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 $\mathrm{Ps}^-$ 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.