98-305 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 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.

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