Department of Physics and Astronomy,

University of Iowa,

Iowa City, IA 52242-1479,

USA

E-mail: william-klink@uiowa.edu

**Representations of the Infinite Dimensional Groups
U(2-infinity) and the Value of the Fine Structure Constant**

**Abstract:**

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.