functions of free field theory $$\Delta^{+}(x)=\int \frac{d^4 p}{(2\pi)^3} \,e^{-ip\cdot x}\theta(p)
\delta(p^2-t)$$ and $$\begin{split}
\Delta^{-}(x) &= \int \frac{d^4 p}{(2\pi)^3}\,e^{ip\cdot
x}\theta(p)\delta(p^2-t) \\
&= \int \frac{d^4 p}{(2\pi)^3}\,e^{-ip\cdot x}\theta(-p)
\delta(p^2-t);
\end{split}$$ and therefore the Feynman propagator function $$\begin{split}
\Delta_{\mbox{\small\rm F}}(x;t) &=
i\theta(x)\Delta^{+}(x;t)+i\theta(-x)\Delta^{-}(x;t)
\\
&=\int\frac{d^4 p}{(2\pi)^4}\frac{e^{-ip\cdot x}} {t-p^2-i\epsilon},
\end{split}$$ where the last expression can be obtained using the representation in eq. (\[eq:theta\]) of the $\theta$–function \[[See e.g. ref. [@Veltman] pp. 50-51.]{}\]. The two–point function $\Pi(q^2)$ appears then to be the Fourier transform of a scalar free field propagating with an arbitrary mass squared $t$ weighted by the spectral function density $\rho(t)$ and integrated over all possible values of $t$, $$\Pi(q^2)=\int d^4 x\, e^{iq\cdot x}\int_{0}^{\infty}dt\,\rho(t)\,
\Delta_{\mbox{\small\rm F}}(x;t).$$ Integrating over $x$ and $p$ results finally in the wanted representation $$\Pi(q^2)=\int_{0}^{\infty}dt\,\rho(t)\frac{1}{t-q^2-i\epsilon}.$$ With $$\Pi(q^2)={\mbox{\rm Re}}\Pi(q^2)+i{\mbox{\rm Im}}\Pi(q^2),$$ and the use of the identity in eq. (\[eq:ppartid\]), it follows that $$\rho(t)\equiv\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t),$$ which identifies the spectral function with the imaginary part of the two–point function.

Notice that the formal manipulations above avoid the question of convergence of the principal value integral $${\mbox{\rm Re}}\Pi(q^2)=\mbox{\rm
PP}\!\int_{0}^{\infty}dt\,\frac{1}{t-q^2}\,\frac{1}{\pi}{\mbox{\rm Im}}\Pi(t).$$ The convergence of the integral in the ultraviolet limit ($t{\rightarrow}\infty$) depends on the behaviour of the spectral function at large $t$–values. When doing above the exchange 