# Living with infinities

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Referee this paper: [arXiv:0903.0568] by Steven Weinberg

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

This is the written version of a talk given 2009 by S. Weinberg in memory of Gunnar Kallen on February 13, 2009.

''I discuss some of Kallen’s work, especially regarding the problem of infinities in quantum field theory, and recount my own interactions with him. In addition, I describe for non-specialists the current status of the problem, and present my personal view on how it may be resolved in the future.''

summarized
paper authored Mar 3, 2009 to hep-th
recategorized Aug 27, 2014

This is a review, so I suppose originality is not a concern. But it would be nice to have the original papers referenced here also.

We can summarise it like this:

"This is the written version of a talk given in memory of Gunnar Kallen on February 13, 2009. I discuss some of Kallen’s work, especially regarding the problem of infinities in quantum field theory, and recount my own interactions with him. In addition, I describe for non-specialists the current status of the problem, and present my personal view on how it may be resolved in the future."

This talk shows the reasonings of different physicists and their attempts to resolve the problem.

## 1 Review

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This is a reminiscence and review of the central contributions of Gunnar Kallen to the early renormalization program, focusing on some issues important in the 1950s: the Lee model unitarity problem, and the analogous problems in ultra-high energy quantum electrodynamics, which are essentially the problems of Landau-pole/triviality, the breakdown of the theory when renormalization constants run to infinity as you go up to an enormous but not infinite energy scale.

The main conclusion from the early work, which was controversial at the time, was that continuum quantum electrodynamics is inconsistent. Today, this is considered uncontroversially true, because we understand these type of triviality breakdowns from an Ising model analogy: in a 4-d Ising model, the couplng at the lattice scale is effectively infinite, but it slowly runs to zero at longer and longer distances when you consider the effective coupling of spin-fluctuations. If you observe any nonzero value of the coupling of spin-fluctuations this reveals the order of magnitude of the lattice scale, which is roughly at the energy where the one loop coupling diverges. The continuum theory breaks down here, which we know already, because in this case we happen to know it is a lattice theory, because that's what we started out with.

Extending the continuum renormalization program beyond the lattice scale gives nonsensical results, you end up with a continuum description with more degrees of freedom than the lattice theory had to begin with, states in the formal continuum description end up getting a negative norm, and scattering processes at high energies violate naive unitary. I only say "naive", because Bender suggest that it might be possible to define a new inner product to rescue unitarity the Lee model, and perhaps in other cases too. But this doesn't affect the discussion significantly.

The modern understanding is that a theory with a Landau pole is only an effective theory in a range of energies, and a full continuum limit can't be taken. The inconsistency of theories beyond the Landau pole is not controversial anymore, now it is taken for granted. This has no bearing on the consistency of the modern renormalization program, it is a failure of taking a complete continuum limit, it is an argument for effective field theories. This limitation doesn't apply to asymptotically free theories, whose renormalization is fully consistent, all the way to the continuum limit. The same goes for asymptotically safe theories, where the fixed point is not at zero coupling, but at some other value.

In the 1950s, however, there was no idea for what could be going on underneath quantum field theory, there was no more fundamental theory, so these types of in-principle limitations were very worrying. By 1960, due to these worries the S-matrix program was active, and already in 1968, string theory appeared from this line of research. Once quasi-realistic string models began to appear, for the first time you could understand what the ultimate theory looks like, and it is not a field theory at all, and there are no problems of taking a continuum limit on local field operators.

Weinberg, however, was a field theorist, and he wants to fix field theory. So he believes that there is the possibility that there is an asymptotically safe theory of quantum gravity, either a fixed point for the renormalization group at large energies, or a perpetually running coupling that never reaches any problematic point.  In line with this program, Weinberg considers seriously the possibility that there is a fixed point for the electromagnetic coupling at large values.

This expectation has no real support, at least not in traditional theories with electric charges. In theories with both electric charges and monopoles, there is an argument for this--- the electric coupling and the magnetic coupling are inversely related, so that the electric coupling can't blow up without the magnetic coupling becoming small, so that perhaps there is a continuum fixed point with both electric and magnetic charges equally strongly coupled. But this idea requires adding monopoles to QED, and it is not clear how to formulate a quantum field theory of fundamental monopoles and charges together.

To add support to the hope that there is an asymptotically safe QED, Weinberg casts doubts on an old great theorem of Kallen's, the demonstration that at least one of the renormalization constants in QED must be infinite. The only argument he gives against the result is that there are some interchanges of integration. But he gives no reason to suspect that the divergence Kallen identified is dependent on orders of integration, this is very implausible for a calculation similar in nature of a Feynman integration, nor does he give a detailed criticism of the original paper which identifies a mistake. The paper is a classic, and is most likely just fine. Still, I should say that Weinberg reviews the argument more than fairly (I personally understood it better after reading Weinberg now than when looking over the original paper some years ago).

Another thing that Weinberg reviews well is Kallen's argument that the wavefunction renormalization for the photon, the "dielectric constant" of the electron-positron vacuum, can only have the effect of reducing the electric charge. This is surely the positivity argument that Schwinger and others advanced in the 1960s against the idea of asymptotic freedom, when it was discovered in non-abelian gauge theories (I personally wondered for a long time what this old argument was, I figured it was some sort of Kallan-ism).

The main motivation for suggesting this paper for review at this early stage of the reviews section seems to be that this paper shows skepticism of the renormalization program. This skepticism is well understood today to have been justified--- renormalization in non-asymptotically-safe theories simply does not work to define a continuum theory. It has no bearing on the correctness of the renormalization procedure in the domain of applicability the effective field theory, and it has no bearing on the correctness of renormalization in asymptotically free theories, where a continuum limit can be taken with no problems of principle at all.

Weinberg's main motivation seems to be in regard to his program of asymptotic safety. I would like to point out something in regard to the asymptotic safety program: it is automatically unnatural. One of the main philosophical shifts in modern renormalization was to stop viewing the running of the coupling to large energies as fundamental and physical, and to attach some mystical significance to where the large coupling ends up. Rather, after Wilson, you view the natural direct of running coupling is toward the infrared, and the starting point in the ultraviolet is generic, not special.

When a theory is asymptotically free (this is a special case of asymptotic safety), and you imagine it close to the continuum limit, you are automatically starting out in the ultraviolet at a value which is finely tuned near zero. For "reasonable" values of the cutoff, like the Planck scale, this is not an issue, because the dependence is logarithmic, and the logarithm is not particularly big at the Planck scale. But Weinberg is suggesting to consider field theories with a continuum limit as fundamental theories, so that the cut off is really going to get pushed to infinity. In this context, the fine tuning problem returns.

To illustrate this, I will give my own pet example of an asymptotically-safe theory--- 5d pure gauge theory. This theory has a lattice coupling which, when small, wants to run to zero in the infrared, because it has positive dimensions, but for large values, on a lattice, it wants to run to infinity in the infrared, due to the confinement. It is possible to choose the coupling at just the magic asymptotically-safe value balanced between the two behaviors, and make the lattice teeny tiny, and keep the coupling from going either way at all for a many decades. It is therefore possible to take a continuum limit. But in this case, the fine-tuning involved is obvious--- if this is coming from something else underneath, the something-else has to have magically chosen the coupling to sit at the magic balance point at short distances.

Of course, with quantum-gravity, presumably Weinberg thinks it is the final theory, and the considerations of continuum limit are somehow philosophically distinguished, and allow you to do this balancing act. I disagree with this position. I think asymptotic safety should be recognized as a fine-tuning.

There are good reasons from string theory not to take asymptotic safety seriously, most significantly holographic degree of freedom counting. But regardless of these other stringy arguments, asymptotic safety is a fine-tuning.

reviewed Jul 28, 2014 by (7,720 points)

I should have said that the stuff about asymptotic safety of 5d gauge theory is AFAIK new, and perhaps requires a review of its own.

Ok, I did a few preliminary calculations some weeks ago to understand what is happening here, but I wanted to do a computer simulation of the thing before saying it in public. I am not sure the continuum theory even ends up rotationally invariant. I'll post when I know a little more.

+1, a very good read for me, thanks.

The argument for our present physical models being effective field theories is usually the Landau pole argument, but what about QCD? Isn't it also a common belief that QCD can be defined in ultraviolet as a field theory?

I figured it might be a good chance to ask since in this talk Weinberg mentioned this again in page 13: "...electrodynamics, the electroweak theory, quantum chromodynamics, and even general relativity-are in fact effective field theories." I wonder what the true justification is.

I think it's a stronger argument, because it is physics, and doesn't depend on mathematical details. People applied, not gravity, but muon theory and strong interactions, to conclude that QED is an effective theory already in the 1960s. The reason the Landau pole was worrisome is because Kallen seemed to demonstrate that it was generic behavior, since positivity demanded positive beta function (so it seemed) and generic estimates demanded that the running of the coupling be to infinity in a finite time. That's why asymptotic freedom was so revolutionary--- it was the first time anyone had an example of a 4d continuum field theory.

But it's the quantum gravity arguments, in particular the holography, that requires something other than field theory at short distances, something that is so tightly restricted by holography that I can't imagine it can be anything other than the most general form of string theory. That's physics, it's the real restriction on a continuum limit, and nobody will find a clever mathematical trick around it, as Bender tries to do regarding the non-unitarity in the Lee model.

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