Quantum Electrodynamics in the Nonlinear Spinor Theory and the Value of Sommerfeld’s Fine-Structure Constant.

Summary

In the nonlinear spinor theory the assumption of a ground state asymmetrical under the isospin group requires the existence of bosons of rest mass zero according to the theorem of Goldstone. Actually the photon plays the rôle of the Goldstone particle. Hence for this assumption to be consistent the eigenvalue equation of the photon must have a solution at mass zero. Therefore this eigenvalue equation should contain a continuous parameter which can be changed until the equation is fulfilled for the rest mass zero of the photon. The average mass of the leptons acts as this parameter; Johnson has emphasized that the equations of quantum electrodynamics are invariant under a scale transformation, if the masses of the leptons are of purely electromagnetic origin. This scale invariance persists in the nonlinear spinor theory, therefore the lepton mass values, from the point of view of quantum electrodynamics, may be multiplied by an arbitrary scale factor. When the average mass of the leptons has been fixed by the condition that the rest mass of the photon should be zero, the coupling constant e ² /ħc = α can be calculated by the methods used by Dhar and Katayama or by Yamazaki for the πN or ηN coupling. The first approximation leads to an average lepton mass of ~ 40 MeV and for the coupling constant to the result α ≈ 0.386 (ϰ_pion/ϰ_nucleon)² ≈ 1/120, which is to be compared with the empirical value α ≈ 1/137.