I'm studying Quantum Field Theory from Weinberg's book, and I'm to the point where he introduces the concept of renormalization. I'd like to know if I'm getting the point that Weinberg makes when speaking about the necessity of renormalization in perturbative calculations. I do get that renormalization is necessary to tame the divergences due to Feynman diagrams containing one or more loops and to extract useful information even from those diagrams. However, in chapter 10, to be more specific in section 10.3, "Field and Mass Renormalization", Weinberg states that if we are to use the usual Feynman rules to calculate S-matrix elements, then we should first re-define the normalization of the fields $\psi_{l}(x)$ so that

$$ \langle {\rm VAC}\,|\,\psi_{l}(0)\,|\ q,\sigma\rangle=(2\pi)^{-3/2}\ u_{l}(q,\sigma)\qquad\quad(*) $$

where $|q,\sigma\rangle$ is an eigenstate of the four-momentum of the complete theory with eigenvalue $q^{\mu}$, and $u_{l}(q,\sigma)$ is the coefficient function appearing in the field $\psi_{l}(x)$. It thus appears that renormalization is necessary in principle to do perturbative calculations, independently of the fact that it is a useful technique to tame the divergences. Now, from the very beginning of the book, Weinberg has been introducing formulae that don't make use of renormalized fields, and everything seemed to go well. The point at which the need for renormalized fields turns up is when he studies poles in momentum-space amplitudes for vacuum-vacuum transitions. Here we learn that the masses of the physical states of the theory, i.e. of the eigenstates of the full four-momentum operator, are the points at which those amplitudes have poles. On the other hand, when deriving the Feynman rules for perturbation theory, we find that the propagators associated to the fields have a pole at the point $p_{\mu}p^{\mu}$, the mass of the free field. Thus the value for the masses in the unperturbed hamiltonian (or in the unperturbed action, in the path-integral formalism) should be taken equal to the value of the physical masses given by the true eigenstates of the full four-momentum. This leads to the necessity of introducing mass renormalization counter-terms in the hamiltonian/action. Moreover, a randomly normalized field would give rise to a discrepancy between the Feynman rules derived through perturbation theory and the Feynman rules derived through the study of the poles of the propagator. Thus one must impose condition $(*)$, and to do so one must introduce field renormalization counter-terms in the hamiltonian/action. Did I get this straight?


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