Barut showed us how it is possible to get a PoincarO invariant n-body equation with a single time. Starting from the Barut equation for n-free particles, we show how to generalize it when they interact through Dirac oscillators with different frequencies.We then particularize the problem to n = 2 and consider the particleantiparticle system whose frequencies are respectively o9 and -e). We indicate how the resulting equation ean be solved by perturbation theoo,, though the spectrum and its comparison with that of the mesons will be given in another publication.
The basis for irreducible representations of groups can have many realizations. Usually those presented in textbooks are the simpler ones based on eigenstates of one-particle Casimir operators. There are of course other types of operators, involving occasionally many particles, that may be invariant under a given group, and their eigenstates then provide a new basis, to which the name of anomalous shall be given, for irreducible representations. This paper shall first illustrate the concept for the simple case of the two-dimensional Euclidean group E(2). The main objective though will be to show that an operator, to which the name of the many-body Dirac oscillator is given, is invariant under the Poincaré group. Thus its eigenstates, which will be discussed explicitly in the case of two bodies, are an anomalous basis for irreducible representations of this group, and the representations are derived explicitly. The many-body Dirac oscillator can then be used with confidence in relativistic problems involving several interacting particles.
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