Relativistic Three-Fermion Wave Equations in Reformulated QED and Relativistic Effects in Muonium Minus

The variational method, within the Hamiltonian formalism of reformulated QED is used to determine relativistic wave equations for a system of three fermions of arbitrary mass interacting electromagnetically. The interaction kernels of the equations are, in essence, the invariant $\mathcal{M}$ matrices in lowest order. The equations are used to obtain relativistic $O({\alpha }^2)$ corrections to the non-relativistic ground state energy levels of the Muonium negative ion (${\mu }^{+}{e}^{-}{e}^{-}$) as well as of ${\mathrm{P}\mathrm{s}}^{-}$ and ${\mathrm{H}}^{-}$, using approximate variational three-body wave functions. The results are compared with other calculations, where available. The relativistic correction for Mu${}^{-}$ is found to be $-1.0773\times {10}^{-4}$ eV.