atyy said:
In a sense, the exposition
http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf by 't Hooft is condensed matter in spirit, because he says: "Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important for our considerations, we replace the continuum of three-dimensional space by a discrete but dense lattice of points."
You overestimate the cogency of motivational handwaving arguments.
t'Hooft is self-contradictory when taken literally in his motivations. By the same token he eliminates (on p.13) structures larger than a certain size and imposes periodic boundary conditions to get a strictly finite system. The resulting strictly finite system is then quantized - but it has no longer asymptotic in- and out-states and hence has no well-defined scattering problem. But the latter was his starting point on p.5!
Moreover, on p. 13 he begins the argument by saying ''it is often forgotten how these answers can be derived rigorously'' - but today there is still no rigorous derivation, only the kind of handwaving he presents. Every derivation of the scattering formula of LSZ (which is the true basis of all perturbative QFT calculations) involves a flat Minkowski space-time and a Poincare invariant field theory!
On p.29 he says ''renormalizability requires our theory to be consistent up to the very tiniest distance scales'', meaning tha continuum limit, as otherwise - in a strictly finte system - there is no renormalizability problem at all! And on the next page, p.30, he states ''renormalizability provides the required coherence of our theories'', Thus he explicitly revokes his earlier motivation to be able to get to the heart of the matter. The bulk of the paper, after all the introductory motivational bla bla that you take for gospel has been said,
the remaining 40 pages, are about the limiting theory, and the techniques used (e.g., dimensional regularization and topological considerations)
assume Poincare invariance. On p.41 he says: ''Unitarity of the S-matrix turns out to be a sensitive criterion to check whether we are performing the continuum limit correctly'' - which makes sense only if the continuum limit is relevant to the standard model - which it indeed is!
Thus t'Hooft's motivational arguments in the first 16 pages cannot be taken literally and
have to be interpreted with many grains of salt!
Note that he got his Nobel prize not for work on a strictly finite quantum field theory in the sense he described on p.13 but for showing that quantizing certain (continuum) classical gauge theories with matter and spontaneously broken symmetry - in a construction with infinitely many degrees of freedom! - one obtains (in renormalized perturbation theory) a renormalizable, Poincare-invariant QFT.
This is very far from what he painted in the introductory quarter of the paper for the purpose of simplifying the true (technically correct) situation. Indeed, it is still unknown today whether a strictly finite lattice-based construction has a good infrared (large distance) and ultraviolet (small distance) limit - so
at present lattices (and their limits) cannot give a good foundation of the standard model.
By the way, t'Hooft discusses on p.65 the fine-tuning problem as something problematic enough that ''it becomes difficult to believe that it [the standard model] represents the real world'' at high enough energies.
