bhobba said:
Conventional re-normalization described in say the following made me wince:
https://arxiv.org/pdf/1208.4700.pdf
'In other words, the bare mass has to diverge in such a way that its divergence cancels the divergent loop correction to yield a finite result. It amounts to shuffling the infinities to unobservable quantities like the bare mass.'
What words like this are trying to describe in prose is the simple mathematical statement of
prop. 16.23:
This says that given a UV cutoff in the form of a sequence of non-singular distributions ##\Delta_{F,\Lambda}## that approximate the true Feynman propagator ##\Delta_F## in that ##\Delta_F = \underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda} ## (
def. 16.20) then
1. while the limit as ##\Lambda \to \infty## of the corresponding effective S-matrices ##\mathcal{S}_\Lambda(g S_{int}) := \exp_{F,\Lambda}\left(\tfrac{1}{i\hbar}( g S_{int})\right)## (
268) need not exist (so we "have a divergence")...
2. ...there is a way to re-adjust, depending on ##\Lambda##, the interaction action functional ##g S_{int}## by adding higher order "counter terms" ##g^2 S_{counter,\Lambda} := \mathcal{Z}_\Lambda(g S_{int}) -g S_{int}## (
remark 16.24) such that the limit of the effective S-matrices applied to the re-defined coupling does exist
$$
\mathcal{S}(g S_{int})
:=
\underset{\Lambda \to \infty}{\lim} \mathcal{S}_{\Lambda} ( \underset{\mathcal{Z}_{\Lambda}(g S_{int})}{\underbrace{g S_{int} + g^2 S_{counter,\Lambda}}} )
$$
The proof of
prop. 16.23 makes transparent how this works, inductively: given counterterms ##Z_{\leq n,\Lambda}## for ##k \leq n## vertices then one looks at the difference between a true S-matrix ##\mathcal{S}(g S_{int})## at the next order ##n+1## and the effective S-matrix at order ##n+1## applied to the re-defined interaction at order ##n##. The local part of that difference has to be the next counterterm ##Z_{n+1,\Lambda}##, while the non-local part has to vanish with ##\Lambda \to \infty## (equation
272).
(By the way, it seems that at this moment my notes are the only place where this proof is actually written out. Previously there was just the faint hints offered in
DFKR14, A.1)
In this way the counterterms manifestly absorb in each order the failure of the effective S-matrix ##\mathcal{S}_\Lambda## to converge to an actual S-matrix ##\mathcal{S}##. Conversely one may like to state this in prose as: "their own divergence cancels the divergence of the loop corrections", where the intuition expressed is that also the limit as ##\Lambda \to \infty## of the counterterms ##\mathcal{Z}_\Lambda## alone does not exist, it's only the combination ## \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda## of S-matrix applied to interaction with counterterms whose limit exists.
##\,##
Generally, there is nothing in the notes that is
fundamentally different from the traditional informal story, it's just that each ingredient of the traditional story is turned into something that makes sense.
For instance UV cutoffs are traditionally discussed in terms of a would-be path integral of the free theory. Since that does not exist, here one asks: What is it that the intuition of the path integral really tries to do for us when we speak of UV cutoffs? And one realizes: It just means to turn the resulting Feynman propagator ##\Delta_F## from a distribution with singularities on the light cone into a non-singular distribution ##\Delta_{F,\Lambda}## (by the evident momentum cutoff, example 16.21) just not done "in the path integral" where it makes not sense, but right there in the formula for the Feynman propagator). And so instead of building up mystery by insisting on a non-existent path integral imagery, we instead admit to reality and simply
define a UV cutoff to be an approximation of the Feynman propagator by non-singular distributions ##\Delta_{F,\Lambda}## as ##\Delta_F = \underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda}## (
Duetsch 10, section 4). With that, everything becomes clear.
This is just the same way of proceeding as what Epstein-Glaser did
in 1973 (following Bogoliubov and Stueckelberg). Instead of insisting that the S-matrix comes from a would-be integration over field histories, they simply said: Let's look for what the tangible
outcome suggested by that imagery is. And they saw that all it means to do for us is to understand that an S-matrix scheme should satisfy causal factorization. This is simple and well defined and all one actually needs to construct pQFT (
remark 15.16).
