Spectral forms for the photon propagation function and the Gell-Mann-Low function

An analysis by Schwinger is extended to prove that the Gell-Mann-Low function $\psi$ is proportional to the spectral function $\sigma$ for the photon propagation function when the latter is expressed as a function of a suitably defined effective charge in the asymptotic region. In terms of the correspondingly defined effective charge variables $\psi \left({{\overline{e}}_M}^2\right)=\left(\frac{d{{\overline{e}}_M}^2}{d{e_M}^2}\right)$. This equality reflects the difference between the effective-charge definitions of Gell-Mann and Low and of Schwinger. Thus, in the new variables, the Gell-Mann-Low equation holds in terms of the function $x^2\sigma \mathrm{}\left(x\right)\mathrm{}$. Moreover, it is possible to define a variable $y\left(x\right)\mathrm{}$ such that $\psi \mathrm{}\left(y\right)=x^2\sigma \mathrm{}\left(x\right)\mathrm{}$. The possibility of using alternative spectral functions to replace $\psi$ is discussed in an appendix.