New Relativistic Wave Equations for Two-Particle Systems

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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 Ψ(r₁,r₂,t)=ψ(r₁,r₂)e-iEt/ℏ, ψ(r₁,r₂)∈S (R³³), and properly define the relativistic reduced mass μ₀, 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 π+ and π-.