Comparison of Point-form and Bethe-Salpeter Formalisms

Compare forms of interactions in two different approaches to relativistic quantum theory: Relativistic quantum mechanics in point form and relativistic quantum field theory. In the formulation of the dynamics of interacting few-particle systems in a relativistic quantum theory one has to satisfy the relativistic transformation properties for observables. In a quantum field theoretical approach, which involves infinitely many degrees of freedom, the correct transformation properties are satisfied from the beginning. In a quantum mechanical framework, where one works with a finite number of (effective) degrees of freedom, one uses mathematical group theory to find constraints for the interaction terms. In both cases results for the mass spectrum of the system of interest can be obtained by solving a dynamical equation. In the quantum field theoretical case one can use the so-called Bethe-Salpeter equation. In the case of relativistic quantum mechanics one gets an eigenvalue equation for the mass operator of the system of interest. The aim of this project is to compare these two approaches first for two-body systems (vector and axial-vector mesons) and then extend the comparison to a three-body system (the nucleon and some of its electromagnetic properties). The results of the calculations on the different levels will show how the different parts of the model interaction in each approach influence the results as well as the comparison itself and how in this respect model independent statements can be made. Furthermore the comparison will allow to study the long-range part of the strong interaction in both approaches especially for the axial-vector mesons, since these particles have a larger extension and so their properties are more sensitive to long-range forces.