  It has been a notably elusive task to find a remotely sensical ansatz for a
calculation of Sommerfeld's electrodynamic fine-structure constant alpha_QED ~
1/137.036 based on first principles. However, this has not prevented a number
of researchers to invest considerable effort into the problem, despite the
formidable challenges, and a number of attempts have been recorded in the
literature. Here, we review a possible approach based on the quantum
electrodynamic (QED) beta function, and on algebraic identities relating
alpha_QED to invariant properties of "internal" symmetry groups, as well as
attempts to relate the strength of the electromagnetic interaction to the
natural cut-off scale for other gauge theories. Conjectures based on both
classical as well as quantum-field theoretical considerations are discussed. We
point out apparent strengths and weaknesses of the most prominent attempts that
were recorded in the literature. This includes possible connections to scaling
properties of the Einstein-Maxwell Lagrangian which describes gravitational and
electromagnetic interactions on curved space-times. Alternative approaches
inspired by string theory are also discussed. A conceivable variation of the
fine-structure constant with time would suggest a connection of alpha_QED to
global structures of the Universe, which in turn are largely determined by
gravitational interactions.
