Quantum field theory and prejudices


It happened somehow, discussing in this and others blogs, that I have declared that today there are several prejudices about quantum field theory. I never justified this claim and now I will try to do this making the arguments as clearest as I can.

The point is quite easy to be realized when we recognize that the only way we know to manage a quantum field theory is small perturbation theory. So, all we know about this matter is obtained through this ancient mathematical technique. You should try to think about our oldest thinkers looking at the Earth and claiming no other continent exists rather than Europe. Whatever variation you get is about Europe. After a lot of time, a brave man took the sea and uncovered America. But this is another story.

There is people in our community that is ready to claim that a strong coupling expansion is not possible at all. I have read this on a beautiful textbook, Peskin and Scroeder. This is my textbook of choice but is plainly wrong about this matter. It is like our ancient thinkers claiming that nothing else is possible other than Europe because this is the best of all the possible worlds. Claims like this come from our renormalization group understanding of quantum field theory. As you may know, such understandings are realized through small perturbation theory and we are taken back to the beginning. How is the other side of the World?

Colombus Well, I am not Columbus but I can say that here we have an entire continent to explore. As happened to Columbus, there are a lot of thinkers against such an idea but maybe it is worth a try taking into account the possible pay-off.

What should one say about renormalization? Renormalization appears in any attempt to use small perturbation theory in quantum field theory. It naturally arises from the product of distributions at the same point. This is not quite a sensible mathematical situation. The question to ask is then: Is this true with any kind of perturbation technique? One could answer: we haven’t any other and so the requirement of renormalizability becomes a key element to have a valid quantum field theory. The reason for this is the same again: The only technique we believe to exist to do computations in quantum field theory is small perturbation technique and this must work to have a sensible theory. Meantime, we have also learned to work with non-renormalizable theories and called them effective theories. All this is well-known matter.

Of course, the wrong point about such a question is the claim that there are no other techniques than small perturbation theory. There is always another face to a medal as the readers of my blog know. But this implies a great cultural jump to be accepted. The same that happened to Columbus.

Chaos and quantum field theory


Dmitry is still on is point trying to prove me wrong (see his post here). Of course, I have a theory mathematically sound and he has nothing and so the discussion is somewhat uneven from the start. He is saying that my point

In order to build a meaningful quantum field theory, the initial conditions should be properly chosen.

is wrong. This means that we can start to develop a meaningful QED without electrons or photons and obtain identical results. A fact that is blatantly wrong. QED spectrum is done with plane waves representing the spectrum of the theory. Without this choice you are at odds with experiments. This is a crucial point for quantum field theory and applies wherever you need to compute a cross section or a decay rate. Choosing different solutions at the start (e.g. by changing initial conditions) will produce a different quantum field theory and this can be at odds with experiments. This is a well-known fact and an example is given through the computations generally done e.g. for the Casimir effect.

So, he continues

“I believe, it is actually wrong. Let us again take a hamiltonian classical system with self-interaction. To get the intermittency (i.e., periodic orbits becoming chaotic trajectories and vise versa), it is enough to fix initial conditions and than vary the coupling constant, as I have explained … Since one has running coupling in a QFT without sweating, one will have intermittency as well in the Schwinger-Dyson equations for Keldysh Green functions. So, strictly speaking, one can only get rid of chaos at the RG fixed point which corresponds to CFT anyway (that is — no particles, nor quasiparticles, just unparticles ;-)).”

I would agree with such an argument if intermittency could be a useful solution to build a quantum field theory and would give us a spectrum to start with. E.g. I would like to see how is the glueball spectrum, to be observed into experiments with a lot of people currently eager to detect it, with a chaotic classical solution like this. You are in serious troubles as you do not even have a Fourier expansion. So, no Fourier expansion no spectrum. Indeed, Dmitry has no such result but just a prejudice: Chaotic classical solutions must be important for Yang-Mills theory. This without a proof that, he claims, should rely on me. But I have already shown that the theory is consistent and complete with integrable solutions and this was my aim. He is just claiming the contrary without providing a serious support to his prejudice.

After this rather questionable facts by his side and having him admitted that I am right as

Currently, there is no formulation of a quantum field theory starting with classical chaotic solutions.

he exposes his “would be”s about such a matter.

Let me comment about this discussion and how all this should be interpreted. Whenever you are smart enough to produce a theory and get it published you will get opinions from two different kind of people.  You will find interested people and criticizing people and this is in the matter of things. Criticizing people could be very useful wherever is able to support arguments with serious evidence. But most of times they will just criticize you on the ground of prejudices they have and these prejudices are those you have just demolished with your theory. So, to move an idea from the status of a prejudice to a status of a theory a strong mathematical and experimental support is needed.  A typical historical example has been the question of aether supporting the propagation of electromagnetic waves. A lot of people kept on believing on that till their death even after a strong evidence for relativity was achieved. This behavior belongs to our community, it was never lost, and we have to cope with it anytime we produce something new. It could imply delay into the acceptance of a theory but it is just human behavior and cannot be changed.

Let me repeat again. My approach is well developed and provides a glueball spectrum (to be observed experimentally), propagators and running coupling making the formulation of a Yang-Mills theory in the infrared complete. Propagators and running coupling are in very good agreement with lattice computations that come out in this somewhat unexpected direction. The same can be said with the spectrum but assuming that the lowest glueball is at about 500 GeV and has been already seen as \sigma resonance or f0(600). The same interpretation should apply to f0(980). This is in agreement with analysis done with dispersion relations by Narison, Mennessier, Ochs (see here) and Minkowski (see here).

What does one have on the side of classical chaotic solutions and quantum field theory? Substantially nothing as also admitted by Dmitry. No theory, no predictions, nothing. So, it can only be classified as a prejudice and a prejudice generally turns out to be wrong. My aim starts and ends when I have showed that my theory is mathematically sound and consistent and I get predictions that could be confirmed or not.  I do not have the burden to prove that, as one of my hypothesis does not like to someone, I have also to formulate my theory without it.

As a conclusion, I would greatly appreciate a formulation of a quantum field theory starting with chaotic solutions that applies to a realistic model of reality. I do not believe in betting but it would be tempting to put a wager on this.

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