arbitrary polynomial is just a constant, and we have got rid of the constant because the on–shell renormalized $\Pi_{\mbox{\tiny\rm R}}$ is defined as $$\Pi_{\mbox{\tiny\rm R}}(q^2)=\Pi(q^2)-\Pi(0)=
\int_{0}^{\infty}
\frac{dt}{t}\,\frac{q^2}{t-q^2-i\epsilon}\,\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t).$$

It is perhaps worth insisting on the fact that asymptotically $$\lim_{t{\rightarrow}\infty}\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t){\Rightarrow}\frac{\alpha}{\pi}
\frac{1}{3};$$ i.e. the electromagnetic spectral function goes to a constant. In fact, it is this constant which fixes the value of the lowest order contribution to the $\beta$–function associated with charge renormalization in QED.

### General Proof of Dispersion Relations

We shall now sketch a proof of the dispersion relation property for two–point functions in full generality. The key of the proof lies in the definition of the time ordered product implicit in eq. (\[eq:tpf\]): $$\mbox{\rm T}(J(x)J(0)^{\dagger}=\theta(x)J(x)J(0)^{\dagger}+
\theta(-x)J(0)^{\dagger}J(x),$$ and the use of translation invariance. The function $\theta(x)$ is the Heaviside function: $\theta(x)=1$ if $x_{0}>0$ and $\theta(x)=0$ if $x_{0}<0$, which has the integral representation $$\label{eq:theta}
\theta(x)=\frac{1}{2\pi i}\int_{-\infty}^{+\infty}dw\,
\frac{ e^{iwx}}{w-i\epsilon}.$$

First we insert a complete set of states $\sum_{\Gamma}\vert\Gamma\rangle
\langle\Gamma\vert$ between the two currents in the T–product definition. This leads to matrix elements of the type $\langle 0\vert
J(x)\vert\Gamma\rangle$ to which we apply translation invariance: $$\langle 0\vert J(x)\vert \Gamma\rangle=\langle 0\vert{{\cal U}}^{-1}{{\cal U}}J(x){{\cal U}}^{-1}{{\cal U}}\vert\Gamma\rangle,$$ where ${{\cal U}}$ is the unitary operator induced 