Meaning of "interpolating field" I have a doubt about an expression largely used in Physics:

"Denote $\phi(x)$ a field that interpolates the particle $A(p)$, i.e. a field such that $\langle A(p)|\phi(x)|0\rangle \neq 0$".

What does it mean in this context interpolation?
 A: It is hard to tell from a single sentence you quoted if the term in that particular context has a different meaning, but interpolating fields are usually introduced as follow.
First of all, one can introduce 'in' and 'out' states as a complete bases of the full interacting Hilbert space. 'In' states are those that "look" like free particle states at $t\rightarrow -\infty$. Similarly for 'out' states. The $S$ matrix is introduce as an unitary operator connecting the two.
Then, one can introduce an 'in' (and similarly 'out') field which is a field that satisfies the free equation of motion and can create 'in' (and 'out') states from the vacuum.
$$
\phi_{\text{in}}(x)=\int\frac{d^3\mathbf{p}}{(2\pi)^3\sqrt{2\omega_p}} \Big(a_{\text{in},\mathbf{p}}e^{-ipx}+a^\dagger_{\text{in},\mathbf{p}}e^{ipx}\Big)
$$
It can be shown that the $S$ matrix would transform 'in' fields into 'out' fields and vice versa:
$$
\phi_{\text{in}}(x)=S \phi_{\text{out}}(x) S^{-1}
$$
The interpolating field $\phi(x)$ is then introduced, in a sense, as the interpolation between the 'in', and 'out' field:
\begin{align}
\lim_{t\rightarrow -\infty}\phi(x)\rightarrow & \sqrt{Z} \phi_{\text{in}}(x) \\
\lim_{t\rightarrow +\infty}\phi(x)\rightarrow & \sqrt{Z} \phi_{\text{out}}(x)
\end{align}
The interpolating field $\phi(x)$ is required to satisfy the full interacting equation of motion. One is then interested in a time evolution operator $U$ that connects these fields:
\begin{equation}
\phi(x)=\sqrt{Z}U^\dagger(x^0)\phi_{\text{in}}(x) U(x^0) \tag{1}
\end{equation}
So that:
$$
\lim_{x^0\rightarrow -\infty} U(x^0) = 1\\
\lim_{x^0\rightarrow +\infty} U(x^0) = S
$$
One can solve for $U$ by plugging the definition in Eqn. (1) into the interacting equation of motion. The solution can be found to be given by the Dyson formula.
(Implicitly, in this derivation one must assume the interaction vanishes at $t\rightarrow\pm \infty$ so there are subtleties in theories with long-range interaction of massless particles.)
