Representations of the Infinite Dimensional Groups U(2-infinity) and the Value of the Fine Structure Constant
A relativistic quantum mechanics is formulated in which all of the interactions are in the four-momentum operator and Lorentz transformations are kinematic. Interactions are introduced through vertices, which are bilinear in fermion and antifermion creation and annihilation operators, and linear in boson creation and annihilation operators. The fermion-antifermion operators generate a unitary Lie algebra, whose representations are fixed by a first order Casimir operator (corresponding to baryon number or charge). Form factors are introduced into the vertices in such a way that the Poincare commutation relations are preserved; such form factors regulate the large mass contributions. Eigenvectors and eigenvalues of the four-momentum operator are analyzed and exact solutions in the strong coupling limit are exhibited. A simple model shows how the fine structure constant might be determined for the QED vertex.