  We seek to introduce a mathematical method to derive the relativistic wave
equations for two-particle system. According to this method, if we define
stationary wave functions as special solutions like
$\Psi(\mathbf{r}_1,\mathbf{r}_2,t)=\psi(\mathbf{r}_1,\mathbf{r}_2)e^{-iEt/\hbar},\,
\psi(\mathbf{r}_1,\mathbf{r}_2)\in\mathscr{S}
(\mathbb{R}^3\times\mathbb{R}^3)$, and properly define the relativistic reduced
mass $\mu_0$, then some new relativistic two-body wave equations can be
derived. On this basis, we obtain the two-body Sommerfeld fine-structure
formula for relativistic atomic two-body systems such as the pionium and pionic
hydrogen atoms bound states, using which, we discuss the pair production and
annihilation of $\pi+$ and $\pi-$.
