the successful application of renormalization group (RG) methods within alternative formulations of QFT illuminates the empirical content of QFT, but not the theoretical content.

The theoretical principles of QFT are remote from the derived S-matrix elements; in LQFT, the fact that renormalization is necessary is one manifestation of this.

LQFT with cutoffs yields valuable insight into the empirical structure of QFT, but not the theoretical content.

Underdetermination of the theory at high energies (or, equivalently, small distance scales) by the results of experiments conducted at relatively low energies is a well-known consequence of the application of RG methods.

one moral that can be drawn from the application of RG methods in LQFT is that the content of the theory (in the form of the Lagrangian that it is appropriate to apply to describe high energy physics) is underdetermined by the empirical evidence obtained from experiments conducted at relatively low energy scales.

The fact that, within LQFT, a range of descriptions of the physics at short distance scales each yield the same predictions for relatively low energies suggests that there could be different theoretical frameworks for QFT which disagree about the physics of short distance scales yet yield the same predictions for relatively low energies. That is, the lessons learned from RG methods ﬁt together nicely with the contention that the empirical evidence for QFT underdetermines the theoretical content of QFT.

But this presumes that the theoretical content of QFT should encompass all length scales. Related to the standard approach towards interpretation—-theories determine entire possible worlds. I can use Fraser as example of dominance of standard approach.

the application of RG methods does not automatically generate a rigorously well-deﬁned theoretical framework for QFT.

Features of RG methods such as the framework for analyzing ﬂows and ﬁxed points in the space of Lagrangians also shed light on the empirical structure of QFT by revealing the relationship between the physical description at high energy scales and the empirical predictions at low energy scales. In this respect, RG methods illustrate the distance between the theoretical and empirical levels in QFT

Another way of making this point that RG methods pertain to the empirical content of QFT rather than the theoretical content is the observation that constructive ﬁeld theorists are now working to adapt RG methods for use in model construction. The goal is to ﬁnd mathematically rigorous analogues of the techniques. (See Rivasseau, 2003, pp. 168–169 for a brief summary.) The reason that constructive ﬁeld theorists are able to exploit RG methods—even though they reject elements of the theoretical content of LQFT—is that RG methods concern the empirical structure of the theory rather than the theoretical content

RG methods demonstrate that theories with that differ radically about the physics of short distance scales (i.e., lagrangians) can all be empirically adequate. This information about the empirical structure of QFT makes a no miracles argument in favor of the approximate truth of the theory that we happen to possess seem very weak.

in the condensed matter case, independent evidence for the existence of atoms plays a pivotal role. At the beginning of the twentieth century, the case had to be made for the existence of atoms. The arguments that made the case were based on various pieces of experimental evidence, including Perrin’s work on Brownian motion (Salmon, 1984). There is no analogue of this evidence in the QFT case; we do not possess evidence of this sort that QFT breaks down at short distance scales or that QFT entails that space is discrete. The evidence for the existence of atoms plays an important role in motivating the introduction of the cutoff in the condensed matter case. That this is a point of disanalogy between the two cases undermines Wallace’s moral that the introduction of a cutoff in QFT is similarly motivated

In the QFT case, one starts with a theory with an inﬁnite number of degrees of freedom, artiﬁcially imposes a lattice, and then employs the statistical continuum limit to remove the lattice. As we have seen, there is a compelling practical justiﬁcation for this procedure: application of the statistical continuum limit (aka RG methods) is an effectual technique for renormalizing the theory. The physical justiﬁcation for the employment of the statistical continuum limit is that the lattice was introduced as an unphysical assumption; taking the statistical continuum limit removes this unphysical assumption and restores the continuum theory with an inﬁnite number of degrees of freedom. In the condensed matter case, the justiﬁcation for taking the statistical continuum limit is exactly the opposite. The theory deﬁned on a lattice is taken to be the physically accurate representation; the continuum theory that results from taking the statistical continuum limit is regarded as the idealization. The justiﬁcation for taking the statistical continuum limit is that, for the purposes of descriptions at distance scales much longer than the lattice, the continuum theory will be a very good approximation. The fact that diametrically opposed physical justiﬁcations are given for imposing the cutoff suggests that we should resist exporting the interpretation of the cutoff that is given in condensed matter physics to QFT. That is, we should not infer that, as in condensed matter physics, in QFT there is some small distance scale at which ‘‘something freezes out the short-distance degrees of freedom.’’ The mere fact that RG methods are applicable in both cases does not support substantive ontological claims of this sort

Wallace’s contention that QFT breaks down at short distance scales in the same manner as condensed matter theory is undermined by the differences in physical justiﬁcations for the cutoffs and the limits in the two cases. It could turn out that this point is related to the above-stated point that RG methods shed light on the empirical content of QFT. The latter point is consonant with the view that RG methods are a useful mathematical tool for extracting approximate predictions in appropriate circumstances (e.g., when a system has a large number of degrees of freedom and the scales are locally coupled), but that these techniques can be successfully applied to physical systems which are dissimilar in many other respects.

The implication in this quote seems to be that useful mathematical tool for extracting approximate predictions that can be applied to wildly different systems => not reflective of reality