${\vec p}=(p_1,p_2,p_3) \in I\!\!R^3$ and suitable
three-variable functions $\phi$, where we have put $\omega({\vec p}) :=
\sqrt{{\vec p}^{ \ 2} + m^2}$. We have used the notation $\vec{\nabla}
:=( \partial_{p_1}, \partial_{ p_2},\partial_{p_3})$. These operators
define a representation of the algebra $\cal G$ which can be integrated to
give a unitary, continuous and irreducible representation of the Poincar\'{e}
group ${\cal P}_{+}^{ \uparrow}$, in the function space $L^2 (I\!\!R^3,
\sigma)$, where $\sigma$ is the measure $d \sigma({\vec p}) :=d\vec{p} \
[2\omega({\vec p}) ]^{-1}$ (see appendix A). The Casimir operators of the
representation (7) are simply $M= m$ (times the identity operator)
and $W=0$.
The representation describing two identical bosons of mass $m>0$ without
interaction, in the Schr\"{o}dinger picture, is given by the symmetrical
tensor product of two copies of the representation (7) (the symmetry
refers to the case where the particles are identical, but this
restriction is not essential and could be removed). The action of the
operators becomes
\begin{eqnarray}
\vec{P} \ \phi(\vec{p}_1, \vec{p}_2) & = & \left( \vec{p}_1 + \vec{p}_2
\right) \ \phi(\vec{p}_1, \vec{p}_2) \nonumber \\
H_0 \ \phi(\vec{p}_1, \vec{p}_2) & = & \left( \omega({\vec p}_1) +
\omega({\vec p}_2) \right) \ \phi(\vec{p}_1, \vec{p}_2) \nonumber \\
\vec{J} \ \phi(\vec{p}_1, \vec{p}_2) & = &- i \left( \vec{p}_1 \wedge
\vec{\nabla}\!_1 + \vec{p}_2 \wedge \vec{\nabla}\!_2 \right) \
\phi(\vec{p}_1, \vec{p}_2) \\
\vec{L}_0 \ \phi(\vec{p}_1, \vec{p}_2) & = & -i \left( \omega({\vec p}_1 )
\vec{\nabla}\!_1 + \omega({\vec p}_2 ) \vec{\nabla}\!_2 \right) \
\phi(\vec{p}_1, \vec{p}_2) \nonumber
\end{eqnarray}
for almost all $({\vec p}_1,{\vec p}_2 ) \in I\!\!R^6$ and suitable six
variable functions $\phi$, where $\vec{\nabla}\!_i :=(
\partial_{({\vec p}_i)_1}, \partial_{({\vec p}_i)_2},\partial_{(
{\vec p}_i)_3})$, $i\in \{1,2\}$. Because of the properties of the
tensor product, (8) defines a representation of the algebra $\cal G$ which
can be integrated to give a unitary and continuous representation of the
Poincar\'{e} group ${\cal P}_{+}^{ \uparrow}$, in the function space
\begin{eqnarray}
{\cal H} \ := \ L^2_{Sym} (I\!\!R^6, \sigma_2), \hspace{6mm}
d\sigma_2({\vec p}_1, {\vec p}_2) := { d{\vec p}_1 \over 2
\omega(\vec{p}_1)} \ { d{\vec p}_2 \over 2 \omega(\vec{p}_2)}
\end{eqnarray}
made of symmetrical functions (i.e. $\phi(\vec{p_1},\vec{p}_2)=\phi
(\vec{p_2},\vec{p}_1)$). Note that $\sigma_2 = \sigma \otimes \sigma$
and recall that $\omega(\vec{p}_i) = \sqrt{\vec{p}_i^{\ 2}+ m^2}$ for
each $i \in \{1,2 \}$. $\cal H$ describes two particles without
interaction (therefore we have put an index 0 to $H_0$ and $ \vec{L}_0$).
The Casimir operator of the representation (8) are given by
\begin{eqnarray}
M^2_0 \ \phi(\vec{p_1},\vec{p}_2) & = & \left[ \left(\omega(\vec{p}_1)+
\omega(\vec{p}_2) \right)^2 \ - \ \left( \vec{p}_1 + \vec{p}_2 \right)^2
\right] \ \phi(\vec{p_1},\vec{p}_2) \\
W^2_0 \ \phi(\vec{p_1},\vec{p}_2) & = &- \left[ \left( (\vec{p}_1 \wedge
\vec{p}_2) \cdot
(\vec{\nabla}\!_1-\vec{\nabla}\!_2) \right)^2 \right. \nonumber \\
& & \ \ \ \ \ -\left.
\left( \omega({\vec p}_1){\vec p}_2 \wedge \vec{\nabla}\!_1 +
\omega({\vec p}_2) {\vec p}_1 \wedge \vec{\nabla}\!_2
\right)^2 \right]
\phi(\vec{p_1},\vec{p}_2) .
\end{eqnarray}
Now we want to modify these operators in order to introduce the interaction.
To remain in the Schr\"{o}dinger picture we keep $\vec{P}$ and $\vec{J}$
unchanged and modify only $H_0$ and $\vec{L}_0$. We introduce a symmetric
operator $\cal O$, the {\em interaction operator}, on which we impose only,
for the moment, the formal commutation properties
\begin{eqnarray}
[{\cal O},\vec{P}]=0 \hspace{20mm} [{\cal O},\vec{J}]=0.
\end{eqnarray}
The interaction representation is defined as follows
\begin{eqnarray}
& \vec{P}, \ \vec{J} & \hspace{5mm} as \ \ in \hspace{3mm} (8) \nonumber \\
H & := & H_0 \ + \ \{{\cal O},H_0 \} \\
\vec{L} & := & \vec{L}_0 \ + \ \{{\cal O},\vec{L}_0 \} \nonumber
\end{eqnarray}
where we have used the notation $\{A,B\} = AB + BA$. The particular form
of the `interaction parts' of (13) together with (12) have the following
nice consequence.
\newtheorem{Christel}{Theorem}
\newtheorem{Faux}[Christel]{Proposition}
\newtheorem{Fau}[Christel]{Lemma}
\begin{Fau} The operators $(13)$ satisfy formally the commutation rules
$(1)$, $(2)$ and $(3)$ of the algebra $\cal G$.
\end{Fau}
{\bf Proof.}
(1) has not been modified; (2) is an immediate consequence of (12); (3)
follows from (12), from the fact that the free algebra $(\vec{P},\vec{J},
H_0, \vec{L}_0)$ satisfies also (3), and from some algebraic calculations
(see [3] or [5] for more details). []
\vspace{2mm}
Note the particular role of (3) : it gives the new operators $H$, $L_1$,
$L_2$ after the change of $L_3$ only. Until now no condition on $\cal O$,
except (12), were needed. But the last commutation rules (4) impose the
following equations
\begin{eqnarray}
0 & = & [\{{\cal O}, H_0 \},L_{0,j}] \ + \ [H_0,\{{\cal O}, L_{0,j} \}]
\ + \ [\{{\cal O}, H_0\}, \{{\cal O},L_{0,j}\}] \\
0 & = & [\{{\cal O}, L_{0,j} \},L_{0,k}] \ + \ [ L_{0,j},\{{\cal O},
L_{0,k} \}] \ + \ [\{{\cal O}, L_{0,j}\}, \{{\cal O},L_{0,k}\}]
\end{eqnarray}
for all $j \neq k \in \{1,2,3\}$. We get six equations for $\cal O$ of
the same form (made of a linear term plus a bilinear term), which must
hold simultaneously. We have now to find solutions to this system.
In order to have a better physical description we make the change of
variables $(\vec{p}_1, \vec{p}_2) \rightarrow (\vec{P}, \vec{Q})$ given
by
\begin{eqnarray}
\vec{P} & = & \vec{p}_1 \ + \ \vec{p}_2 \nonumber \\
\vec{Q} & := & {1 \over 2} ( \vec{p}_1 - \vec{p}_2 ) \ - \ {1 \over 2}\
\vec{P} \ {\omega(\vec{p}_1) \ - \ \omega(\vec{p}_2) \over \
M_0(\vec{p}_1, \vec{p}_2) + \omega(\vec{p}_1) + \omega(\vec{p}_2) }
\end{eqnarray}
where $ M_0(\vec{p}_1, \vec{p}_2) = \left[ (\omega(\vec{p}_1)+\omega
(\vec{p}_2))^2 - \vec{P}^2 \right]^{1/2}$ is the function appearing in
(10). The variable $\vec{P}$ is the total momentum while $\vec{Q}$ is
the relative momentum, defined as the momentum in the centre-of-mass
frame (more precisely, $\vec{Q}$ is a Lorentz transformation of ${1
\over 2} ( \vec{p}_1 - \vec{p}_2 )$ leading to the centre-of-mass frame,
see appendix B for more explanations).
Under this change of variables the free representation (8) becomes
\begin{eqnarray}
\vec{P} \ \phi(\vec{P},\vec{Q}) & = & \vec{P} \ \phi(\vec{P},\vec{Q})
\hspace{10mm} \mbox{(multiplication by the variable } \vec{P} \mbox{) }
\nonumber \\
H_0 \ \phi(\vec{P},\vec{Q}) & = & \Omega ({\vec P},{\vec Q} ) \
\phi(\vec{P},\vec{Q}) \nonumber \\
\vec{J} \ \ \phi(\vec{P},\vec{Q}) & = & \left( \vec{J}\,^{\vec{P}} +
\vec{J}\,^{\vec{Q}} \right) \ \phi(\vec{P},\vec{Q}) \\
\vec{L}_0 \ \phi({\vec P},{\vec Q}) & = & \left( -i\Omega({\vec P},
{\vec Q}) \vec{\nabla}\!_{\vec{P}} \ - \ { \vec{P} \wedge \vec{J}
\,^{\vec{Q}} \over 2\omega(\vec{Q}) + \Omega ({\vec P},{\vec Q} )}
\right) \ \phi({\vec P}, {\vec Q} ) \nonumber
\end{eqnarray}
for almost all $(\vec{P},\vec{Q}) \in I\!\!R^6$ and suitable functions
$\phi$, where we have put
\begin{eqnarray}
\Omega({\vec P}, {\vec Q}) & := & \sqrt{{\vec P}^2 + 4\omega({\vec Q})^2}
\ = \ \sqrt{{\vec P}^2 + 4\vec{Q}^2 + 4m^2} \nonumber \\
\vec{J}\,^{\vec{P}} & := & -i \vec{P} \wedge \vec{\nabla}\!_{\vec{P}} \\
\vec{J}\,^{\vec{Q}} & := & -i \vec{Q} \wedge \vec{\nabla}\!_{\vec{Q}}
\nonumber
\end{eqnarray}
(see the calculations in appendix B). As for (8), these operators define a
representation of the algebra $\cal G$ which can be integrated to give a
unitary and continuous representation of the Poincar\'{e} group
${\cal P}_{+}^{ \uparrow}$, in the function space (9), which in these
variables becomes
\begin{eqnarray}
{\cal H} = L^2_{ev. \vec{Q}} (I\!\!R^3 \times I\!\!R^3,\mu), \hspace{6mm}
d\mu({\vec P}, {\vec Q}) := { d{\vec Q} \ d{\vec P} \over
2 \omega(\vec{Q}) \Omega({\vec P}, {\vec Q})} \ = \ { d\sigma({\vec Q})
\ d{\vec P} \over
\Omega({\vec P}, {\vec Q})}
\end{eqnarray}
made of even functions w.r.t. $\vec{Q}$ (i.e. $\phi(\vec{P},\vec{Q}) =
\phi(\vec{P},-\vec{Q})$), see appendix B.
Let us discuss the representation (17). In the non-relativistic case,
it is possible to find relative variables which are invariant under the
proper Galilean transformations. In the relativistic case we have not
managed to find such relative variables, and the Lorentz generators
contain a term involving $ \vec{J}\,^{\vec{Q}}$, i.e. acting on $\vec{Q}$.
Note that only the angular variables of $\vec{Q}$ are concerned, because
$\vec{Q}^2 = {1 \over 4} \left[ ( \vec{p}_1 - \vec{p}_2 )^2 - (\omega
(\vec{p}_1) \ - \ \omega(\vec{p}_2))^2 \right]$ is an invariant quantity.
This can also be seen from the Casimir operators, which are now only
functions of $\vec{Q}^2$ and $(\vec{J}\,^{\vec{Q}})^2$ :
\begin{eqnarray}
M^2_0 \ \phi( {\vec P}, {\vec Q}) & = & 4 \omega(\vec{Q})^2 \
\phi({\vec P}, {\vec Q}) \\
W^2_0 \ \phi( {\vec P}, {\vec Q}) & = & 4 \omega(\vec{Q})^2 \
(\vec{J}\,^{\vec{Q}})^2 \ \phi({\vec P}, {\vec Q}).
\end{eqnarray}
To find solutions of (14), (15) we must write $\cal O$ more explicitly.
The most general form compatible with (12) is
\begin{eqnarray}
{\cal O} \ \phi(\vec{P},\vec{Q}') & := & \int_{I\!\!R^6} d \mu( \vec{P}',
\vec{Q}') \ \ \phi(\vec{P}',\vec{Q}')
\ { \eta(\vec{P},\vec{Q},\vec{P}',\vec{Q}') \over \Omega(\vec{P},
\vec{Q} ) + \Omega(\vec{P}', \vec{Q}' ) } \ \delta ( \vec{P}-\vec{P}')
\nonumber \\
& = & \int_{I\!\!R^3} { d\sigma (\vec{Q}') \over \Omega({\vec P},
{\vec Q'})} \ \phi(\vec{P},\vec{Q}')
\ { h(\vec{P}, \vec{Q}, \vec{Q}') \over \Omega(\vec{P}, \vec{Q} ) +
\Omega(\vec{P}, \vec{Q'} ) }
\end{eqnarray}
where $h(\vec{P}, \vec{Q}, \vec{Q}') := \eta(\vec{P},\vec{Q},\vec{P},
\vec{Q}')$ is an arbitrary kernel (for the moment) which is even w.r.t
$\vec{Q}$ and $\vec{Q}'$, that is $h(\vec{P}, \vec{Q}, \vec{Q}') =
h(\vec{P}, -\vec{Q}, \vec{Q}') = h(\vec{P}, \vec{Q}, -\vec{Q}')$,
and which satisfies the symmetry condition
\begin{eqnarray}
h(\vec{P},\vec{Q}, \vec{Q}')^{\textstyle * } \ = \
h(\vec{P},\vec{Q}',\vec{Q})
\end{eqnarray}
where the star * means complex conjugation, all these conditions being
necessary for $\cal O$ to be a symmetric operator on $\cal H$. The Dirac
$\delta$ pseudo-function assures that ${\cal O}$ (formally) commutes with
$\vec{P}$ and the last condition of (12) leads (formally) to
\begin{eqnarray}
\left( \vec{J}\,^{\vec{P}}+ \vec{J}\,^{\vec{Q}}+ \vec{J}\,^{\vec{Q}'}
\right) \ h(\vec{P}, \vec{Q}, \vec{Q}') \ = \ 0.
\end{eqnarray}
The denominateur in (22) is necessary for $h$ to be the kernel of the
interaction part of $H$. In fact, the Hamiltonian (13) can be written
\begin{eqnarray}
H \ \phi(\vec{P},\vec{Q}') \ = \ \Omega(\vec{P}, \vec{Q}) \ \phi(\vec{P},
\vec{Q}) \ + \ \int_{I\!\!R^3} { d\sigma (\vec{Q}') \over \Omega
({\vec P},{\vec Q'} )} \ \phi(\vec{P},\vec{Q}' ) \ h(\vec{P}, \vec{Q},
\vec{Q}')
\end{eqnarray}
(note that $h$ has no physical dimension) while $\vec{L}$ is given by
\begin{eqnarray}
\vec{L} \ \phi(\vec{P},\vec{Q}') & = & \vec{L}_0 \phi({\vec P},
{\vec Q}) \ + \ \int { d\sigma(\vec{Q}') \over
\Omega(\vec{P}, \vec{Q}') } \ \left[\vec{L}_0 \phi(\vec{P}, \vec{Q}')
\right] \ { h(\vec{P}, \vec{Q}, \vec{Q}') \over\Omega(\vec{P},
\vec{Q}')} \\
& - & i \vec{P} \int { d\sigma(\vec{Q}') \over \Omega(\vec{P},
\vec{Q}') } \ \phi(\vec{P}, \vec{Q}') \left[ { Dh(\vec{P}, \vec{Q},
\vec{Q}') \over \| \vec{P} \| \Omega(\vec{P}, \vec{Q}') } \ -
\ { h(\vec{P}, \vec{Q}, \vec{Q}') \over \Omega(\vec{P}, \vec{Q}')^2 }
\right] \nonumber \\
& + & \int { d\sigma(\vec{Q}') \over \Omega(\vec{P}, \vec{Q}') }
\ \phi(\vec{P}, \vec{Q}') \nonumber \\
& & \hspace{5mm} \times \ { \vec{P} \wedge \left[ { \textstyle
\vec{J}\,^{\vec{Q}} \over \textstyle 2\omega(\vec{Q}) + \Omega
({\vec P},{\vec Q} )} + { \textstyle \vec{J}\,^{\vec{Q}'}
\over \textstyle
2\omega(\vec{Q}') + \Omega ({\vec P},{\vec Q}')} \right] \over
\Omega(\vec{P}, \vec{Q})
+\Omega(\vec{P}, \vec{Q}')} \ h(\vec{P}, \vec{Q}, \vec{Q}') \nonumber .
\end{eqnarray}
The substitution of (22) in (14) and (15) leads to six non-linear,
integro-differential equations for $h$. It seems difficult to find a
general solution $h \neq 0$ unless we restrict ourselves to solutions
satisfying the folowing condition
\begin{eqnarray}
\vec{J}\,^{\vec{Q}} \ h(\vec{P}, \vec{Q}, \vec{Q}') \ = \ 0
\end{eqnarray}
which leads to $\vec{J}\,^{\vec{P}} h = \vec{J}\,^{\vec{Q}'}h = 0$
because of (21) and (22). Thus (27) is equivalent to the requirement
that $h(\vec{P}, \vec{Q}, \vec{Q}')$ is function of the norms
$\|\vec{P}\|$, $\| \vec{Q}\|$, $\| \vec{Q}'\|$ only. With this condition,
the six equations (14) and (15) coincide. Let us introduce the
differential operator $D$ given by
\begin{eqnarray}
Dh(\vec{P},\vec{Q}, \vec{Q}') \ := \ { \Omega(\vec{P}, \vec{Q} ) \
\Omega(\vec{P},\vec{Q}' ) \over \Omega(\vec{P},\vec{Q}) \ + \
\Omega(\vec{P},\vec{Q}') } \
\partial_{\|\vec{P}\|} \ h(\vec{P},\vec{Q},\vec{Q}' ).
\end{eqnarray}
In appendix C we show that (14), (15) and (27) lead to the single equation
\begin{eqnarray}
0 & = & 2Dh(\vec{P},\vec{Q}, \vec{Q}') \ + \ \int { d\sigma(\vec{Q''})
\over \Omega({\vec P},{\vec Q''})^2 } \left\{ - { \|\vec{P}\|
\over \Omega(\vec{P}, \vec{Q}'')} \ h(\vec{P},\vec{Q}, \vec{Q}'')
h(\vec{P},\vec{Q}'', \vec{Q}') \right. \nonumber \\
& + & \left. { \over } \ Dh(\vec{P},\vec{Q}, \vec{Q}'')
h(\vec{P},\vec{Q}'', \vec{Q}')\ + \ h(\vec{P},\vec{Q}, \vec{Q}'')
Dh(\vec{P},\vec{Q}'', \vec{Q}')
\right\}.
\end{eqnarray}
This is the fundamental equation which guarantees the relativistic
structure of the theory. Let us sum up what we have found in a proposition.
\vspace{2mm}
\begin{Faux} Let $h(\vec{P},\vec{Q}, \vec{Q}')$ be a function of
$\| \vec{P}\|$, $\| \vec{Q}\| $, $\| \vec{Q}'\|$ only which satisfies
$(23)$ and $(29)$. Then the operators $(13)$, with $\cal O$ given by
$(22)$, are symmetric and satisfy formally the commutation rules $(1)$
to $(4)$ of the algebra $\cal G$.
\end{Faux}
\vspace{12mm}
{\large {\bf 3. Existence of solutions of the fundamental equation}}
\vspace{6mm}
In a first part we give the complete set of solutions of (29) lying in
a ball of a Banach space, neglecting the conditions (23) and (27).
In a second part we introduce a particular subset of solutions,
which satisfy also (23) and (27), and we establish some of their
analytic properties.
\vspace{2mm}
\newline
{\bf Definition 1} Let $\cal K$ be a compact set of $I\!\!R^3$ with
Lebesgue-measure $|{\cal K}| \neq 0$. We denote by $\cal B$ the Banach
space of continuous and bounded functions $h(\vec{P},\vec{Q}, \vec{Q}')$
with support in $(\vec{P},\vec{Q}, \vec{Q}') \in I\!\!R^3 \times
{\cal K} \times {\cal K}$ for which $Dh(\vec{P},\vec{Q},\vec{Q}')$
exists and is also bounded and continuous, given the norm
\begin{eqnarray}
|h|_{\cal B} \ : = \ \ \|h\|_{\infty} + \|Dh \|_{\infty} \nonumber
\end{eqnarray}
where the differential operator $D$ is given by (28).
The following result assures the existence of a large class of solutions
of (29).
\begin{Faux} There exists $K_1 \in (0,\infty)$ such that, for all $c \in
C^0_0({\cal K} \times{\cal K})$ satisfying $ \|c\|_{\infty}0$ and let $\cal K$ be the ball of $I\!\!R^3$
of centre 0 and radius $R$. Let $\cal V$ be the set of functions
$c(\vec{Q},\vec{Q}') \in C^2_0({\cal K} \times{\cal K})$ which satisfy
$c(\vec{Q}, \vec{Q}')^{\textstyle * } = c(\vec{Q}',\vec{Q})$, which
depend only on the norms $\|\vec{Q}\|$ and $\| \vec{Q}'\|$, and such that
\begin{eqnarray}
\| c \|_{\infty} \ \ \leq \ \ {m^3 \over 2 \ |{\cal K}|} .
\end{eqnarray}
Note that $\cal V$ depends on the two positive parameters $m$ and $R$.
\vspace{4mm}
Let us recall that $c$ is the kernel of the interaction hamiltonian in
the centre-of-mass frame, as function of the momenta. The set $\cal V$
is made of kernels $c$ with compact support and bounded by a constant
depending on the support. Moreover the functions $c(\vec{Q}, \vec{Q}')$
must depend on $\|\vec{Q}\|$ and $\| \vec{Q}'\|$ only. All these
requirements take us away from physical applications.
Each $c \in \cal V$ satisfies the hypothesis of proposition 3 and
moreover satisfies
\begin{eqnarray}
\| c \|_{L^2( I\!\!R^6)} \ \ \leq \ \ {m^3 \over 2}
\end{eqnarray}
where $L^2( I\!\!R^6)$ refers to the ordinary Lebesgue measure (not
the $\sigma_2$ measure). The following set of properties hold on $\cal V$.
\begin{Faux} Let $c \in {\cal V}$.
The function $h$ deduced from $c$ by $(34)$ (or $(35)$) has the following
properties:
$1)$ $h$ satisfies $(23)$,
$2)$ $h(\vec{P},\vec{Q},\vec{Q}')$ depends only on the norms
$\|\vec{P}\|$, $\|\vec{Q}\|$, $\| \vec{Q}'\|$,
$3)$ for all $\vec{Q},\vec{Q}' \in I\!\!R^3$, the function $\vec{P}
\rightarrow h(\vec{P},\vec{Q}, \vec{Q}')$ is a function of $\vec{P}^2$
analytic in a $\ I\!\!\!\!C$-neighbourhood of $I\!\!R_{+}$,
$4)$ for all $\vec{P}^2$ in some $ \ I\!\!\!\!C$-neighbourhood of
$I\!\!R_{+}$, the function $\vec{Q},\vec{Q}' \rightarrow h(\vec{P},
\vec{Q},\vec{Q}')$ has compact support and belongs to $C^2(I\!\!R^6)$,
$5)$ the partial derivatives of $h(\vec{P},\vec{Q},\vec{Q}')$, of any
order w.r.t. $\vec{P}^2$ and of order $\leq 2$ w.r.t. $\vec{Q},\vec{Q}'$,
belong to $\cal B$,
$6)$ $h$ and $Dh$ satisfy the estimates
\begin{eqnarray}
\sup_{\vec{P} \in I\!\!R^3} \int_{I\!\!R^6} d\vec{Q} d\vec{Q}'
|h(\vec{P},\vec{Q},\vec{Q}')|^2 & \leq & \left({2 \over 3}\right)^2
m^6 \nonumber \\
\sup_{\vec{P} \in I\!\!R^3} \int_{I\!\!R^6} d\vec{Q} d\vec{Q}'
|Dh(\vec{P},\vec{Q},\vec{Q}')|^2 & \leq & \left( {1 \over 20}
\right)^2 m^6.
\end{eqnarray}
\end{Faux}
{\bf Proof.} First we establish the convergence of $h =
\sum_{n=0}^{\infty} h_n$, where the $h_n$ are introduced by (36),
by another way. From the estimate on $b$ given by (33) it follows that
$|h_n|_{\cal B}< {\cal N}_n {K'}_1^{n-1} |c|^n_{\cal B}$, where
${\cal N}_n$ is the number of terms of $h_n$. To get an estimate on
${\cal N}_n$ we consider the trivial equation : $x=u+x^2$ with $u \in
I\!\!\!\!C$ and small $|u|$, which is a caricature of the equation (32).
The fixed-point theorem leads to a solution $x = \sum_{n=1}^{\infty}
{\cal N}_nu^n$, convergent for $|u|<1/4$ and unique in the disc
$|x|<1/2$, where the ${\cal N}_n$ are given by ${\cal N}_1=1$
and ${\cal N}_n=\sum_{i=1}^{n-1} {\cal N}_i {\cal N}_{n-i}$ for all
$n>1$. On the other hand, an elementary calculation gives $x= {1
\over 2}(1- \sqrt{1-4u})$ for $|u| \leq 1/4$ as the unique solution near
the origin. By expansion of this solution in power series we obtain
explicit expressions for ${\cal N}_n$, from which follows easily the
estimate ${\cal N}_n < 4^n$ for all $n \geq 1$. By summing up these
estimates we get
\begin{eqnarray}
|h_n|_{\cal B} \ < \ {K'}_1^{n-1} \ {\cal N}_n \ |c|_{\cal B}^n \ <
\ { 1 \over K'_1}
(4K'_1 |c|_{\cal B})^n
\end{eqnarray}
which assures the convergence of $h = \sum_{n=0}^{\infty} h_n$ provided
$4K'_1 |c|_{\cal B}<1$, in agreement with the proof of proposition 3.
\vspace{2mm}
{\em Proof of $1)$}. Because $A(0)=c$ we have $A(0)(\vec{Q},\vec{Q}')
=A(0)(\vec{Q}',\vec{Q})^*$. If $g \in {\cal B}$ satisfies $g(\vec{P},
\vec{Q},\vec{Q}')=g(\vec{P},\vec{Q}',\vec{Q})^*$, so does $b(g,g)$ and
$A(g)$. By an induction argument it follows that $A^n(0)$ satisfies this
symmetry for all $n$, and so does the uniform limit $lim_{n \rightarrow
\infty}A^n(0)$.
\vspace{2mm}
{\em Proof of $2)$}. If $f(\vec{P},\vec{Q},\vec{Q}')$, $g(\vec{P},\vec{Q},
\vec{Q}')$ are only function of $\|\vec{P}\|$, $\|\vec{Q}\|$, $\|\vec{Q}'
\|$, so are $b(f,g)(\vec{P},\vec{Q},\vec{Q}')$ and $Db(f,g)(\vec{P},
\vec{Q},\vec{Q}')$. Thus, if $c(\vec{Q},\vec{Q}')$ is only function of
$\|\vec{Q}\|, \ \|\vec{Q}'\|$, then $h_n(\vec{P},\vec{Q},\vec{Q}')$ is
only function of $\|\vec{P}\|$, $\|\vec{Q}\|$, $\|\vec{Q}'\|$ for all
$n$. This property passes obviously to the uniform limit $h=lim_{N
\rightarrow \infty}\sum_{n=0}^N h_n$.
\vspace{2mm}
{\em Proof of $3)$}. Let us fix $(\vec{Q}, \vec{Q}') \in I\!\!R^6$ and
put $x=\|\vec{P} \|$. In $h_n$, $x$ appears only in kernels $\sqrt{ x^2
+ 4\omega(\vec{\xi})^2}$ or as limit of integrals $\int_0^x dy$ over a
variable $y$ appearing only in kernels $\sqrt{ y^2 + 4\omega
(\vec{\chi})^2}$ or as limit of integrals $\int_0^y dz$, and so on, for
various variables $\vec{\xi},\vec{\chi}$, etc... All these kernels are
analytic in a strip $\{x \in I\!\!\!\!C |\ |Im x|<2m\}$. Moreover the
derivation of $( x^2 + 4\omega(\vec{\xi})^2)^{k/2}$, for any $k \in
Z\!\!\!Z$, gives a function bounded by a constant times $( x^2 +
4\omega(\vec{\xi})^2)^{k/2}$. Thus the integrability is preserved. From
all these properties follows that $h_n(\vec{P},\vec{Q}, \vec{Q}')$ is
an analytic function of $\| \vec{P} \|$ in the strip given above, for
all $(\vec{Q},\vec{Q}') \in I\!\!R^6$.
To study the convergence for $n \rightarrow \infty$ we note that
$\partial_{\| \vec{P} \|}^j h_n$ satisfies the bound (40) multiplied
by a constant due to the derivations of the kernels, and by the factor
$(n-1)^j$ for the number of terms caused by $j$ derivations, according
to the Leibniz formula. These new factors have no influence on the
convergence, and thus $\partial_{\| \vec{P} \|}^j h_n$ converges to an
element of $\cal B$, which is nothing else than $\partial_{\| \vec{P}
\|}^j h$ because of the uniform convergence.
The analyticity is obtained by giving a small fixed imaginary part to $\|
\vec{P} \|$, taking $\| \vec{P} \| + i \eta$ in the definition of $b$
(30), with $ | \eta | $ small enough to have again the estimate (33).
The same argument as before, based on (40) which is still valid, shows
the convergence of $ h_n$ and $\partial_{\| \vec{P} \|}^j h_n$, for all
$n$, uniformly in $\eta$.
To see that $h$ is an analytic function of $\vec{P}^2$ (and not only of
$\| \vec{P} \|$) we note that, if $f(\vec{P},\vec{Q}, \vec{Q}')$,
$g(\vec{P},\vec{Q}, \vec{Q}') \in {\cal B}$ are functions of
$\| \vec{P} \|$, $\| \vec{Q} \|$, $\| \vec{Q}' \|$ that are even in
$\| \vec{P} \|$, the same is true for $b(f,g)(\vec{P},\vec{Q}, \vec{Q}')$.
A glance to (36) suffices to convince ourselves that all the terms of
$h_n$, and thus the limit $h$ itself, are even functions of $\| \vec{P} \|$.
\vspace{2mm}
{\em Proof of $4)$}. The functions $h_n$ can be written as
\begin{eqnarray*}
h_n(\vec{P},\vec{Q}, \vec{Q}') & = & \int d\sigma(\vec{Q}''_1) \cdots
d\sigma(\vec{Q}''_{n-1}) \ c(\vec{Q}, \vec{Q}''_1) \ c(\vec{Q}''_1,
\vec{Q}''_2) \cdots \\
& & \ \cdots c(\vec{Q}''_{n-1}, \vec{Q}') \ \Theta_n(\vec{P},\vec{Q},
\vec{Q}';\vec{Q}''_1, \ldots, \vec{Q}''_{n-1})
\end{eqnarray*}
where the kernels $\Theta_n$ can be deduced from (34), (30), (31) by an
easy (and tedius) calculation. This expression shows clearly that the
support property of $c$ leads to a support property for all $h_n$, and
thus for $h$, w.r.t. $\vec{Q}, \vec{Q}'$.
For $j=1$ or 2, the $j$th derivation of $h_n$ w.r.t. $\vec{Q}$ or
$\vec{Q}'$ is bounded by the estimate (40) multiplied by $(\| \partial
c \|_{\infty}/\| c \|_{\infty})^j$ or $\| \partial^j c \|_{\infty}/\|
c \|_{\infty}$, by a constant due to the derivations of the kernels
$\sqrt{ z + 4\vec{Q}^2}$, $\sqrt{ z' + 4\vec{Q}'^2}$ (for various $z,z'
\in \ I\!\!\!\!C$ with $Rez, Rez'>0$) w.r.t. $\vec{Q}$ or $\vec{Q}'$,
and the factor $(n-1)^j$ for the number of terms caused by $j$
derivations, according to the Leibniz formula. As before, these new
factors have no influence on the convergence, and then these functions
converges to an element of $\cal B$, which is nothing else than the
corresponding derivation of $ h$.
\vspace{2mm}
{\em Proof of $5)$}. We have already seen in the proofs of 2) and 3) that
the derivation of $h$ at any order w.r.t. $\| \vec{P} \|$ and at order
$\leq 2$ w.r.t. $\vec{Q},\vec{Q}'$, belong to $\cal B$. The derivations
w.r.t. $\vec{P}^2$ are bounded by the derivations w.r.t. $\| \vec{P} \|$
for large $\| \vec{P} \|$; for small $\| \vec{P} \|$ the boundedness is
assured by the analyticity of $h$ w.r.t. $\vec{P}^2$.
\vspace{2mm}
{\em Proof of $6)$}. Let us introduce the norm
\begin{eqnarray}
\|f\|_{*} \ := \ m^{-3} \ \sup_{\vec{P} \in I\!\!R^3} \
\sqrt{ \int_{I\!\!R^6} d\vec{Q} \ d\vec{Q}' \
|f(\vec{P},\vec{Q},\vec{Q}')|^2 }
\end{eqnarray}
for all $f \in {\cal B}$. From an easy estimate involving the
Cauchy-Schwartz inequality we get
\begin{eqnarray}
\| b(f,f)\|_{*} & < & {\sqrt{3} \over 8} \ \left( \|f\|_{*}^2+
\|Df\|_{*}^2 \right) \nonumber \\
\|Db(f,f)\|_{*} & < & {\sqrt{3} \over 16} \ \left( \|f\|_{*}^2+
\|Df\|_{*}^2 \right)
\end{eqnarray}
for all $f \in {\cal B}$. Now recall that $h = \lim_{n \rightarrow \infty}
A^n$. Let us suppose that for some $n \in I\!\!N$ we know that
$\|A^n\|_{*}\leq 2/3$ and $\|DA^n\|_{*}\leq 1/20$ (because of (38) this
is true for $n=0$). Then from (38) and (42) we obtain
\begin{eqnarray*}
\|A^{n+1}\|_{*} & = & \|c+b(A^n,A^n)\|_{*} \ \leq \ \|c\|_{*} +
\|b(A^n,A^n)\|_{*} \\
& \leq & {1 \over 2} + {\sqrt{3} \over 8} \ ( \|A^n\|_{*}^2+
\|DA^n\|_{*}^2) \ \leq \ {1 \over 2} + {\sqrt{3} \over 8} \left(
{4 \over 9} + {1 \over 20^2} \right) \ < \ {2 \over 3} \\
\|DA^{n+1}\|_{*} & = & \|Db(A^n,A^n)\|_{*} \ \leq \ {\sqrt{3} \over 16}
\ ( \|A^n\|_{*}^2+ \|DA^n\|_{*}^2) \\
& \leq & {\sqrt{3} \over 16} \left({4 \over 9} + {1 \over 20^2} \right)
\ < \ {1 \over 20}.
\end{eqnarray*}
It follows that these bounds hold for all $n$. Let us take the limit
$n \rightarrow \infty$. Because the convergence of $A^n$ and $DA^n$ is
uniform, the sequences $\|A^n\|_{*}$ and $\|DA^n\|_{*}$ converge too
(taking the integral in (41) as a Riemann integral). Thus the above
estimates still hold in the limit, which establishes (39). []
\vspace{12mm} \newpage
{\large {\bf 4. Unitarity representations of the Poincar\'{e}
group}}\vspace{6mm}
Until now, the representations (13) of the algebra $\cal G$ were
only formal, in the sense that the domains of the operators and of
the commutators were not specified. The proposition 4 brings
the results which allow us to show that these algebra representations
are well defined and can be integrated to give unitary continuous
representations of ${\cal P}_{+}^{ \uparrow}$.
\begin{Christel}
Let $c \in {\cal V}$. Let $\cal O$ be the interaction operator $(22)$
with the kernel $h$ deduced from $c$ in proposition $3$. Then $\{ H,
\vec{P},\vec{J},\vec{L} \}$, given by $(13)$, are the generators of a
unitary continuous representation of ${\cal P}_{+}^{ \uparrow}$.
\end{Christel}
\vspace{3mm}
{\bf Proof.} According to theorem 5 of [13], $ H$, $\vec{P}$, $\vec{J}$,
$\vec{L}$ are the generators of a unitary continuous representation of
${\cal P}_{+}^{ \uparrow}$ if the following three conditions are
satisfied: 1) they are self-adjoint, 2) they admit a common invariant
dense domain $\cal D$ on which the commutation rules of the algebra hold,
3) $\cal D$ is a domain of essential self-adjointness for the operator
$\Delta = m^{-2}(H^2 + {\vec P}^2) + \vec{J}^2 + {\vec L}^2$. This
scheme gives the three natural steps of the proof.
\vspace{2mm}
{\em 1st step: proof that $ H$, $\vec{P}$, $\vec{J}$, $\vec{L}$ are
self-adjoint. } It is clear for $\vec{P}$ and $\vec{J}$. To study $H$
and $\vec{L}$ we need a general estimate of the norm of vectors like
\begin{eqnarray*}
\phi_{\xi}(\vec{P}, \vec{Q}) \ := \ \int { d\sigma(\vec{Q'}) \over
\Omega(\vec{P}, \vec{Q'}) }
\ \phi(\vec{P}, \vec{Q'}) \ \xi(\vec{P}, \vec{Q}, \vec{Q'})
\end{eqnarray*}
for all $\phi \in {\cal H}$ and for kernels $\xi$ such that
\begin{eqnarray}
|\!|\!| \xi |\!|\!|^2 \ \ := \ \ \sup_{\vec{P} \in I\!\!R^3} \ \
\int { d\sigma(\vec{Q}) \over \Omega(\vec{P}, \vec{Q}) } \
\int { d\sigma(\vec{Q}'') \over \Omega(\vec{P}, \vec{Q}'') } \
\left| \xi(\vec{P}, \vec{Q}, \vec{Q}'') \right|^2.
\end{eqnarray}
is well defined. From the Cauchy-Schwarz inequality and the Fubini
theorem we obtain
\begin{eqnarray}
\| \phi_{\xi} \|^2 & = & \int { d\vec{P} d\sigma(\vec{Q}) \over
\Omega(\vec{P}, \vec{Q}) } \left| \int { d\sigma(\vec{Q'}) \over
\Omega(\vec{P}, \vec{Q'}) } \ \phi(\vec{P}, \vec{Q'}) \ \xi(\vec{P},
\vec{Q}, \vec{Q'}) \right|^2 \nonumber \\
& \leq & \int { d\vec{P} d\sigma(\vec{Q}) \over \Omega(\vec{P},
\vec{Q}) } \left[ \int { d\sigma(\vec{Q'}) \over \Omega(\vec{P},
\vec{Q'}) } \left|\phi(\vec{P}, \vec{Q'})\right|^2 \right]
\left[ \int { d\sigma(\vec{Q''}) \over \Omega(\vec{P}, \vec{Q''}) }
\left| \xi(\vec{P}, \vec{Q}, \vec{Q''}) \right|^2 \right] \nonumber \\
& \leq & \ |\!|\!| \xi |\!|\!|^2 \ \| \phi \|^2 .
\end{eqnarray}
For $H$, given by (25), we find the bound on the operator norm
$\|H-H_0\|_{op} \leq |\!|\!|h|\!|\!|$. From (39) it follows that
$|\!|\!|h |\!|\!| \leq m/6$. Thus $H-H_0$ is a bounded operator, and
$H$ is self-adjoint on the domain of $H_0$.
For $\vec{L}$ we consider (26) without the last term (because
$\vec{J}\,^{\vec{Q}}h = \vec{J}\,^{\vec{Q}'}h = 0$). The estimate (44)
leads to
\begin{eqnarray}
\|(L_j-L_{0,j}) \phi \| \ \leq \ |\!|\!|h/\Omega|\!|\!| \
\|L_{0,j} \phi \| \ + \ (|\!|\!|h/\Omega|\!|\!| +
|\!|\!|Dh/\Omega|\!|\!|) \ \| \phi \|
\end{eqnarray}
for all $j \in \{1,2,3\}$. From the Kato-Rellich theorem (sect. X.2 of
[14]) it follows that $\vec{L}$ is self-adjoint on the domain of
$\vec{L}_0$ provided $|\!|\!|h/\Omega|\!|\!|<1$. This condition is
indeed satisfied because from (39): $|\!|\!|h/\Omega|\!|\!|
\leq |\!|\!|h|\!|\!|/2m<1/12$. (Note that the condition
$ |\!|\!|h|\!|\!|/2m<1$ implies the operator norm bound $\|H-H_0\|_{op}
<2m$, from which the positivity of $H$ follows).
\vspace{3mm}
\newline
{\em 2nd step: the invariant domain and the commutation rules. }
Let $\cal D$ be the following domain
\begin{eqnarray}
{\cal D} & := & \left\{ \phi \in {\cal H} \ \left| \ \Omega(\vec{P},
\vec{Q})^{\ell} \ P_1^{n_1} P_2^{n_2} P_3^{n_3} \
L_{0,1}^{m_1} L_{0,2}^{m_2} L_{0,3}^{m_3} \
J_1^{j_1} J_2^{j_2} J_3^{j_3} \
\ \phi(\vec{P},\vec{Q}) \in {\cal H} \right. \right. \nonumber \\
& & \left. \ { \over } \mbox{ for all} \ \
\ell,n_1,n_2,n_3,m_1,m_2,m_3,j_1,j_2,j_3 \in I\!\!N \right\} .
\end{eqnarray}
$\cal D$ is dense (because it contains $C^{\infty}_0(I\!\!R^6)$) and
is clearly left invariant by $H_0$, $\vec{P}$, $\vec{J}$ and $\vec{L}_0$.
To show that it is also invariant under $H$ and $\vec{L}$ it is
sufficient to show that it is invariant under $\cal O$. Let $\phi \in
{\cal D}$ and let us apply an operator $\Omega(\vec{P},\vec{Q})^{\ell}
P_1^{n_1} P_2^{n_2} P_3^{n_3} L_{0,1}^{m_1} L_{0,2}^{m_2} L_{0,3}^{m_3}
J_1^{j_1} J_2^{j_2} J_3^{j_3} $ to the vector ${\cal O}\phi$. From 2),
proposition 4, we get 0 unless $j_1=j_2=j_3=0$. In that case, from the
Leibniz rule we find
\begin{eqnarray}
& & \Omega(\vec{P},\vec{Q})^{\ell} \ P_1^{n_1} P_2^{n_2} P_3^{n_3} \
L_{0,1}^{m_1} L_{0,2}^{m_2}
L_{0,3}^{m_3} \ {\cal O}\phi(\vec{P},\vec{Q}) \ \ = \nonumber \\
& & \sum_{\vec{\alpha} + \vec{\beta} = \vec{m} } C_{\vec{\alpha},
\vec{\beta}}
\int { d\sigma(\vec{Q}') \over \Omega( \vec{P}, \vec{Q}')} \
\left[ { \over } \Omega(\vec{P},\vec{Q}')^{\ell+\beta_1+\beta_2+
\beta_3+1} P_1^{n_1} P_2^{n_2} P_3^{n_3}
L_{0,1}^{\alpha_1} L_{0,2}^{\alpha_2} L_{0,3}^{\alpha_3} \
\phi(\vec{P},\vec{Q}') \right] \nonumber \\
& & \hspace{5mm} \times \ \left\{ \left({\Omega(\vec{P},\vec{Q})
\over \Omega(\vec{P},\vec{Q}')}\right)^{\ell+\beta_1+\beta_2+\beta_3}
\partial_{P_1}^{\beta_1} \partial_{P_2}^{\beta_2}
\partial_{P_3}^{\beta_3} {h(\vec{P},\vec{Q},\vec{Q}') \over
\Omega(\vec{P},\vec{Q}')(\Omega(\vec{P},\vec{Q})+\Omega(\vec{P},
\vec{Q}')) } \right\}
\end{eqnarray}
for some coefficients $C_{\vec{\alpha}, \vec{\beta}}$. Because $\phi
\in {\cal D}$ the expression in [ ] is a vector of $\cal H$. By using
the estimate (44) we find that (47) belongs to $\cal H$ if the
expression in $\{ \ \}$ has a well defined $|\!|\!| . |\!|\!|$ norm.
This is the case because of 3), 4) and 5) of proposition 4, because
the derivations of $\Omega(\vec{P}, \vec{Q}')^{-1} (\Omega(\vec{P},
\vec{Q})+\Omega(\vec{P},\vec{Q}'))^{-1}$ w.r.t. $\vec{P}$ give
bounded functions, and because
\begin{eqnarray}
{\Omega(\vec{P},\vec{Q}) \over \Omega(\vec{P},\vec{Q}')} \ \leq \
{ \| \vec{P} \| \over \Omega(\vec{P},\vec{Q}')} \ +
\ { 2 \omega(\vec{Q}) \over \Omega(\vec{P},
\vec{Q}')} \ \leq \ 1 \ + \ { \omega(\vec{Q}) \over m}
\end{eqnarray}
gives a bounded contribution, due to the compact support of $\vec{Q}
\mapsto h(\vec{P},\vec{Q}, \vec{Q}')$. Thus $\cal D$ is an invariant
domain for all operators $H$, $\vec{P}$, $\vec{J}$ and $\vec{L}$.
By working on $\cal D$ the commutation rules for $\{H, \vec{P}, \vec{J},
\vec{L} \}$ can be deduced from those for $\{H_0, \vec{P}, \vec{J},
\vec{L}_0 \}$ without having to take care of their domain. Thus we have
only to do formal calculations, and thus to apply proposition 2.
\vspace{3mm}
\newline
{\em 3th step: proof that $\cal D$ is a domain of essential
self-adjointness for the operator $\Delta$. } We show that $\Delta$ is
self-adjoint, using theorems by Nelson and Kato-Rellich.
Because $\{ \vec{P}, H_0, \vec{J}, \vec{L}_0 \}$ are the infinitesimal
generators of a unitary continuous representation of a Lie group it
follows from a theorem by Nelson (see [13], theorem 3) that all these
operators and $\Delta_0 = m^{-2}(H^2_0 + {\vec P}^2) + \vec{J}^2 +
{\vec L}^2_0$ are self-adjoint and that there exists a domain
${\cal D}_0$ which is a common invariant domain and a common core for
all of them. However the largest common invariant domain is given by
(46), thus ${\cal D}$ is also a common core. By the Kato-Rellich
theorem (theorem X.12 of [14]), $\Delta$ is essentially self-adjoint
on $\cal D$ if
\begin{eqnarray}
\| ( \Delta - \Delta_0) \phi \| \ \ \leq \ \ k_1 \| \Delta_0 \phi
\| \ + \ k_2 \| \phi \|
\end{eqnarray}
for all $\phi \in {\cal D}$, for some $0From (44) it follows that $\| (H^2 - H_0^2) \phi \| \leq |\!|\!| \Xi
|\!|\!| \ \| \phi \|$, provided $|\!|\!| \Xi |\!|\!|$ is defined,
which gives a contribution to the second term of (49) only. To see the
existence of $|\!|\!| \Xi |\!|\!|$ we use first (48) to obtain
\begin{eqnarray*}
{ \left(\Omega(\vec{P}, \vec{Q}) + \Omega(\vec{P}, \vec{Q}')
\right)^2 \over 2\omega(\vec{Q}) \Omega(\vec{P}, \vec{Q}) \
2\omega(\vec{Q}') \Omega(\vec{P}, \vec{Q}')} \ \ < \ \ {3 \over 2m^2}
\end{eqnarray*}
which gives $|\!|\!| (\Omega+\Omega')h |\!|\!|< m^2 \sqrt{3 /2}
\|h \|_{*}$ (which is bounded by $m^2 \sqrt{2/3}$ from (39)) and then
the Cauchy-Schwarz inequality to get $|\!|\!| \int d\sigma''
hh/\Omega'' |\!|\!| \leq |\!|\!|h |\!|\!|^2 < m^2/36$, from (39) again.
Because in (49) the coefficient of $\|\phi \|$ has not to be bounded,
we omit to estimate it explicitly henceforth. Now let us put
$\vec{L}_h := \vec{L}- \vec{L}_0$. We have to control the
action of $\vec{L}^2 - \vec{L}_0^2 = \vec{L}_0 \cdot \vec{L}_h +
\vec{L}_h \cdot \vec{L}_0 + \vec{L}_h^2$ on vectors $\phi \in {\cal D}$.
Replacing $\phi$ by $\vec{L}_0 \phi$ in (45) leads to the estimate
\begin{eqnarray}
\|\vec{L}_h \cdot \vec{L}_0 \phi\| \ < \ |\!|\!|h/\Omega|\!|\!| \
\|\vec{L}_0^2 \phi \| \ + \ (|\!|\!|h/\Omega|\!|\!| +
|\!|\!|Dh/\Omega|\!|\!|) \ \sum_{j=1}^3 \| L_{0,j}\phi \|
\end{eqnarray}
On the other hand, applying $\vec{L}_0$ to (26) and neglecting the
contribution of $\vec{J}^{\vec{Q}}$ gives
\begin{eqnarray*}
\vec{L}_0 \cdot \vec{L}_h \phi(\vec{P}, \vec{Q}) & = & \int
{d\sigma(\vec{Q}') \over \Omega(\vec{P}, \vec{Q}')} \
{\Omega(\vec{P}, \vec{Q}) \over \Omega(\vec{P}, \vec{Q}')} \ \left[
\vec{L}_0^2 \phi(\vec{P}, \vec{Q}') \right] \
{ h(\vec{P}, \vec{Q}, \vec{Q}') \over \Omega(\vec{P}, \vec{Q}')} \\
& - & i \int {d\sigma(\vec{Q}') \over \Omega(\vec{P}, \vec{Q}')} \
{\Omega(\vec{P}, \vec{Q}) \over \Omega(\vec{P}, \vec{Q}')}
\ \left[ \vec{P} \cdot \vec{L}_0 \phi(\vec{P}, \vec{Q}') \right]
\ \times \\
& & \left[ - 3 { h(\vec{P}, \vec{Q}, \vec{Q}') \over \Omega(\vec{P},
\vec{Q}')^2} + \left( {2 \over \Omega(\vec{P}, \vec{Q}')} +
{1 \over \Omega(\vec{P}, \vec{Q})} \right)
{ Dh(\vec{P}, \vec{Q}, \vec{Q}') \over \|\vec{P} \|} \right] \\
- \ \Omega(\vec{P}, \vec{Q}) \int & d\sigma(\vec{Q}') &
\phi(\vec{P}, \vec{Q}') \ \left( 2 + \| \vec{P} \| \
\partial\!_{ \| \vec P \|} \right) \ \left[ { Dh(\vec{P}, \vec{Q},
\vec{Q}') \over \| \vec{P} \| \Omega(\vec{P}, \vec{Q}')^2} -
{ h(\vec{P}, \vec{Q}, \vec{Q}') \over \Omega(P,q')^3} \right]
\end{eqnarray*}
for all $\phi \in {\cal D}$. Note that in the last term the factor
$ \|\vec{P} \| \partial\!_{ \| \vec P \|} Dh \|\vec{P} \|^{-1}$ which
is well defined, in norm $\|.\|_{*}$, because $h$ is an analytic
function of $\vec{P}^2$ with all derivatives in $\cal B$ (points 3)
and 5) of proposition 4). From the estimate (44) we get
\begin{eqnarray}
\|\vec{L}_0 \cdot \vec{L}_h \phi\| & \leq & |\!|\!|\Omega
h/\Omega'^2|\!|\!| \ \|\vec{L}_0^2 \phi \| \ + \ 3(|\!|\!|\Omega
h/\Omega'^2|\!|\!| + |\!|\!|\Omega Dh/\Omega'^2|\!|\!|)
\ \sum_{j=1}^3 \| L_{0,j}\phi \| \nonumber \\ & & + \ k_3 \|\phi\|
\end{eqnarray}
for some $k_3 \in (0, \infty)$. By applying $\vec{L}_h$ to (26) and
by using (53) we get an estimate of the last term
\begin{eqnarray}
\|\vec{L}_h^2\phi\| & \leq & |\!|\!|h/\Omega|\!|\!| \ \|\vec{L}_0
\cdot \vec{L}_h \phi \| \ + \ (|\!|\!|h/\Omega|\!|\!| +
|\!|\!|Dh/\Omega|\!|\!|) \ \sum_{j=1}^3 \| L_{h,j}\phi \|
\nonumber \\
& \leq & \ |\!|\!|\Omega h/\Omega'^2|\!|\!| \ \|\vec{L}_0^2
\phi \| \ + \ \left[ { \over }3 |\!|\!|h/\Omega|\!|\!| \ \left(
|\!|\!|\Omega h/\Omega'^2|\!|\!| + |\!|\!|\Omega Dh/\Omega'^2|\!|\!|\
\right) \right. \nonumber \\
& & + \left. |\!|\!|h/\Omega|\!|\!| + |\!|\!|Dh/\Omega|\!|\!|
{ \over } \right] \ \sum_{j=1}^3 \| L_{0,j}\phi \| \ + \ k_4 \|\phi\|
\end{eqnarray}
for some $k_4 \in (0, \infty)$. Finally by collecting the estimates (52),
(53), (54) we obtain
\begin{eqnarray}
\| (\Delta - \Delta_0) \phi \| & \leq &
\left( \left|\!\left|\!\left| { h \over \Omega }
\right|\!\right|\!\right|
+ \left|\!\left|\!\left| { Dh \over \Omega } \right|\!\right|\!\right|
\right) \|\vec{L}_0 ^2 \phi \|
\ + \ \left[ 2\left|\!\left|\!\left| { h \over \Omega }
\right|\!\right|\!\right| + 2\left|\!\left|\!\left|
{ Dh \over \Omega } \right|\!\right|\!\right| \right. \nonumber \\
& & + \ \left. 3 \left( 1+ \left|\!\left|\!\left| { h \over \Omega }
\right|\!\right|\!\right| \right)
\left( \left|\!\left|\!\left| { \Omega h \over {\Omega'}^2 }
\right|\!\right|\!\right| + \left|\!\left|\!\left| { \Omega Dh \over
{\Omega'}^2 } \right|\!\right|\!\right| \right)
\right] \ \ \sum_{j=1}^3 \| L_{0,j} \phi \| \nonumber \\
& & + \ \ k_5 \ \| \phi \|
\end{eqnarray}
for some $k_5 \in (0, \infty)$ and all $\phi \in {\cal D}$. It remains
to find a bound on $\|L_{0,j} \phi\|$, $\|\vec{L}_0^2 \phi\|$ of the
type $k_{6}\| \Delta_0 \phi\| + k_{7} \|\phi \|$ for constants $k_{6}$
and $k_{7}$. Following Nelson ([13], proof of lemma 6.1) we find
\begin{eqnarray*}
\sum_{j=1}^3 \| L_{0,j} \phi \| & \leq &
\sqrt{3} \left[ \sum_{j=1}^3 \left( \| L_{0,j}\phi \|^2 + m^{-2}
\| P_j \phi \|^2 + \| J_j \phi \|^2 \right) + m^{-2} \left\|H_0\phi
\right\|^2 \right]^{1/2} \\
& = & \sqrt{3} \ \left(\phi,\Delta_0\phi \right)^{1/2}
\ < \ \sqrt{3} \ \left(\phi,({\scriptstyle {1 \over 2}} \Delta_0^2 +
\Delta_0 + {\scriptstyle {1 \over 2}}) \phi\right)^{1/2} \\
& = & \sqrt{3/2} \ \|(\Delta_0+1) \phi \| \ \leq \ \sqrt{ 3/2} \
\left(\| \Delta_0 \phi \| + \|\phi\| \right).
\end{eqnarray*}
The bound on $\|\vec{L}_0^2\phi\|$ needs more development. Following [13],
proof of lemma 6.1 we try to write $\Delta_0^2- (\vec{L}_0^2)^2$ as a sum
of positive and negative operators. For that we use the commutation rules
(1) to (4), which imply
\begin{eqnarray*}
\vec{P}^2\vec{L}^2 + \vec{L}^2\vec{P}^2 & = & 2\sum_{j=1}^3 P_j
\vec{L}^2 P_j \ - \ 6 H^2 \\
\vec{P}^2\vec{J}^2 + \vec{J}^2\vec{P}^2 & = & 2\sum_{j=1}^3 P_j
\vec{J}^2 P_j \ - \ 6 \vec{P}^2 \\
\vec{J}^2\vec{L}^2 + \vec{L}^2\vec{J}^2 & = & 2\sum_{j=1}^3 J_j
\vec{L}^2 J_j \\
H^2\vec{L}^2 + \vec{L}^2H^2 & = & 2H \vec{L}^2 H \ - \ 2\vec{P}^2 .
\end{eqnarray*}
These relations imply that $\Delta_0^2- (\vec{L}_0^2)^2 +
2m^{-2}(2 \vec{P}^2 +3H_0^2) $ is positive. Then
$$ (\vec{L}_0^2)^2 \ < \ \Delta_0^2 + 2m^{-2}(2 \vec{P}^2 +3H_0^2)
\ < \ \Delta_0^2 +6\Delta_0 + 9 \ = \ (\Delta_0 + 3 )^2 $$
and finally $\|\vec{L}_0^2 \phi\| < \|(\Delta_0 +3 ) \phi\| \leq
\|\Delta_0 \phi\| + 3 \|\phi\|$. By collecting all these results the
estimate (55) can be written as (49) with
\begin{eqnarray}
k_1 & = &
\left(\left|\!\left|\!\left|{ h\over \Omega} \right|\!\right|\!\right|
+ \left|\!\left|\!\left|{ Dh\over \Omega} \right|\!\right|\!\right|
\right) \nonumber \\
& + & \sqrt{3 \over 2} \ \left[ 2\left|\!\left|\!\left|{ h\over \Omega}
\right|\!\right|\!\right| + 2\left|\!\left|\!\left|{ Dh\over \Omega}
\right|\!\right|\!\right| + 3\left(1+ \left|\!\left|\!\left|{ h\over
\Omega} \right|\!\right|\!\right| \right)
\left( \left|\!\left|\!\left|{ \Omega h\over \Omega'^2}
\right|\!\right|\!\right| + \left|\!\left|\!\left|{ \Omega Dh\over
\Omega'^2} \right|\!\right|\!\right| \right) \right]
\end{eqnarray}
Now from (39) it follows that
\begin{eqnarray}
\left|\!\left|\!\left|{ h\over \Omega} \right|\!\right|\!\right|
\ \leq \ {\left|\!\left|\!\left| h \right|\!\right|\!\right| \over 2m}
\ \leq \ {\|h\|_{*} \over 16} \ \leq \ {1 \over 24} \hspace{12mm}
\left|\!\left|\!\left|{ Dh\over \Omega} \right|\!\right|\!\right|
\ \leq \ {\|Dh\|_{*} \over 16} \ \leq \ {1 \over 320}
\end{eqnarray}
where $\| \ \|_{*}$ is the norm (41), and, with $\Omega(\vec{P},
\vec{Q})/\Omega(\vec{P},\vec{Q}') \leq 2 \omega(\vec{Q})
\omega(\vec{Q}') /m^{-2}$ from (52), we get
\begin{eqnarray}
\left|\!\left|\!\left|{ \Omega h\over \Omega'^2}
\right|\!\right|\!\right| \ \leq \ {\left|\!\left|\!\left|
\omega \omega' h \right|\!\right|\!\right| \over m^3}
\ \leq \ {\|h\|_{*} \over 8} \ \leq \ {1 \over 12} \hspace{12mm}
\left|\!\left|\!\left|{ \Omega Dh\over \Omega'^2}
\right|\!\right|\!\right| \ \leq \ {\|Dh\|_{*}
\over 8} \ \leq \ {1 \over 160}
\end{eqnarray}
The substitution of (57) and (58) in (56) gives $k_1 < 1/2 <1$. []
\vspace{12mm}
{\large {\bf 5. Bound states and relativistic Schr\"{o}dinger
equation}}\vspace{5mm}
We have obtained a family of unitary representations of the Poincar\'{e}
group by doing perturbations of the two free boson model. We have now to
investigate the physics of these models, in order to see the effective
differences between them. In a quantum theory, the physics of a model
reduces to two problems, the search for the bound states and the
construction of the scattering matrix.
We begin with the first problem. In a relativistic theory, the bound
states are related to the discrete part of the spectrum of the mass
operator $M$. More precisely a bound state has a mass $m_B$ solution
to the eigenvalue problem $M \phi = m_B \phi$, with $\phi \in D(M)$,
and $\phi \neq 0$. In our models this equation is better written for
$M^2$ and from (50) it takes the form
\begin{eqnarray}
M^2 \phi(\vec{P},\vec{Q}) & = & 4(\vec{Q}^2 + m^2) \ \phi(\vec{P},
\vec{Q}) \ + \ \int { d \sigma(\vec{Q}') \over
\Omega(\vec{P},\vec{Q}')} \ \phi(\vec{P},\vec{Q}')
\ \Xi(\vec{P},\vec{Q},\vec{Q}') \nonumber \\
& = & \ m_B^2 \ \phi(\vec{P},\vec{Q})
\end{eqnarray}
with $\phi \in D(M^2)$, $\phi \neq 0$, where the kernel $\Xi$ is given
by (51). Here $\vec{P}$ plays the role of a parameter. The function
$\phi$, when it exists, is continuous on $\vec{P}$ (see the proof of
proposition 6 below). So one can take $\vec{P}=0$ in (59) (choosing the
centre-of-mass frame) and this equation reduces to
\begin{eqnarray}
4(\vec{Q}^2 + m^2) \ \varphi(\vec{Q}) \ + \
\int {d\vec{Q}' \over 4(\vec{Q}'^2+m^2)} \ \varphi(\vec{Q}')
\ \Xi(0,\vec{Q},\vec{Q}') \ = \ m_B^2 \ \varphi(\vec{Q})
\end{eqnarray}
where $\varphi(\vec{Q}) = \phi(0,\vec{Q})$. Now for $\vec{P}=0$, $M^2$
is just $H^2$. Thus the square root of the operator involved in (60)
exists and leads to
\begin{eqnarray}
2 \sqrt{\vec{Q}^2+m^2} \ \varphi(\vec{Q}) \ + \
\int {d\vec{Q}' \over 4(\vec{Q}'^2+m^2)} \
\varphi(\vec{Q}') \ c(\vec{Q},\vec{Q}') \ \ =
\ \ m_B \ \varphi(\vec{Q}).
\end{eqnarray}
The operator $M$, the positive square root of the operator $M^2$
apearing in (59), satisfies the following properties.
\vspace{1mm}
\begin{Faux}
For all $c \in {\cal V}$ the operator $M$ is essentially self-adjoint
on the domain $\cal D$ given by $(46)$. Moreover, the spectrum of $M$
is contained in $(0, \infty)$ where $(0,2m)$ contains at most a finite
number of eigenvalues $m_B$, which are solutions to the eigenvalue
equation $(61)$.
\end{Faux}
\vspace{3mm}
{\bf Remark.} The equation (59) (or its reduced forms (60), (61))
can be conside\-red as a {\em relativistic Schr\"{o}dinger equation}
because it plays the role of the Schr\"{o}dinger equation in Quantum
Mechanics: it generates the discrete structure of the bound states.
\vspace{3mm} \newline
{\bf Historical remark.} Looking for a relativistic equation for the
Hydrogen atom, Dirac rejected the equation: $(\sqrt{ - \Delta +m^2}
-e^2 /r)f = Ef$, which is the first natural attempt to a relativistic
generalization of the Schr\"{o}dinger equation. The reason was that
it is not invariant under Lorentz transformations (see the discussion
in [16]). Now, the analogue of this equation in our models is (61),
which indeed cannot satisfy such an invariance, because it has been
obtained in a given referential frame (what is more, $\vec{Q}$ is
not an ordinary momentum, because $\| \vec{Q} \|$ is Lorentz
invariant). In fact the relativistic equation is (59) and however,
its invariance is not obvious: it is invariant under the
transformations of the variables $(\vec{P},\vec{Q})$ deduced from the
generators $\vec{L}$ given by (26).
\vspace{2mm} \newline
{\bf Proof of proposition 6.}
Note that the spectrum of $M_0^2$, given by (20), covers the interval
$[4m^2, \infty)$ and is absolutely continuous.
$M^2_0$ is clearly essentially self-adjoint on $\cal D$. From (44) we
have the bound on the operator norm $\|M^2-M_0^2\|_{op} \leq |\!|\!|
\Xi |\!|\!|$ which is bounded by $ |\!|\!| \Xi |\!|\!| \leq
m^2(\sqrt{2/3} + 1/36)$ from the estimates just after (51). It follows
that $M^2$ is essentially self-adjoint on $\cal D$, and so is $M$.
Because $H$ and $\vec{P}$ commute they admit a simultaneous spectral
measure $d{\cal E}(\rho,\vec{P})$ where $\rho>0$ is the spectral variable
associated with $H$. In the spectral representation $M$ becomes the
multiplication operator by $(\rho^2 - \vec{P}^2)^{1/2}$. Because $M$
commutes with the Poincar\'{e} representation, the invariant subspaces
are limited in the $(\rho, \vec{P})$ space by half hyperbolas $\rho =
( \vec{P}^2 + K^2 )^{1/2} $ with $K >0$, and so is the support of
$d{\cal E}$.
Let us introduce the family of Hilbert spaces
\begin{eqnarray}
{\cal H}_{\!\vec{P}} := L^2(I\!\!R^3, d\sigma_{\!\!\vec{P}})
\hspace{5mm} \mbox{ where } \hspace{5mm}
d\sigma_{\!\!\vec{P}}(\vec{Q}) :=
d\sigma(\vec{Q})\Omega(\vec{P},
\vec{Q})^{-1}.
\end{eqnarray}
and $\vec{P} \in I\!\!R^3$ plays the role of three parameters.
The original Hilbert space $\cal H$ is the direct integral
\begin{eqnarray}
{\cal H} \ \ = \ \ \int_{I\!\!R^3 } \ \bigoplus \
{\cal H}_{\!\!\vec{P}} \ d{\vec P}
\end{eqnarray}
(see sect. 1.5 of [19]). We also consider the operators
$H_{0,\vec{P}}$ and $H_{\!\vec{P}}$, the restrictions of $H_0$ and $H$
to ${\cal H}_{\!\!\vec{P}}$, still given by the same formulas as
before, but with fixed $\vec{P}$. That is, $H_{\!\vec{P}}$ is given by
\begin{eqnarray*}
H_{\!\vec{P}} \varphi(\vec{Q}) \ = \ H_{0,\vec{P}} \varphi(\vec{Q})
\ + \ \int d \sigma_{\!\!\vec{P}}(\vec{Q}') \ \varphi(\vec{Q}')
\ h(\vec{P},\vec{Q},\vec{Q}')
\end{eqnarray*}
for suitable $\varphi$, where $H_{0,\vec{P}} \varphi(\vec{Q}) =
\Omega(\vec{P}, \vec{Q}) \varphi(\vec{Q})$.
>From (39) the integral $\int |h(\vec{P},\vec{Q},\vec{Q}')|^2
d\sigma_{\!\!\vec{P}}(\vec{Q}) d\sigma_{\!\!\vec{P}}(\vec{Q}')$ is well
defined. Thus $H_{\!\vec{P}}$ is a compact perturbation of $H_{0,\vec{P}}$.
By the `classical Weyl theorem' (sect. XIII.4 of [15]), the essential
spectrum of $H_{\!\vec{P}}$ is that of $H_{0,\vec{P}}$, that is the
interval $[\Omega(\vec{P},0), \infty)$, and it may exist a finite number
of eigenvalues in $(0, \Omega(\vec{P},0))$ of finite multiplicities.
To see how the possible eigenvalues depend on $\vec{P}$ we note that
$\vec{P}^2 \rightarrow H_{\!\vec{P}}-H_{0,\vec{P}}$ is an analytic
family of compact operators in a $ \ I\!\!\!\!C^2$-neighbourhood of
$I\!\!R_{+}$. This property holds because by giving a small
imaginary value to $\vec{P}^2$, the integral $\int |h(\vec{P},\vec{Q},
\vec{Q}')|^2 d\sigma_{\!\!\vec{P}}(\vec{Q}) d\sigma_{\!\!\vec{P}}
(\vec{Q}')$ is still defined, because of 4), proposition 4. Thus
$H_{\!\vec{P}}$ is a `holomorphic family of type (A)' according to [17],
and by theorems 1.7 and 1.8 of sect. VII of [17] the eigenvalues
(if any) are continuous functions of $\vec{P}$ and the eigenvectors
can be chosen continuous. In the $( \rho,\vec{P})$-space the eigenvalues
form continuous curves. Now, from the previous discussion about the
support of $d{\cal E}(\rho,\vec{P})$, these curves can only be half
hyperbolas $\rho = ( \vec{P}^2 + m_B^2 )^{1/2} $, for a finite number
of $m_B \in (0,2m)$ solutions to the eigenvalue equation for $\vec{P}=0$,
that is to (61). Going back to the operator $M^2_{\!\vec{P}} =
H_{\!\vec{P}}^2 - \vec{P}^2$ we conclude that its spectrum is contained
in the interval $[4m^2, \infty)$ possibly completed with a finite number
of eigenvalues $m_B$ independant of $\vec{P}$, of finite multiplicities
and lying in the interval $(0,4m^2)$.
Because the spectrum of $M^2_{\!\vec{P}}$ is independant of $\vec{P}$,
it is also the spectrum of $M^2$ (but now the dimension of the eigenspaces,
as subspaces of $\cal H$, is infinite). []
\vspace{12mm}
{\large {\bf 6. Scattering states and scattering matrix}}\vspace{5mm}
We continue to investigate the physical content of our models by
exploiting now the scattering theory. In Quantum Field Theory the
notion of interacting particle is ambiguous. Particles can only be
recognized in the asymptotic states, where the interaction is no more
efficient. In our models the construction of asymptotic states
describing two free particles will confirm that they effectively
concern two particle systems. Moreover the asymptotic completeness,
when it holds, will imply that only two particle states are present
(apart from the bound states) in the Hilbert state space.
A vector $\psi \in {\cal H}$ is said to be a `two particle scattering
state' if its time evolution, for very large time (or very small time),
is not distinguishable from the two free particle evolution, that is,
if there exist $\psi_{out} $ (or $\psi_{in}$) in ${\cal H}$ such that
$\psi= U^{-} \psi_{out}$ (or $\psi = U^{+} \psi_{in}$), where the
`wave operators' $U^{+}$, $U^{-}$ are given by
\begin{eqnarray}
U^{\pm} \ := \ s-\lim_{t \rightarrow \mp \infty} \ e^{ \textstyle itH}
\ e^{ -\textstyle itH_0} .
\end{eqnarray}
Here $H_0$ is the two free particle Hamiltonian (17). When the operators
(64) exist, they map any vector of $\cal H$ on a two-particle scattering
state, and so the study of two particle scattering can be undertaken.
\vspace{1mm}
\begin{Faux}
For all $c \in {\cal V}$ the operators $U^{\pm}$ exist on all $\cal H$.
\end{Faux}
\vspace{3mm}
{\bf Proof.}
We apply Cook's method
(theorem XI.4 of [18]), with $A=H$ and $B=H_0$. Note that $D(H)=D(H_0)$
because $H-H_0 $ is bounded. Let us consider the function
\begin{eqnarray*}
F(t,\vec{P},\vec{Q}) \ := \ \int { d \sigma(\vec{Q}') \over
\Omega(\vec{P},\vec{Q}')} \ \phi(\vec{P},\vec{Q}')
\ h(\vec{P},\vec{Q}, \vec{Q}') \ e^{\textstyle -it
\Omega(\vec{P},\vec{Q}')}
\end{eqnarray*}
for $\phi \in C_0^{\infty}(I\!\!R^6)$ satisfying $\phi(\vec{P},(0,Q_2))
=\partial_{Q_1}\phi(\vec{P},(0,Q_2))=0$. Such vectors $\phi$ generate
a dense subspace of $\cal H$. By performing two integrations by parts
w.r.t. $Q'_1$ we get
\begin{eqnarray}
& & F(t,\vec{P},\vec{Q}) = \nonumber \\
& & - {1 \over 32 t^2} \int d\vec{Q}'
e^{\textstyle -it \Omega(\vec{P},\vec{Q}')}
\partial_{Q_1'} \left( {\Omega(\vec{P},\vec{Q}') \over Q'_1} \
\partial_{Q_1'} { \phi(\vec{P},\vec{Q}')
h(\vec{P},\vec{Q}, \vec{Q}') \over Q'_1 \omega(\vec{Q}') } \right).
\end{eqnarray}
We recall that $Q_1 \rightarrow h(\vec{P},\vec{Q}, \vec{Q}')$ belongs to
$C^2_0(I\!\!R)$ by 4), proposition 4. Thus the r.h.s. of (65) is a
bounded function of $\vec{P},\vec{Q}$ with compact support and then
\begin{eqnarray*}
t \mapsto \left[ \int { d\vec{P} \ d \sigma(\vec{Q}) \over
\Omega(\vec{P},\vec{Q})} \ \left|F(t,\vec{P},\vec{Q})\right|^2
\right]^{1/2} \ = \ \left\|
(H-H_0) e^{ \textstyle -itH_0} \phi \right\|
\end{eqnarray*}
is in $L^1([1, \infty))$, as required by Cook's method. []
\vspace{2mm}
Proposition 7 leads to the existence of two particle states, defined as
the states contained in the range of the operators $U^{\pm}$. Now two
particle scattering processes can be obtained by computing the matrix
elements of the scattering operator $S:=U^{- *}U^+$. However the
question of the existence of other states (apart from the bound states)
naturally arises, which is related to the asymptotic completness problem.
We will give a partial answer to this question under an new condition
on $c(\vec{Q},\vec{Q}')$, the interaction kernel in the centre-of-mass
frame. In Quantum Mechanics, the asymptotic completeness problem is
stated once the separation of the centre-of-mass motion has been done.
In the relativistic case, this separation cannot be performed, but we
can restrict ourselves to a particular frame, given by the choice of a
fixed $\vec{P} \in I\!\!R^3$. As in the proof of proposition 6 we
consider the Hilbert space decomposition (62), (63) and the operators
$H_{\!\vec{P}}$, $H_{0,\vec{P}}$, $\vec{L}_{\!\vec{P}}$,
$\vec{J}_{\!\vec{P}}$ and $M_{\!\vec{P}}$, acting on suitable
subspaces of ${\cal H}_{\!\!\vec{P}}$, still given by the
same formulas as before, but with fixed $\vec{P}$. Now for each
$\vec{P}$ the 'wave operators'
\begin{eqnarray}
U^{\pm}_{\!\vec{P}} \ = \ s-\lim_{t \rightarrow \mp \infty} \
e^{ \textstyle itH_{\!\vec{P}}}
\ e^{ -\textstyle itH_{0,\vec{P}}}
\end{eqnarray}
can be formed on ${\cal H}_{\!\!\vec{P}}$ (the existence of the strong
limit is easily established by repeating the proof of proposition 7,
with fixed $\vec{P}$). The question of the asymptotic completeness will
be discussed in ${\cal H}_{\!\!\vec{P}}$. According to [18], it consists
of two statements: 1) the Hamiltonian $H_{\!\vec{P}}$ has no singular
continuous spectrum and 2) the ranges of $U^{\pm}_{\!\vec{P}}$ coincide
with the subspace of ${\cal H}_{\!\!\vec{P}}$ corresponding to the
absolutely continuous part of the spectrum of $H_{\!\!\vec{P}}$. Only
the second statement, called the 'completeness of the wave operators',
will be considered here. We note that it is sufficient to insure the
unitarity of the scattering operator $S_{\!\vec{P}}:=U_{\!\vec{P}}^{- *}
U_{\!\vec{P}}^+$.
Let us take an interaction kernel in the centre-of-mass $c \in {\cal V}$
satisfying moreover
\begin{eqnarray}
c \in C^3_0(I\!\!R^6)
\end{eqnarray}
which implies that the integration operator of kernel $c$ is trace class
(see appendix D). We will see that the condition (67) implies the
completeness of the wave operators $U^{\pm}_{\!\vec{P}}$ for all
$\vec{P} \in I\!\!R^3$.
\vspace{1mm}
\begin{Faux}
For all $c \in {\cal V}$ satisfying (67) and for all $\vec{P} \in
I\!\!R^3$, the ranges of the operators $U_{\!\vec{P}}^{\pm}$ coincide
with the absolutely continuous subspace of $H_{\!\vec{P}}$ in
${\cal H}_{\!\!\vec{P}}$ (in other words, the wave operators are
complete). Then the scattering operator $S$ is unitary.
\end{Faux}
\vspace{3mm}
{\bf Proof.}
In appendix D we show that any $c \in {\cal V}$ satisfying (67) is the
kernel of a trace class operator in ${\cal H}_{\!\!\vec{P}=0}$, which
implies that the ranges of $U_{\!\vec{P}=0}^{\pm}$ coincide with the
absolutely continuous subspace of $H_{\!\vec{P}=0}$ (Kato-Rosenblum
theorem, corollary 2, sect. 6.2, [19]). The proposition affirms that
this is also the case for all $\vec{P} \neq 0$. This holds if the
operator $V_{\!\vec{P}} :=H_{\!\vec{P}}-H_{0,\vec{P}}$ is trace class
in ${\cal H}_{\!\!\vec{P}}$ for all $\vec{P} \in I\!\!R^3$. Now
$V_{\!\vec{P}}$ is an integral operator with kernel $h_{\!\vec{P}}
(\vec{Q},\vec{Q}') := h(\vec{P},\vec{Q},\vec{Q}')$ (for fixed
$\vec{P}$). The proof of point 4), proposition 4 shows that
$(\vec{Q},\vec{Q}') \mapsto h(\vec{P},\vec{Q},\vec{Q}')$, for fixed
$\vec{P}$, lies in the same class $C^p_0(I\!\!R^6)$ as $c$. Thus
$h_{\!\vec{P}} \in C^3_0(I\!\!R^6)$, and appendix D can be used again,
to conclude the proof. []
\vspace{2mm}
The full asymptotic completeness (with absence of singular continuous
spectrum) needs probably more restrictions on $c$ (see [6] for the
2-d case).
\vspace{12mm}
{\large {\bf 7. An example and the non-triviality}}\vspace{5mm}
To establish the non-triviality we consider an example for which $c$ is
the kernel of a one dimensional range operator. This has the advantage
of leading to explicit results.
Let $\zeta\in C^3_0([0, \infty))$ with support in $[0,R]$, with uniform
norm bounded by
\begin{eqnarray}
\|\zeta\|_{\infty} \ \leq \ \sqrt{ 3 \over 8 \pi } \ \left({m \over R}
\right)^{3 / 2}
\end{eqnarray}
and bounded below by
\begin{eqnarray}
\zeta(r) \ \geq \ \left\{ \begin{array}{l} \sqrt{ 1 \over 4 \pi }
\ \left({\textstyle m \over \textstyle R} \right)^{3 / 2}
\hspace{10mm} \mbox{ if } \ r< {\textstyle R \over \textstyle 2}
\\ 0 \hspace{10mm} \mbox{ if } \ r \geq
{\textstyle R \over \textstyle 2} \end{array} \right.
\end{eqnarray}
where $R$ and $m$ are the parameters introduced in (37).
\vspace{1mm}
\begin{Faux}
Let $c$ be the function given by
\begin{eqnarray}
c(\vec{Q},\vec{Q}') \ = \ \lambda \ \zeta(\|\vec{Q}\|) \
\zeta(\|\vec{Q}'\|)
\end{eqnarray}
for all $(\vec{Q},\vec{Q}') \in I\!\!R^6$, $\zeta$ as above and
$\lambda \in I\!\!R$. For all $|\lambda| \leq 1$ the function $c$
belongs to $\cal V$ and thus is the interaction kernel in the
centre-of-mass frame of a continuous unitary representation of
the Poincar\'{e} group. Moreover the two-particle wave operators
are defined and are complete. The scattering operator $S$ and the
mass operator $M$ satisfy
i) $S \neq Id$
ii) $\lambda =-1 $ and $m/R>2$ imply that $M$ has an eigenvalue $m_B
\in (0,2m)$.
\end{Faux}
\vspace{3mm}
{\bf Proof.}
The function $c$ of (70) belongs to $\cal V$ because it has the right
support and because (68) leads to (37). Thus from theorem 5, $c$ is the
interaction kernel in the centre-of-mass frame of a continuous unitary
representation of the Poincar\'{e} group. Moreover $c$ satisfies (67).
Then the existence and the completeness of the wave operators follow
from propositions 7 and 8. In particular, the scattering operator $S$ is
defined and unitary.
\vspace{1mm}
\newline
{\em Proof of i)}. Note that $\cal H$ carries two different unitary
continuous representations of the Poincar\'{e} group, the free
representation generated by $\{ H_0, \vec{P}, \vec{J},\vec{L}_0
\}$ and the interaction representation generated by $\{ H, \vec{P},
\vec{J}, \vec{L} \}$. Because $H_0$ and $\vec{P}$ commute they admit
a simultaneous spectral measure $d{\cal E}_0(E,\vec{P})$ where $E>0$
is the spectral variable associated with $H_0$. In the spectral
representation $M_0$ becomes the multiplication operator by
$(E^2 -\vec{P}^2)^{1/2}$. Because $M_0 \geq 2m$ the support of
$d{\cal E}_0$ is $\{(E,\vec{P}) | E^2 -\vec{P}^2 \geq 4m^2 \}$.
For such $(E,\vec{P})$ let us consider the space $d{\cal E}_0(E,
\vec{P}) {\cal H} =: {\cal H}_{\!E,\vec{P}}$ which appears in the
spectral decomposition of ${\cal H}_{\!\!\vec{P}}$ which diagonalizes
$H_{0,\vec{P}}$
\begin{eqnarray*}
{\cal H}_{\!\!\vec{P}} \ \ = \ \ \int_{sp(H_{0,\vec{P}})} \ \bigoplus
\ {\cal H}_{\!E,\vec{P}} \ dE
\end{eqnarray*}
where $sp(H_{0,\vec{P}})$ is the spectrum of $H_{0,\vec{P}}$, that is
the interval $[\Omega(\vec{P},0), \infty)$. The operator $S_{\!\vec{P}}$
restricted to ${\cal H}_{\!E,\vec{P}}$ is denoted $S_{\!E,\vec{P}}$
and is called the 'scattering matrix'. The action of $\vec{L}_0$ on the
variables $(E,\vec{P})$ is easily computed. For $\vec{\beta} \in
I\!\!R^3$, $\|\vec{\beta}\| <1$, the Lorentz boost $L_0(\vec{\beta})$
of speed $\vec{\beta}$ (from which $\vec{L}_0$ are the infinitesimal
generators) acts as follows
\begin{eqnarray}
L_0(\vec{\beta}) \ {\cal H}_{\!E,\vec{P}} \ \ = \ \
{\cal H}_{\Lambda_{\vec{\beta}} (E,\vec{P})}
\end{eqnarray}
where $\Lambda_{\vec{\beta}}$ is the Lorentz matrix given in (79),
appendix B. It follows from (71) that the relation between scattering
matrix is
\begin{eqnarray}
L_0(\vec{\beta}) \ S_{\!E,\vec{P}} \ L_0(\vec{\beta})^{-1} \ \ =
\ \ S_{\Lambda_{\vec{\beta}} (E,\vec{P})}
\end{eqnarray}
which shows that $S$ is known once $S_{\!E,0}$ is given. Let us
restrict ourselves to ${\cal H}_{\!\vec{P}=0}$ on which the interaction
operator $V_0=H_{\vec{P}=0}-H_{0,\vec{P}=0}$ has one-dimensional range,
that is $V_0 \varphi = \lambda (\eta, \varphi)_0 \eta$ where
$(.,.)_0$ is the scalar product of ${\cal H}_{\!\vec{P}=0}$ and
$\eta(\vec{Q}) = \zeta(\| \vec{Q} \|)$. According to [19], theorem 3
of sect. 6.7, the scattering operator gives for all $v \in
{\cal H}_{\!E,\vec{P}=0}$
\begin{eqnarray}
\left( v,(S_{\!E,0} - I) v \right)_{E,0} \ = \
- 2 \pi i \lambda \ D(E+i0)^{-1} \ \left| \left(\eta_E, v
\right)_{E,0} \right|^2
\end{eqnarray}
where
\begin{eqnarray}
D(z) \ = \ 1 \ + \ \lambda \left(\eta, (H_{0,0}-z)^{-1} \eta
\right)_0
\end{eqnarray}
for all $z \in \ I\!\!\!\!C$, $Imz>0$. In (73), $(.,.)_{E,0}$ is the
scalar product of ${\cal H}_{\!E,\vec{P}=0}$ and $\eta_E$ is the
restriction of $\eta$ to ${\cal H}_{\!E,\vec{P}=0}$. In fact $\eta_E=
\zeta({1 \over 2} \sqrt{E^2-4m^2})$ is a constant in ${\cal H}_{\!E,
\vec{P}=0}$ so that $(\eta_E, v)_{E,0}= \zeta({1 \over 2} \sqrt{E^2-
4m^2}) (1, v)_{E,0}$, which is not zero for all $2m< E <\sqrt{R^2+
4m^2}$ (see the definition of $\eta$) and for all $v$ in a large set
of ${\cal H}_{\!E,\vec{P}=0}$. On the other hand, for $E$ as above let
us compute the limit needed in (73)
\begin{eqnarray*}
D(E+i0) \ = \ 1 & - & \lambda i \pi^2 \ {\sqrt{E^2-4m^2} \over 2E}
\ \left| \zeta \left({1 \over 2} \sqrt{E^2-4m^2}\right) \right|^2 \\
& + & {\pi \over 2} \ p. v. \int_{2m}^{2\sqrt{R^2+m^2}} \ {du \over u}
\ {\sqrt{u^2-4m^2} \over u-E } \
\left| \zeta \left({1 \over 2} \sqrt{u^2-4m^2}\right) \right|^2
\end{eqnarray*}
which has obviously a non zero imaginary part. Then $S_{\!E,0} - I$ is
non-zero for all such $E$. Because the representation in $\cal H$
generated by $\{ H_0, \vec{P}, \vec{J},\vec{L}_0 \}$ is continuous,
if follows from (72) that the scattering operator $S$ differs from
identity.
\vspace{1mm}
\newline
{\em Proof of ii)}. From proposition 6 with (70) and $\lambda =-1$ the
equation (61) can be written as follows
\begin{eqnarray*}
\varphi(\vec{Q}) \ = \ K \ {\zeta(\|\vec{Q}\|) \over
2\omega(\vec{Q})- m_B}
\end{eqnarray*}
where $K$ is a constant. This function belongs to $D(M_{\vec{P}=0})
\subset {\cal H}_{\vec{P}=0}$ and is an eigenfunction of $M_{\vec{P}=0}$
provided the implicit equation
\begin{eqnarray}
1 \ = \ \int {d\vec{Q} \over 4 (\vec{Q}^2+m^2)} \
{\zeta(\|\vec{Q} \|)^2 \over 2\sqrt{\vec{Q}^2+m^2}-m_B} .
\end{eqnarray}
admits a solution $0 < m_B< 2m$. To study this last question we consider
the r.h.s. of (75) as a function of the variable $m_B$. For $m_B=0$ we
get from (68)
\begin{eqnarray*}
\int {d\vec{Q} \over 4 (\vec{Q}^2+m^2)} \ {\zeta(\|\vec{Q} \|)^2 \over
2\sqrt{\vec{Q}^2+m^2}} & \leq & \|\zeta\|_{\infty}^2 \ \int_{\cal K}
{d\vec{Q} \over 8 (\vec{Q}^2+m^2)^{3/2}} \\
& \leq & {m^3 \over 2 |{\cal K}|} \ {|{\cal K}| \over 8 m^3} \ = \ {1
\over 16} \ < \ 1.
\end{eqnarray*}
Now it follows from (69) that for $m_B=2m$ the r.h.s. of (75) is
majorated as follows
\begin{eqnarray*}
\int {d\vec{Q} \over 4 (\vec{Q}^2+m^2)} \ {\zeta(\| \vec{Q} \|)^2
\over 2\sqrt{\vec{Q}^2+m^2}-2m} & = & { \pi \over 2} \ \int_0^R dr
\ d(r)^2 \ { \sqrt{r^2 + m^2} + m \over r^2+m^2} \\
\geq \ {m^3 \over 8 R^3} \int_0^{R/2m} dr { 1 +\sqrt{r^2 + 1}
\over r^2+1 } & = & \left({m \over 2R } \right)^3 \left(
\arctan{R \over 2m} + \arg \sinh {R \over 2m} \right)
\end{eqnarray*}
which is $>1$ for $m/R>2$. Thus (75) has a (unique) solution $0 < m_B
< 2m$, which is an eigenvalue of (61) and then, from proposition 6,
an eigenvalue of $M$. []
\vspace{12mm}
{\large {\bf 8. Conclusion}}\vspace{5mm}
Let us sum up what we have found. We have constructed a family of
unitary, continuous representations of the Poincar\'{e} group in
four space-time dimensions, as perturbations of the two free boson
model. The scattering operator $S$ can be performed and is unitary and
non trivial in some cases.
The physical content of this mathematical construction is based on Wigner's
famous interpretation [21], according to which an elementary particle is
described by an irreducible, unitary, and continuous representation of
${\cal P}_{+}^{\uparrow}$.
In the construction of $S$ we have verified the existence of scattering
states, i.e. states which for large (or small) time approach the tensor
product of two irreducible unitary representations of
${\cal P}_{+}^{\uparrow}$, interpreted as describing two free particles.
This permits us to affirm that our models really describe relativistic
quantum systems of two particles. Moreover the non triviality of $S$ allows
us to claim that the interaction between the particles can be effective.
The absence of particle creation or anhilation makes these
models interesting for low energy physics, in particular for the bound
state problem. The existence of a bound state equation similar to the
ordinary Schr\"{o}dinger equation of Quantum Mechanics (and which can be
called a 'relativistic Schr\"{o}dinger equation') is probably the
strongest result of the paper.
However these results have been obtained for interaction operators $\cal O$
with ranges contained in the subspace of zero angular momentum (the
so-called 's-wave subspace'). To get a better characterization of these
interactions, let us compute the change of the second Casimir operator,
the Pauli-Lubanski operator, given by (6). In the representation (13)
it becomes
\begin{eqnarray*}
W^2 \ \ = \ \ W^2_0 \ + \ \{ \{ {\cal O}, \vec{W}_0 \},
\vec{W}_0 \} \ \ + \ \ \{ {\cal O}, \vec{W}_0 \}^2
\end{eqnarray*}
where $W^2_0$ is given by (21) and $\vec{W}_0 = -H_0 \vec{J} + \vec{P}
\wedge \vec{L}_0$. Now condition (27) implies $ \{ {\cal O},
\vec{W}_0 \} = 0$ and thus
\begin{eqnarray*}
W^2 \ \ = \ \ W^2_0.
\end{eqnarray*}
In contradiction with what happens to the Casimir operator $M$, the
introduction of these interactions has no effect on the second Casimir
operator, the Pauli-Lubanski operator $W^2$.
\vspace{12mm}
{\large {\bf Appendix A. One-particle representation}}\vspace{6mm}
We show that the operators (7) are the infinitesimal generators of
a unitary, continuous and irreducible representation of the universal
covering of the Poincar\'{e} group ${\cal P}_{+}^{ \uparrow}$. Such a
representation is a good candidate for a theory describing a quantum
relativistic spinless particle of mass $m>0$. The Poincar\'{e}
representation describing one particle of mass $m>0$ without spin in
given by the following lemma (which is essentially given in [21]).
\begin{Fau} The operators $(7)$ are the infinitesimal generators of
a unitary, conti\-nuous and irreducible representation of the universal
covering of the Poincar\'{e} group ${\cal P}_{+}^{ \uparrow}$ in the
space $L^2(I\!\!R^3, \sigma)$.
\end{Fau}
\vspace{3mm}
We recall that $d \sigma({\vec p}) =d{\vec p} \ [2\omega({\vec p})
]^{-1}$ where $\omega({\vec p}) = \sqrt{{\vec p}^{ \ 2} + m^2}$.
\vspace{3mm} \newline
{\bf Proof.} Let us consider the following transformations of functions
$\phi \in L^2(I\!\!R^3, \sigma)$ :
\begin{eqnarray}
( \tau,{\vec \xi}, \vec{\alpha}, \vec{\beta}) \cdot
\ \phi({\vec p}) \ \ := \ \ e^{\textstyle i( \tau \omega({\vec p}) +
{\vec \xi} \cdot {\vec p})}
\ \phi( R_{\vec{\alpha}} \lambda_{\vec{\beta}}{\vec p})
\end{eqnarray}
for all $\tau \in I\!\!R$, $\vec{\xi}, \vec{\alpha}, \vec{\beta} \in
I\!\!R^3$, $\| \vec{\beta} \| <1$, and almost all $\vec{p} \in
I\!\!R^3$, where $R_{\vec{\alpha}}$ is the matrix of the rotation
of $I\!\!R^3$ of the Euler angles $\vec{\alpha}$ and
$ \lambda_{\vec{\beta}} \ \vec{p}$ is the `spatial' component of the
pure Lorentz transform of $\left(\omega(\vec{p}), \vec{p}\right)$ of
velocity $\vec{\beta}$ (i.e. the `spatial' component of (79) (see
below) for such four-vector).
Let us show that the transformation (77) defines a unitary,
continuous and irreducible representation in $L^2(I\!\!R^3, \sigma)$
of the Poincar\'{e} group ${\cal P}_{+}^{ \uparrow}$,
the infinitesimal generators of which are given by (7).
The unitarity and the group law are easily seen by introducing a
fourth variable $p^0$ together with a Dirac-delta function in the
measure, that is by replacing $L^2(I\!\!R^3, \sigma)$ by
$L^2(I\!\!R^4, \Sigma)$ where $d\Sigma(p^0,\vec{p}) =
dp^0dp_1dp_2dp_3 \delta({p^0}^2-\vec{p}^{ \ 2} -m^2) \theta(p^0)$,
$\theta$ being the Heaviside function. Because of the $\Sigma$
measure, a function $\phi(p^0,\vec{p})$ is not distinguishable
from $\phi(\omega(\vec{p}), \vec{p})$. The transformation (78)
becomes, after Fourier transformation
$$( \tau, {\vec \xi},\vec{\alpha}, \vec{\beta}) \cdot \
\tilde{\phi}(t,{\vec x}) \ \ = \ \ \tilde{\phi}\left( ( \tau,
{\vec \xi}, \vec{\alpha}, \vec{\beta})^{-1} \cdot (t,{\vec x})
\right) $$
(where $\tilde{\phi}$ is the Fourier transform of $\phi$) that is,
given in terms of the ordinary Poincar\'{e} transformation on
space-time. The group law is now obvious. The unitarity follows
from the Poincar\'{e} invariance of the measure $\Sigma$.
The continuity of the representation is easily seen in $L^2(I\!\!R^3,
\sigma)$, by standard analysis methods. The irreducibility is obtained
as follows. Let us take any $\phi\in L^2(I\!\!R^3, \sigma)$, $\phi
\neq 0$. Let us show that the only vector orthogonal to
$\{e^{\textstyle i \vec{\xi} \cdot \vec{p}} \phi(\lambda_{\vec{\beta}}
\ \vec{p}) \ |$ $ \vec{\xi},\vec{\beta} \in I\!\!R^3 \}$ is 0. By the
Fourier theory
\begin{eqnarray}
\int_{I\!\!R^3} d\sigma( \vec{p}) \ \psi(\vec{p})^* \
e^{\textstyle i \vec{\xi} \cdot \vec{p} } \phi(\lambda_{\vec{\beta}}
\ \vec{p}) \ = \ 0 \ \ \mbox{ for all } \ \ \vec{\xi} \in I\!\!R^3
\end{eqnarray}
implies that $\psi(\vec{p})=0$ for almost all $\vec{p}$ such that
$\phi(\lambda_{\vec{\beta}} \ \vec{p}) \neq 0$. Without loss of
generality we may assume that $\phi(0) \neq 0$. For all $\vec{p} \in
I\!\!R^3$ the vector $\vec{\beta} = - \ \vec{p} / \omega( \vec{p})$
satisfies $\lambda_{\vec{\beta}} \ \vec{p}=0$ (see appendix B). Thus
if moreover we impose that (78) holds for all $\vec{\beta} \in
I\!\!R^2$ we get $\psi=0$.
After an elementary calculation the operators (7) turn out to be the
infinitesimal generators of the representation (77). []
\vspace{12mm}
{\large {\bf Appendix B. The relative momentum}}\vspace{6mm}
We explain the change of variable $(\vec{p}_1, \vec{p}_2) \rightarrow
(\vec{P}, \vec{Q})$ given in (16) and establish somes useful formulas.
The meaning of $\vec{P}=\vec{p}_1+\vec{p}_2$, the total momentum, is
clear. The notion of relative momentum is more subtle. Physically it
must be related to the momentum of one of the particles in the
centre-of-mass frame.
Let $p_1= \left( \omega(\vec{p}_1), \vec{p}_1 \right) $ and
$p_2=(\omega(\vec{p}_2), \vec{p}_2)$ be the energy-momenta
of two free particles, of the same mass to simplify. We define
first
\begin{eqnarray*}
P= \left( P^0,\vec{P} \right) = p_1+p_2 \hspace{15mm} q =
\left( q^0, \vec{q} \right) ={1 \over 2}(p_1-p_2).
\end{eqnarray*}
Note the useful relation $\vec{P} \cdot \vec{q} = P^0q^0$. Let
$\Lambda_{\vec{\beta}}$ be a Lorentz transformation such that
$\Lambda_{\vec{\beta}}P = (M_0, \vec{0} \ )$, where $M_0 =[(P^0)^2
-\vec{P}^2]^{1/2}$. Physically $\Lambda_{\vec{\beta}}$ is
associated with a change of referential frame, which put the two
particles of momenta $\vec{p}_1, \vec{p}_2$ on their centre-of-mass
frames. Then we define the relative momentum $\vec{Q}$ as the
spatial projection of $\Lambda_{\vec{\beta}} \ q$. Note that the
relative momentum just defined depends on the way to reach the
centre-of-mass frame, which is not unique (up to rotations of
$I\!\!R^3)$.
To get more explicit expressions we need the formula of a pure Lorentz
transform $\Lambda_{\vec \beta}$ of velocity $\vec{\beta} \in I\!\!R^3$,
$\| \vec{\beta} \| <1$. It acts on a four-dimensional vector $(p^0,
\vec{p}) \in I\!\!R \times I\!\!R^3$ as follows
\begin{eqnarray}
\Lambda_{\vec \beta} \left( \begin{array}{c} p^0 \\ \vec{p}
\end{array} \right) =
\left( \begin{array}{c} \gamma (p^0+ \vec{\beta} \cdot \vec{p}) \\
\\ \vec{p} + \vec{\beta} \left[ \gamma p^0 + { \textstyle
\gamma -1 \over \textstyle \beta^2 }
\vec{p} \cdot \vec{\beta}\right] \end{array} \right)
\end{eqnarray}
where we have put $\beta = \| \vec{\beta} \|$ and $\gamma =
(1-\beta^2)^{-1/2} $. The condition that the spatial component of
$\Lambda_{\vec{\beta}} P$ vanishes gives
\begin{eqnarray*}
\vec{P} \ + \ \vec{\beta} \left[ \gamma P^0 \ + \ {\gamma -1 \over
\beta^2 } \ \vec{P} \cdot \vec{\beta}\right] \ = \ 0 .
\end{eqnarray*}
If we choose the direction of $\vec{\beta}$ parallel to $\vec{P}$
(this condition leads to a particular way to reach the centre-of-mass
frame), we get the unique solution $\vec{\beta} =- \vec{P} / P^0$.
After some calculation one gets $\Lambda_{-\vec{P}/ P^0} \ q = (0,
\vec{Q})$ where
\begin{eqnarray*}
\vec{Q} \ = \ \vec{q} \ - \ {\vec{P} \over P^0} \left[ {P^0 q^0
\over M_0} \ - \ {P^0 - M_0 \over M_0 } \ { (P^0)^2 \over \vec{P}^2}
\ {\vec{q} \cdot \vec{P} \over P^0} \right]
\end{eqnarray*}
which, after some easy development, gives finally the formula (16).
Let us calculate the operators of the two free boson representation (8)
in these variables, to establish (17). The norm $\| \vec{Q} \|$ is
obtained by the properties of the Lorentz transform, which gives
$ \vec{Q}^2 = (q^0)^2-\vec{q}^{ \ 2}$. From the trivial identity
$(P^0)^2-\vec{P}^2 = 4m^2 + 4[(q^0)^2-\vec{q}^{ \ 2}]$ it follows
\begin{eqnarray*}
M_0 \ = \ 2 \omega(\vec{Q})
\end{eqnarray*}
and then
\begin{eqnarray*}
P^0 \ = \ \omega(\vec{p}_1) + \omega(\vec{p}_1) \ = \ M_0^2 +
\vec{P}^2 \ = \
\sqrt{ \vec{P}^2 + 4 \vec{Q}^2 + 4m^2} \ = \ \Omega(\vec{P},\vec{Q})
\end{eqnarray*}
in agreement with (20) and (18). The angular momentum $\vec{J}$ is
easily obtained because a rotation of the system $\vec{p}_1, \vec{p}_2$
leads to the same rotation of $\vec{P}, \vec{Q}$ (this is an obvious
consequence of the formula (16)). The Lorentz generator $L_{0,1}$ is
obtained as follows. Under an infinitesimal Lorentz transform of
hyperbolic angle $\gamma$ in the first direction (of unit vector
$\vec{e}_1$) the fundamental quantities become
\begin{eqnarray*}
& \vec{P} \rightarrow \vec{P} + \gamma P^0 \vec{e}_1 \ \ \ \ \ \ \ \ \
& \vec{q} \rightarrow \vec{q} + \gamma q^0 \vec{e}_1 \\
& P^0 \rightarrow P^0 + \gamma P_1 \ \ \ \ \ \ \ \ \
& q^0 \rightarrow q^0 + \gamma q_1 \ .
\end{eqnarray*}
Under such a transformation $\vec{Q}$ becomes
\begin{eqnarray*}
\vec{Q} & \rightarrow & \vec{q} + \gamma q^0 \vec{e}_1 - {q^0 +
\gamma q_1 \over M_0 + P^0 + \gamma P_1 } \ \left( \vec{P} +
\gamma P^0 \vec{e}_1 \right) \\
& & \approx \ \ \vec{Q} \ + \ \gamma \ {1 \over M_0 + P^0 } \
\left( (\vec{P} \cdot \vec{Q}) \vec{e}_1- Q_1 \vec{P} \right)
\end{eqnarray*}
where we have performed a first order development and used the relation
$q^0M_0 = \vec{P} \cdot \vec{Q}$. The formula for $L_{0,1}$ given in (17)
follows from the calculation
\begin{eqnarray*}
L_{0,1} \phi(\vec{P}, \vec{Q}) \ = \ -i \partial_{\gamma} \left.
\phi \left(\vec{P}+ \gamma P^0 \vec{e}_1, \vec{Q}+\gamma \ { (\vec{P}
\cdot \vec{Q})
\vec{e}_1- Q_1 \vec{P} \over M_0 + P^0 } \ \right) \right|_{\gamma=0}.
\end{eqnarray*}
$L_{0,2}$ and $L_{0,3}$ are obtained in the same way.
The Jacobian of the transformation $(\vec{p}_1, \vec{p}_2) \rightarrow
(\vec{P}, \vec{Q})$ is calculated in two steps. First we perform the
transformation $(\vec{p}_1, \vec{p}_2) \rightarrow (\vec{P}, \vec{q}\ )$,
with Jacobian 1, and then the transformation $(\vec{P}, \vec{q} \ )
\rightarrow (\vec{P}, \vec{Q})$ with Jacobian (using the formula (16)
for $\vec{Q}$)
\begin{eqnarray*}
\left| \det \left( { \partial Q_i \over \partial q_j } \right)
\right|^{-1}
\ = \ \left[ 1 \ - \ \vec{P} \cdot \vec{\nabla}\!_{\vec{q}}
\ { q^0 \over M_0 + P^0 } \right]^{-1}
\end{eqnarray*}
which, after a long and tedious calculation, gives $ 4 \omega
(\vec{p}_1) \omega(\vec{p}_2) [M_0 P^0]^{-1}$. Thus the measure
$\sigma_2$ becomes $d\sigma_2(\vec{p}_1,\vec{p}_2) = d \mu(\vec{P},
\vec{Q})= d\vec{P} d\sigma(\vec{Q}) \Omega(\vec{P},\vec{Q}) ^{-1}$,
as in (19).
\vspace{12mm}
{\large {\bf Appendix C. The fundamental equation}}\vspace{6mm}
The fundamental equation (29) is the condition on a kernel $h(\vec{P},
\vec{Q}, \vec{Q}')$, depending only on the norms $\|\vec{P} \|$,
$\|\vec{Q} \|$, $\|\vec{Q}' \|$, for the relation (14) and (15) to
hold. This norm dependence leads to (27) and thus to $L_{0,j}
{\cal O} = L_{0,j}^{\vec{P}} {\cal O}$ and ${\cal O} L_{0,j} =
{\cal O} L_{0,j}^{\vec{P}}$ where $ L_{0,j}^{\vec{P}} := -i
\Omega(\vec{P}, \vec{Q}) \partial_{P_j}$ for all $j \in \{1,2,3 \}$.
\vspace{1mm}
Let us calculate the jth equation of (14). The linear part in $\cal O$
can be written
$$A_j \ := \ [ \{H_0, {\cal O} \},L_{0,j}] + [H_0, \{L_{0,j},
{\cal O}\}] \ = \ 2(H_0 {\cal O} L_{0,j} - L_{0,j} {\cal
O} H_0 + iP_j {\cal O} )$$
where we have used $[H_0,L_{0,j}] = iP_j$ and the fact that $P_j$ and
$\cal O$ commute. By applying this operator to a vector $\phi \in
{\cal D}$ (the domain given by (46)) we obtain, in obvious symbolic
notation
\begin{eqnarray*}
A_j \phi(\vec{P},\vec{Q}) & = &
-2i\Omega\int d\sigma(\vec{Q}') { \partial_{P_j} \phi h \over \Omega
+ \Omega'} \ + \ 2 i\Omega\partial_{P_j}
\int d\sigma(\vec{Q}') { \phi h \over \Omega + \Omega'} \\
+ & 2 iP_j & \int
{d\sigma(\vec{Q}') \over \Omega' }{\phi h \over \Omega + \Omega'} \
= \ 2 i \int d\sigma(\vec{Q}') \phi \left( \Omega \partial_{P_j}
{ h \over \Omega + \Omega'} \
+ { P_j \over \Omega' } {h \over \Omega + \Omega'} \right) \\
& = & \ 2i \int d\sigma(\vec{Q}') \ \phi(\vec{P},\vec{Q}')
\ { \Omega(\vec{P},\vec{Q}) \ \partial_{P_j} h(\vec{P},\vec{Q},
\vec{Q}') \over \Omega(\vec{P},\vec{Q}) + \Omega(\vec{P},\vec{Q}') } .
\end{eqnarray*}
To study the bilinear part, we rewrite (25) and (26) :
\begin{eqnarray*}
\{ H_0, {\cal O} \} \ \phi(\vec{P},\vec{Q}) & = & \int
{d\sigma(\vec{Q}') \over \Omega(\vec{P},\vec{Q}') }
\ \phi(\vec{P},\vec{Q}') \ h(\vec{P},\vec{Q},\vec{Q}') \\
\{ L_{0,j}, {\cal O} \} \ \phi(\vec{P},\vec{Q}) & = & -i \int
{d\sigma(\vec{Q}') \over \Omega(\vec{P},\vec{Q}') } (\partial_{P_j}
\phi(\vec{P},\vec{Q}')) \ h(\vec{P},\vec{Q},\vec{Q}) \\
- & i \Omega(P,q) & \int d\sigma(\vec{Q}')
\phi(\vec{P},\vec{Q}') \ \partial_{P_j} { h(\vec{P},\vec{Q},\vec{Q}')
\over \Omega(\vec{P},\vec{Q}')(\Omega(\vec{P},\vec{Q}) +
\Omega(\vec{P},\vec{Q}')) } \ .
\end{eqnarray*}
Thus the bilinear part in $\cal O$ gives, in symbolic notation
\begin{eqnarray*}
& &B_j \phi(\vec{P},\vec{Q}) \ := \ [ \{ H_0, {\cal O}\}, \{ L_0,
{\cal O} \} ] \ \phi(\vec{P},\vec{Q}) \ = \\
& & \int {d\sigma'' \over \Omega''} h^{Q,Q''} \left[ -i \int {d\sigma'
\over \Omega' }(\partial_j \phi') h^{Q'',Q'} \ - \ i \Omega'' \int
d\sigma' \phi'\partial_j { h^{Q'',Q'} \over \Omega'(\Omega''+\Omega')}
\right. \\
& & \hspace{10mm} + \ \left. i\partial_j \int {d\sigma'
\over \Omega' }\phi' h^{Q'',Q'} \right] \ + \ i \Omega \int d\sigma''
\left(\partial_j{ h^{Q,Q''} \over \Omega''(\Omega + \Omega'')} \right)
\int {dq' \over \Omega' } \phi' h^{Q'',Q'} \ \\
& &= \ i \int d\sigma'' \int d\sigma' \phi' \left(- \ h^{Q,Q''}
\partial_j { h^{Q'',Q'} \over \Omega'(\Omega'' + \Omega')} \ + \
{ h^{Q,Q''} \over \Omega''} \partial_j { h^{Q'',Q'} \over \Omega'}
\right. \\
& & \hspace{80mm} + \ \left. { \Omega \over \Omega'} \left( \partial_j
{ h^{Q,Q''} \over \Omega''(\Omega + \Omega'')} \right)
h^{Q'',Q'} \right) \\
& & = \ i \int { d\sigma(\vec{Q}') \over \Omega(\vec{P}, \vec{Q}') }
\ \phi(\vec{P}, \vec{Q}')
\int { d\sigma(\vec{Q}'') \over \Omega(\vec{P}, \vec{Q}'')^2 }
\left\{ - \ { P_j \over \Omega(\vec{P}, \vec{Q}'')} \ h(\vec{P},
\vec{Q}, \vec{Q}'') h(\vec{P}, \vec{Q}'', \vec{Q}')
\right. \\
& & \hspace{40mm} + \ { \Omega(\vec{P}, \vec{Q}) \Omega(\vec{P},
\vec{Q}'') \partial_{P_j} h(\vec{P}, \vec{Q}, \vec{Q}'')
\over \Omega(\vec{P}, \vec{Q}) + \Omega(\vec{P}, \vec{Q}'') } \
h(\vec{P}, \vec{Q}'', \vec{Q}') \\
& & \hspace{40mm} + \ \left. h(\vec{P}, \vec{Q}, \vec{Q}'') \
{ \Omega(\vec{P}, \vec{Q}'') \Omega(\vec{P}, \vec{Q}') \partial_{P_j}
h(\vec{P}, \vec{Q}'', \vec{Q}') \over \Omega(\vec{P}, \vec{Q}'') +
\Omega(\vec{P}, \vec{Q}') } \right\} .
\end{eqnarray*}
Because $h$ depends on $\vec{P}$ only through $\| \vec{P} \|$ we can
replace everywhere $\partial_{P_j}$ by $P_j\| \vec{P} \|^{-1}
\partial_{\| \vec{P} \|}$. The condition $A_j+B_j=0$, which must hold
for all $\phi \in {\cal D}$, leads to the fundamental equation (29),
for all $j \in \{1,2,3\}$.
The equations (15) are slightly more difficult to establish. By using
$\vec{J}^{\vec{Q}} {\cal O} =0$ and by putting $\vec{J}^{\vec{P}}
:=\vec{J}-\vec{J}^{\vec{Q}}$ the linear part of the first
equation of (15) can be written, in symbolic notations :
\begin{eqnarray*}
& & C\phi(\vec{P}, \vec{Q}) \ := \ 2 \left( L_{0,1}^{\vec{P}}
{\cal O} L_{0,2}^{\vec{P}} - L_{0,2}^{\vec{P}} {\cal O}
L_{0,1}^{\vec{P}} -i {\cal O} J_3^{\vec{P}} \right) \phi(\vec{P},
\vec{Q}) \ = \\
& &- 2\Omega \int d\sigma' \left[ \left( \partial_2
\phi' \right) \partial_1 - \left( \partial_1 \phi' \right)
\partial_2 \right] {h \over \Omega+ \Omega' } \ - \ 2\int
{d\sigma' \over \Omega'} \left[ \left( P_1 \partial_2
- P_2 \partial_1 \right) \phi' \right] {h
\over \Omega+ \Omega' } \\
& & = \ - { 2i \over \| \vec{P} \| }\ \int { d\sigma(\vec{Q}')
\over \Omega(\vec{P}, \vec{Q}') }
\ \left(J_3^{\vec{P}} \phi(\vec{P}, \vec{Q}') \right) \
D h(\vec{P}, \vec{Q}, \vec{Q}')
\end{eqnarray*}
The bilinear part of this equation gives
\begin{eqnarray*}
& & D\phi(\vec{P}, \vec{Q}) \ := \ \left( {1 \over 2} (C {\cal O} +
{\cal O} C) + L_{0,1}^{\vec{P}} {\cal O}^2 L_{0,2}^{\vec{P}} -
L_{0,2}^{\vec{P}} {\cal O}^2 L_{0,1}^{\vec{P}} +i {\cal O}^2
J_3^{\vec{P}} \right) \phi(\vec{P}, \vec{Q}) = \\
& & = \ - { i \over \| \vec{P} \| } \int { d\sigma' \over \Omega' }
J_3^{\vec{P}} \phi' \int { d\sigma'' \over \Omega'' } \left(
D h^{\vec{Q},\vec{Q}''} { h^{\vec{Q}'',\vec{Q}'} \over \Omega' +
\Omega''} \ + \ { h^{\vec{Q},\vec{Q}''} \over \Omega + \Omega''}
D h^{\vec{Q}'',\vec{Q}'} \right) \\
& & \ \ ( -i)^2 \int { d\sigma' \over \Omega' } \ \Omega\Omega'
\left[(\partial_2\phi')\partial_1 -(\partial_1\phi')\partial_2
\right] \int { d\sigma'' \over \Omega'' } \
{ h^{\vec{Q},\vec{Q}''} \over \Omega + \Omega''} \ { h^{\vec{Q}'',
\vec{Q}'} \over \Omega' + \Omega''} \\
& & \ \ + \ \int { d\sigma' \over \Omega' } \ \left[(P_1 \partial_2
-P_2\partial_1)\phi'\right] \int { d\sigma'' \over \Omega'' } \
{ h^{\vec{Q},\vec{Q}''} \over \Omega + \Omega''} \ { h^{\vec{Q}'',
\vec{Q}'} \over \Omega' + \Omega''} \\
& & = \ - \ { i \over \| \vec{P} \| } \int { d\sigma' \over \Omega'}
\ J_3^{\vec{P}} \phi' \int { d\sigma'' \over \Omega''^2 } \left(
D h^{\vec{Q},\vec{Q}''} h^{\vec{Q}'',\vec{Q}'} \ + \
h^{\vec{Q},\vec{Q}''}D h^{\vec{Q}'',\vec{Q}'} \right) \\
& & \ \ \ \ + \ i \int { d\sigma' \over \Omega' } \
J_3^{\vec{P}} \phi' \int { d\sigma'' \over \Omega''^3 } \
h^{\vec{Q},\vec{Q}''} h^{\vec{Q}'',\vec{Q}'}
\end{eqnarray*}
The condition $C+D=0$, which must hold for all $J_3^{\vec{P}}\phi$, leads
to the fundamental equation (29) again. The two other equations of (15)
lead to the same result.
\vspace{12mm}
{\large {\bf Appendix D. A criterium for trace-class operators}}
\vspace{6mm}
Let ${\cal H}_s = L^2(I\!\!R^3, s)$ be a Hilbert space where $s$ is
any positive $\sigma$-finite measure, and let $F$ be a integral
operator on ${\cal H}_s$ of kernel $f : I\!\!R^3 \times I\!\!R^3
\mapsto I\!\!\!\!C$ that is
\begin{eqnarray}
(Fu)(x) \ = \ \int_{I\!\!R^3} \ f(x,y) \ u(y) \ ds(y)
\end{eqnarray}
for all suitable $u \in {\cal H}_s$.
\begin{Fau}
Let $K$ be a compact subset of $I\!\!R^3$ and $f \in L^2(I\!\!R^3
\times I\!\!R^3,s \otimes s)$ such that for almost all $x \in I\!\!R^3$
the function $y \mapsto f(x,y)$ has support in $K$. Let us suppose that
the three derivatives
\begin{eqnarray*}
\partial_{y_1} f(x,y), \ \ \ \partial_{y_2}\partial_{y_1} f(x,y) \ ,
\ \ \ \partial_{y_3}\partial_{y_2}\partial_{y_1} f(x,y)
\end{eqnarray*}
exist in $L^2(I\!\!R^3 \times I\!\!R^3,s \otimes s)$.
Then $F$, defined by (80), is trace class.
\end{Fau}
{\bf Proof.} We follow [20], sect. XI.9.32. We will use that a product
of two Hilbert-Schmidt operators is a trace class operator. Let $k>0$
be such that $K \subset [-k,k]^3$. Let $u \in {\cal H}_s$. From three
integrations by parts we get
\begin{eqnarray*}
(Fu)(x) & = & \int_{[-k,k]^3} \ \left[\partial_{y_3}\partial_{y_2}
\partial_{y_1} f(x,y) \right] \ \int_{-k}^{y_1} \int_{-k}^{y_2}
\int_{-k}^{y_3} u(t) \ dt \ ds(y) \\
& = & \int_{I\!\!R^3} \ \left[\partial_{y_3}\partial_{y_2}
\partial_{y_1} f(x,y) \right] \ \int_{-k}^{y_1} \int_{-k}^{y_2}
\int_{-k}^{y_3} \chi_{[-k,k]^3}(y) \ u(t) \ dt \ ds(y)
\end{eqnarray*}
We have written $F$ as a product of two Hilbert-Schmidt operators, the
first one of kernel $(x,y) \mapsto \partial_{y_3}\partial_{y_2}
\partial_{y_1} f(x,y)$ and the second one of kernel $(x,y) \mapsto
\chi_{[-k,k]^3}(x)$ $\chi_{[-k,x_1]}(y_1) \chi_{[-k,x_2]}(y_2)
\chi_{[-k,x_31]}(y_3)$. Thus $F$ is trace class. []
\vspace{12mm}
{\large{\bf Acknowledgment}}\vspace{6mm}
I would like to thank the Professors W. O. Amrein, P. Bader, J. Bros,
J. Fr\"{o}hlich, G. Gallavotti, J.-J. Loeffel, R. Stora and S. V.
Varadarajan for fruitful discussions and usefull suggestions. I am
gratefull to them for their interest in this work and for their
encouragements.
\vspace{12mm}
{\large {\bf References}}
\vspace{6mm}
\newline
[1] P. A. M. Dirac, {\em Forms of relativistic dynamics}, Rev Mod Phys
21 (1949), 392-399
\vspace{3mm}\newline
[2] D. G. Currie, {\em Interaction contra classical relativistic
Hamiltonian particle mechanics}, Journ. Math. Phys. 4 (1963).
See also E. C. G. Sudarshan, N: Mukunda, {\em Classical dynamics: a
modern perspective}, John Wiley and Sons, (1974)
\vspace{3mm}\newline
[3] E. Frochaux, {\em Non-trivial representations of the special
relativity group}, Forum Math. 9 (1997), 75-102
\vspace{3mm} \newline
[4] E. Frochaux, {\em Two relativistic boson models in the
Schr\"{o}dinger picture in three space-time dimensions}, Journ. Math.
Phys. 37 (1996), 2979-3000
\vspace{3mm} \newline
[5] E. Frochaux, {\em A relativistic quantum equation for $N \geq 2$
bosons in two space-time dimensions}, Helv Phys Acta 68 (1995), 47-63
\vspace{3mm} \newline
[6] E. Frochaux, A. Roessl, {\em A relativistic generalization of
Quantum Mechanics in two space-time dimensions}, in preparation
\vspace{3mm}\newline
[7] J. Glimm, A. Jaffe, {\em Quantum Physics}, Springer-Verlag, (1987)
\vspace{3mm}\newline
[8] D. Iagolnitzer, {\em Scattering in quantum field theory},
Princeton series in Physics, Priceton University press (1993),
and references therein
\vspace{3mm}\newline
[9] E. Frochaux, {\em A variational proof of the existence of a
bound state in a relativistic quantum model}, Helv. Phys. Acta 61
(1998), 923-957.
{\em The bound states of the ${\cal P}( \varphi)_2$ relativistic
quantum field models with weak coupling obtained by the
variational perturbation method}, Nucl. Phys. B 389 (1993), 666-702
\vspace{3mm} \newline
[10] E. Frochaux, {\em The bound states in Quantum Field Theory:
review of some analytic problems raised by the variational
perturbation method}, Helv. Phy. Acta 66 (1993), 567-613
\vspace{3mm} \newline
[11] E. Frochaux, {\em How a relativistic Schr\"{o}dinger equation
arises in Quantum Field Theory}, in preparation
\vspace{3mm} \newline
[12] E. Zeidler, {\em Nonlinear Functional Analysis and its
Applications}, Vol.1, Springer-Verlag, New York Berlin Heidelberg
Tokyo (1985)
\vspace{3mm} \newline
[13] E. Nelson, {\em Analytic vectors}, Annals of Mathematics,
Vol 70 (1959), 572-615
\vspace{3mm} \newline
[14] M. Reed, B. Simon, {\em Methods of modern mathematical physics,
Vol II, Fourier analysis and self-adjointness}, Academic press,
New York - San Francisco - London (1975)
\vspace{3mm} \newline
[15] M. Reed, B. Simon, {\em Methods of modern mathematical physics,
Vol VI, Analysis of operators}, Academic press, New York -
San Francisco - London (1978)
\vspace{3mm} \newline
[16] J. Sucher, {\em Relativistic invariance and the square-root
Klein-Gordon equation}, Jour. Math. Phys. 4 (1963), 17-23
\vspace{3mm} \newline
[17] T. Kato, {\em Perturbation theory for linear operators},
Grundlehren der mathematischen Wissenschaften 132, Springer (1966)
\vspace{3mm} \newline
[18] M. Reed, B. Simon, {\em Methods of modern mathematical physics,
Vol III, Scattering theory}, Academic press, Boston - San Diego -
New York - London - Sydney - Tokyo - Toronto (1979)
\vspace{3mm} \newline
[19] D. R. Yafaev, {\em Mathematical scattering theory, general
theory}, Translations of mathematical monographs, Amer. Math. Soc,
Providence, Rhode Island (1992)
\vspace{3mm} \newline
[20] N. Dunford, J. T. Schwartz, {\em Linear operators: Part II:
Spectral theory}, Wiley Classical Library, Interscience Publishers
(1988)
\vspace{3mm} \newline
[21] E. P. Wigner {\em Unitary representations of the inhomogeneous Lorentz
group}, Annals of Mathematics, Vol 40 (1939), 149-204
\end{document}