# The act of renormalization

A question that has been bothering me for a while, is why are we as a physics community so fine with remormalization in QFT? Experimentally (QED especially) the field is VERY precise, however, looking at the mathematical side of renormalization, it doesn't look... very logical. We get divergent integrals, but then say that we weren't measuring the right process, or our constants need to normalized, etc, etc.

Where can I find work being done to make QFT more mathematically consistent? I'm just a newbie when it comes to research in QG (and like GR more than QFT), but this question has been bothering me for months, and I'm sure either missing something, or will have to just learn to 'accept' the mathematically inconsistencies of QFT!

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I would claim that renormalisation (or the need to do so) is not a problem of QFT itself, but rather a problem of constructing a specific realisation. In principle there are axiomatic frameworks like the Wightman axioms which specify QFT, but do not even mention renormalisation. It is to my knowledge an open question whether gauge theories can fit that framework, or if there are even such interacting QFTs in 4 spacetime dimensions. So in this sense our usual QFTs and in particular the standard model might not have a solid mathematical foundation. But if this would be the case, we could just regard renormalisation as an extension of (canonical) quantisation, which is not unique anyway.

ohwilleke
Gold Member
The physics community is fine with renormalization because it give us the right answers experimentally to high precision, despite the fact that the coefficients in the equations are independent of the experimental inputs.

Renormalization was not mathematically established as valid in a rigorous way when it was introduced as a technique, and my understanding from a recent blog conversation with a graduate student in a relevant subfield of mathematical physics is that this rigorous proof continues to be elusive and that its absence of a barrier to further progress in mathematical quantum physics. https://4gravitons.wordpress.com/2016/01/01/who-needs-non-empirical-confirmation/

arivero
Gold Member
looking at the mathematical side of renormalization, it doesn't look... very logical.
Well, imagine a function and that I want to get a measure of how it grows in a point x. I can try to define the quantity F(x+delta)/delta but then I get an infinity when delta goes to zero. I can give up and say that it makes not sense to consider "growing" of functions, or substract a counterterm, F(x)/delta, that also is infinity when delta goes to zero. But substracting both infinities I get a finite quantity that I can call the grow, or better, the fluxion, of F at the point x.

Staff Emeritus
2019 Award
Please, please stop for a second. This (divergent integrals) is not renormalization. This is regularization.

The problem of divergences is not something specific to QFT, or even QM. You get it classically - "what is the self-energy of an electron?" It basically comes about every time you have inverse powers of r and try and calculate something that goes down to r = 0. The answer has always been "we don't believe our theories are correct at these small scales" - even in the classical era. That may not be completely comforting, but it's probably true.

Jeff Rosenbury and FieldTheorist
atyy
For relativistic QFTs that can be consistent at arbitrarily high energies, https://www.amazon.com/dp/1107005094/?tag=pfamazon01-20&tag=pfamazon01-20.

However, we believe (without proof) that the physically important QFTs like the standard model of particle physics are not valid at all energies. In this case, one uses lattice gauge theory http://arxiv.org/abs/hep-lat/0211036. Much of the standard model can be rigourously formulated with lattice gauge theory, but whether chiral interactions with non-Abelian gauge fields can be done this way is still unknown.

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julian
Gold Member
Putting renormalization on a more rigorous mathematical footing was claimed by the introduction by Kreimer’s method of Feynman diagram renormalization via Hopf algebra of rooted trees.

arivero
Paul Colby
Gold Member
Renormalization is done almost as an after thought in classical physics. It's common to take the observed mass of the electron, treat the electron as a point particle of this mass and, and simply ignore the infinite energy in the point coulomb field.