a stable scalar particle in some nontrivial interacting field theory. The equation of motion for this system may be written as $$\Box \phi(x) + m^2 \phi(x) = -j(x).$$ This equation of motion is determined from a Lagrangean density ${\cal L}(\phi,j)$. Furthermore, let us insist on studying the renormalized field, mass and current. This means that the correlation function for $\phi$ will have a pole with residue of unity at the physical mass, $m$: $$\langle \phi, \vec{p} | \phi(x) | {\rm vac} \rangle  =
e^{ip\cdot x}$$ which implies that $$\lim_{q^2 \rightarrow m^2}\, (q^2 - m^2) \,  \int {\rm d}^4 x \,
e^{i q \cdot x}
\, \langle {\rm vac}| {\rm T}[\phi(x) \phi(0)] |{\rm vac} \rangle
\,
= \, i .
\label{renorm}$$

We are concentrating on the renormalized quantities because un-renormalized properties are not observable and depend on the details of the renormalization procedure. Ultimately we will be interested in the spectral decomposition of the propagator in terms of the physical states of the system and this is directly related to the [*renormalized*]{} fields and sources.

It is worth noting that the renormalization conditions put quite stringent constraints on matrix elements of the renormalized source. In particular they imply that source does 