Relativistic Electron-Pair Systems and the Structure of Neutral Mesons

In an effort to obtain a semiclassical model for the neutral $\pi$ meson, a Bohr-Sommerfeld type of system with the proton replaced by a positron is investigated in the limit of high velocities. It is found that as a result of the relativistic increase in the electromagnetic field between the two moving charges, a natural minimum approach distance occurs equal to one-half of the classical "shell-electron" radius. At this separation, a new set of quantized states becomes possible which is found to be energetically unstable. The lowest state possesses an energy approximately equal to the ${\pi }^0$ meson energy. The relativistic states are characterized further by the greatly increased importance of perihelion precession, which accounts for one-half of the total angular momentum in the extreme relativistic case. When the effect of precession on the intrinsic magnetic moment is taken into account, the total energy of the system is found to be $263m_0{c}^2$, in close agreement with the observed ${\pi }^0$ meson mass. The lifetime of the system against annihilation into two gamma rays is calculated on the basis of the close analogy to singlet positronium. Its value is found to be 2.06×${10}^{-16\mathrm{}}$ sec, in good agreement with the latest value of the observed ${\pi }^0$ meson lifetime. The implications for the structure of other nuclear particles and their interactions are briefly discussed.