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Relativistic quantum mechanics of few-particle systems
The basic problem is to construct operators that provide a unitary representation of the inhomogeneous Lorentz group, also called the Poincare group. These operators are used to transform quantum mechanical state vectors from one inertial frame to another. It is relatively straightforward to construct models for two-particle systems, as well as coupled two-particle systems, which satisfy exactly the requirements of special relativity. For systems with three or more particles it is also necessary to take into account the requirement of cluster separability, sometimes also called the principle of macroscopic locality. This is the requirement that when the system is separated into subsystems the mathematical description should reduce to the description of the subsystems. This is especially important in scattering problems where initially and finally the system is separated into two or more fragments.
The requirements of special relativity place restrictions on the possible interactions between particles, however there is still a lot of freedom left in choosing interactions. Most people believe that quantum field theory provides the basis for constructing interactions. More explicitly, it is generally believed that interactions are due to the exchange of particles. The most important part of the electromagnetic interaction between two charged particle is due to the exchange of a photon, while the longest range part of the strong interaction between two nucleons is due to the exchange of a pion. Recently Professor Fuda has been developing systematic techniques for using particle exchange models as the basis for constructing Poincare invariant, quantum mechanical models of few particle systems. In particular a relativistic one boson exchange model of the two-nucleon system has been constructed which allows for the exchange of pi, eta, rho, omega, delta, and sigma mesons.
More recently a particle exchange model for the pion-nucleon system has been developed which takes into account coupling to the inelastic channels. This exchange model fits the scattering data up to a pion lab energy of 700 MeV. The model has been used to calculate the photoproduction of pions from the nucleon. It is presently being extended to make possible the calculation of other mesons (eta, rho, omega, etc.) from the nucleon as well as the electroproduction of mesons from the nucleon. It is anticipated that these calculations will provide useful information on the electromagnetic excitation of the baryon resonances. They may also help in the search for the so-called "missing resonances". These are resonances that are predicted by the quark model, but don't show up in pion-nucleon elastic scattering.