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.

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