Plane Wave Solutions and the Energy Difference between Pions

It is shown that the plane wave (2) is a solution of the d’Alembert wave equation (6) provided that Eq. (7) is true. The pion masses are given by Eq. (9) with the expectation values (12) and (13). It is known that an equation of type (13) produces a Lamb shift from m theory when the hydrogenic wave functions are used. So this is an explanation of the rest energy difference between the neutral and charged pions.

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