User vladimir kalitvianski - MathOverflow

Answer by Vladimir Kalitvianski for What are some correct results discovered with incorrect (or no) proofs?

2011-04-03T17:58:27Z

Renormalizations in QFT

Renormalizations as discarding perturbative corrections to masses and charges were not easily accepted, even by their inventors, because of being obviously anti-mathematic. It remains to be a prescription, lucky in some rare cases and wrong in the others.

In Physics we use a perturbation theory where the perturbation is supposed to be small but it is "big" in QFT. First we write down a non perturbed Hamiltonian, let's say:

$\hat H_0 = -\frac{\hbar^2}{2m_e}\frac{d^2}{dx^2} + \hat{V}_0 (x)$ (1)

Everything in it is quite physical including the electron mass. Then we "develop" our theory and include, as we think, a small interaction that has also a kinetic and a potential term:

$\hat H_{int} = -\epsilon\frac{d^2}{dx^2} + \hat{V}_1 (x)$ (2)

The kinetic term shifts the particle mass, it is obvious. But our mass is already good in (1) and any its shifting worsens agreement with experiment. Discarding this correction "restores" the right kinetic part of the Hamiltonian, and taking $\hat{V}_1$ into account improves agreement with experiment. So the discarding practice became a part of QFT calculations.

Appearance of a kinetic perturbative term is due to our misunderstanding interactions. Some part of interactions cannot be treated perturbatively but should be present in the zeroth-order approximation. Discarding is a very bad practice. For (2) it may luckily work, but for other our guesses of interactions it can be more complicated and be just "non renormalizable".

Although shown on a simplest example, the renormalizations in QFT have nothing else in their meaning but repairing a wrongly guessed Hamiltonian via repairing the corresponding solutions. Normally it is difficult to see explicitly that some part of guessed interaction, namely a "self-action" term, is of a kinetic nature. That is why presently they "explain" renormalizations differently.

A correct theory development should not include kinetic perturbative terms. Then the perturbative series will be reasonable, in my opinion.

Answer by Vladimir Kalitvianski for Mathematical explanation of the failure to quantize gravity naively

2011-02-25T16:12:06Z

"I'm wondering what actually fails mathematically. "

It is the physical model that fails. The physical model leads to certain equations that do not have physical (as well as mathematical) solutions. Even zeroth-order approximation fails.

This property does not belong exclusively to QG. Any QFT fails because of bad physical model. It is seen first in the zeroth-order approximation (it is too "far" form the exact solution); next it is seen as divergences of perturbative corrections (IR and UV infinities).

Normally one cannot proceed without patching and repairing the originally obtained solutions. Sometimes such "doctoring" numbers works but it is just a huge luck and in the end this luck turns into a bad luck - people (even mathematicians!) get used to patching and repairing solutions (???!!!) and try to apply such "prescriptions" to all models. Fortunately it does not work in many cases so not all people get smug.

There are several ideologies of justifying "renormalization" but even they fail. Nevertheless many keep smiling and "explaining" even in that case ("effective theory" ideology) like a failed magician.

It is practically impossible to find a reasonable attitude to all that. I was trying to understand and explain "success" of renormalizations myself and I even succeeded but my work needs further development.

Comment by Vladimir Kalitvianski

2011-02-26T11:45:01Z

A question to jeremy: can the tree level QED be considered as "prefect effective theory"? I mean, the first Born approximation results such as Mott, Bhabha, Kelin-Nishina cross sections, are they OK in you opinion?