I was motivated to ask this question by the equality claimed in equation 10.3.3 of Weinberg's volume 1 of QFT books.

My interpretation of that,

If $O_s$ is a quantum field of spin $s$, $\psi_s$ is the free field of spin $s$, $|p,\sigma\rangle$ is a one-particle state of some interacting theory and $|0\rangle$ is the vacuum state, then there should exist a constant $N$ such that,

$\langle 0|O_s|p,\sigma\rangle = \frac{N}{(2\pi)^3} \langle 0|\psi_s|p,\sigma\rangle$.

If I understand his equation 10.3.6 then that seems to say that the field $O_s$ will be said to have been "renormalized" if $N$ can be set to $1$.

So is one saying that all effect of interactions are absorbed into an overall factor at the level of matrix elements?

That sounds very surprising to me - and it seems that Weinberg claims that it follows merely from the fact that $O_s$ and $\psi_s$ have to transform under the same irreducible representation of the Poincare group.

I would be glad if someone can elaborate this point.

Also what happens if one replaces $|p,\sigma\rangle$ by multi-particle states (it's not clear to me as to what the complete set of labels is that will be required to index the continuum of multiparticle states, and clearly total momentum or/and the invariant mass is not enough).

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