Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 028, 8 pages      quant-ph/0512228      https://doi.org/10.3842/SIGMA.2005.028

Representations of U(2∞) and the Value of the Fine Structure Constant

William H. Klink
Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA

Received September 28, 2005, in final form December 17, 2005; Published online December 25, 2005

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). Eigenvectors and eigenvalues of the four-momentum operator are analyzed and exact solutions in the strong coupling limit are sketched. A simple model shows how the fine structure constant might be determined for the QED vertex.

Key words: point form relativistic quantum mechanics; antisymmetric representations of infinite unitary groups; semidirect sum of unitary with Heisenberg algebra.

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