# Insights Interview with Mathematician and Physicist Arnold Neumaier - Comments

Tags:
1. Jan 3, 2018

### A. Neumaier

Wilson's perspective is intrinsically approximate; changing the scale changes the theory. This is independent of renormalized perturbation theory (whether causal or not), where the theory tells us that there is a vector space of parameters from which to choose one point that defines the theory. The vector space is finite-dimensional iff the theory is renormalizable.

In the latter case we pick a parameter vector by calculating its consequences for observable behavior and matching as many key quantities as the vector space of parameters has dimensions. This is deemed sufficient and needs no mathematical axiom of choice of any sort. Instead it involves experiments that restrict the parameter space to sufficiently narrow regions.

In the infinite-dimensional case the situation is similar, except that we would need to match infinitely many key quantities, which we do not (and will never) have. This just means that we are not able to tell which precise choice Nature is using.

But we don't know this anyway for any physical theory - even in QED, the best theory we ever had, we know Nature's choice only to 12 digits of accuracy or so. Nevertheless, QED is very predictive.

Infinite dimensions do not harm predictability elsewhere in physics. In fluid dynamics, the solutions of interest belong to an infinite-dimensional space. But we are always satisfied with finite-dimensional approximations - the industry pays a lot for finite element simulations because its results are very useful in spite of their approximate nature. Thus there is nothing bad in not knowing the infinite-dimensional details as long as we have good enough finite-dimensional approximations.

2. Jan 3, 2018

### A. Neumaier

Yes, this is quite similar to causal perturbation theory.

3. Jan 3, 2018

### A. Neumaier

This is nice in that it illustrates the concepts on free scalar fields, so that one can understand them without all the technicalities that come later with the renormalization. I don't have yet a good feeling for factorization algebras, though.

4. Jan 3, 2018

### Urs Schreiber

I'd eventually enjoy a more fine-grained technical discussion of some of these matters, to clear out the issues. But for the moment I'll leave it at that.

By the way, not only may we view the the string perturbation series as a way to choose these infinitely many renormalization parameters for gravity by way of other data, but the same is also true for "asymptotic safety". Here it's the postulate of being on a finite-dimensional subspace in the space of couplings that amounts to the choice.

5. Jan 4, 2018

### Urs Schreiber

The space of choices of renormalization parameters at each order is not a vector space, but an affine space. There is no invariant meaning of "setting to zero" these parameters, unless one has already chosen an origin in these affine spaces. The latter may be addressed as a choice of renormalization scheme, but this just gives another name to the choice to be made, it still does not give a canonical choice.

You know this, but here is pointers to the details for those readers who have not see this:

In the original Epstein-Glaser 73 the choice at order $\omega$ happens on p. 27, where it says "we choose a fixed auxiliary function $w \in \mathcal{S}(\mathbb{R}^n)$ such that...". With the choice of this function they build one solution to the renormalization problem at this order (for them a splitting of distributions) which they call $(T^+, T^-)$. With this "origin" chosen, every other solution of the renormalization at that order is labeled by a vector space of renormalization constants $c_\alpha$ (on their p. 28, after "The most general solution"). It might superficially seem the as if we could renormalize canonically by declaring "choose all $c_\alpha$ to be zero". But this is an illusion, the choice is now in the "scheme" $w$ relative to which the $c_\alpha$ are given.

In the modern reformulation of Epstein-Glaser's work in terms of extensions of distributions in Brunetti-Fredenhagen 00 the analogous step happens on p. 24 in or below equation (38), where at order $\omega$ bump functions $\mathfrak{w}_\alpha$ are chosen. The theorem 5.3 below that states then that with this choice, the space of renormalization constants at that order is given by coefficients relative to these choices $\mathfrak{w}_\alpha$.

One may succintly summarize this statement by saying that the space of renormalization parameters at each order, while not having a preferred element (in particular not being a vector space with a zero-element singled out) is a torsor over a vector space, meaning that after any one point is chosen, then the remaining points form a vector space relative to this point. That more succinct formulation of theorem 5.3 in Brunetti-Fredenhagen 00 is made for instance as corollary 2.6 on p.5 of Bahns-Wrochna 12.

Hence for a general Lagrangian there is no formula for choosing the renormalization parameters at each order. It is in very special situations only that we may give a formula for choosing the infinitely many renormalization parameters. Three prominent such situations are the following:

1) if the theory is "renormalizable" in that it so happens that after some finite order the space of choices of parameters contain a unique single point. In that case we may make a finite number of choices and then the remaining choices are fixed.

2) If we assume the existence of a "UV-critical hypersurface" (e.g. Nink-Reuter 12, p. 2), which comes down to postulating a finite dimensional submanifold in the infinite dimensional space of renormalization parameters and postulating/assuming that we make a choice on this submanifold. Major extra assumptions here. If they indeed happen to be met, then the space of choices is drastically shrunk.

3) We assume a UV-completion by a string perturbation series. This only works for field theories which are not in the "swampland" (Vafa 05). It transforms the space of choices of renormalization parameters into the space of choices of full 2d SCFTS of central charge 15, the latter also known as the "perturbative landscape". Even though this space received a lot of press, it seems that way too little is known about it to say much at the moment. But that's another discussion.

There might be more, but the above three seem to be the most prominent ones.

6. Jan 4, 2018

### Urs Schreiber

I wrote:

Hm, I guess Arnold will argue that we can construct choices for these auxiliary functions. There won't be a canonical choice but at least constructions exist and we don't need to appeal to non-constructive choice principles. Okay, I suppose I agree then!

7. Jan 5, 2018

### A. Neumaier

Yes.

More specifically, there is no significant difference between choosing from a finite number of finite-dimensional affine spaces in the renormalizable case and choosing from a countable number of finite-dimensional affine spaces in the renormalizable case. The same techniques that apply in the first case to pick a finite sequence of physical parameters (a few dozen in the case of the standard model) that determine a single point in each of these spaces can be used in the second case to pick an infinite sequence of physical parameters that determine a single point in each of these spaces. Here a parameter is deemed physical if it could be in principle obtained from sufficiently accurate statistics on collision events or other in principle measurable information.

Any specific such infinite sequence provides a well-defined nonrenormalizable perturbative quantum field theory. Thus there is no question of being able to make the choices in very specific ways. As in the renormalizable case, experiments just restrict the parameter region in which the theory is compatible with experiment. Typically, this region constrains the first few parameters a lot and the later ones much less.This is precisely the same situation as when we have to estimate the coefficients of a power series of a function $f(x)$ from a finite number of inaccurate function values given together with statistical error bounds.

Last edited: Jan 5, 2018
8. Jan 22, 2018 at 8:04 AM

### A. Neumaier

See also this thread from Physics Stack exchange, where solutions of an ''unrenormalizable'' QFT obtained by reparameterizing a renormalizable QFT are discussed.

Last edited: Jan 22, 2018 at 8:32 AM