 98305 Frochaux E.
 New representations of the Poincar\'e group
describing two interacting bosons
(110K, LaTeX)
Apr 27, 98

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Abstract. New representations of the Poincar\'{e} group are given, which describe two
bosons with interaction in four spacetime 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 mparc 96545.
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98305.tex