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Representations of $${\text U}(2\infty)$$ and the value of the fine structure constant. (English) Zbl 1092.81067
Summary: 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.
##### MSC:
 81V10 Electromagnetic interaction; quantum electrodynamics 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 22D10 Unitary representations of locally compact groups 81T27 Continuum limits in quantum field theory 81T15 Perturbative methods of renormalization applied to problems in quantum field theory
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