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

Attempts at a determination of the fine-structure constant from first principles: A brief historical overview