Asymptotic Form of the Electron Propagator and the Self-Mass of the Electron

The electron self-mass problem is discussed in the context of ordinary renormalized quantum electro-dynamics. All perturbation contributions to the renormalized self-energy part $\Sigma \mathrm{}\left(p\right)\mathrm{}$, which diverge logarithmically or remain constant in the limit when $p\gg m$, are summed. The resulting $\Sigma \mathrm{}\left(p\right)\mathrm{}$ vanishes in the limit $\frac{p^2}{m^2}\to \infty$ and yields a value for $\delta m$ which is finite and equal to $m$. To obtain this result it is only assumed that the exact photon Green's function at small distances behaves like the bare propagator, which is the case if the eigenvalue equation for the bare coupling constant has a finite root. It is shown that in spite of the fact that the resulting mechanical mass $m_0$ vanishes identically, no conservation equation is obtained for any axial-vector current. Hence no Goldstone bosons appear in ordinary quantum electrodynamics when it is summed to all orders.