The Hyperfine Structure of Hydrogen

The two-body formalism of Schwinger is modified to consider the case of the hydrogen atom. The proton's anomalous moment is treated by adding a Pauli-type term to the Lagrangian. A perturbation theory based upon the Green's function is developed and the first-order correction to the Fermi hyperfine splitting of the ground state is calculated. The method of calculation used is that of Karplus and Klein in their positronium work. Aside from the usual renormalizations encountered, an extra infinity appears in the calculation associated with the assumption of a point anomalous magnetic moment for the proton. On the hypothesis that the proton's moment is actually distributed, cutoffs are inserted. The modified hyperfine formula leads to a new value of $\alpha :\frac{1}{\alpha }=137.0378\mathrm{}$ for a cutoff at the meson length and $\frac{1}{\alpha }=137.0374\mathrm{}$ for a cutoff at the proton length.