Why physicists cannot renormalize all divergent integrals?

[Anixx]

TL;DR Summary

Why the physicists have troubles with infinities in many physical theories, such as quantum gravity? Why cannot they just use divergent integrals and regularize or renormalize them in the end so to obtain finite values?

Why the physicists have troubles with infinities in many physical theories, such as quantum gravity? Why cannot they just use divergent integrals and regularize or renormalize them in the end so to obtain finite values?
I mean, operations on divergent integrals are not a problem, and techniques for regularization are known.

 

[MathematicalPhysicist]

This is suitable for beyond the SM forum, isn't it?

 

[PeterDonis]

This is suitable for beyond the SM forum, isn't it?

Divergent integrals appear in current quantum theories, so the question is suitable for this forum.

 

[Anixx]

Can anyone give an example of non-renormalizable divergency in physics?
My question: https://physics.stackexchange.com/q...ics/613960?noredirect=1#comment1382001_613960

 

[PeterDonis]

Can anyone give an example of non-renormalizable divergency in physics?

The quantum field theory of a massless spin-2 field is known to be non-renormalizable.

 

[atyy]

Although the formal proof is lacking in many cases, physicists currently use the effective field theory framework of Kenneth Wilson (since the 1970s) to make sense of both "renormalizable" and "non-renormalizable" theories quantum field theories. Thus we can make sense of the quantum theory of gravity (spin-2) as a quantum theory that makes correct and sensible predictions for low energies, although it still fails at high energies.

https://arxiv.org/abs/1209.3511
The effective field theory treatment of quantum gravity
John F. Donoghue

There are also analogs of this interpretation of renormalization and effective theories in simple quantum mechanics.

https://arxiv.org/abs/quant-ph/0503074
On the limit cycle for the 1/r^2 potential in momentum space
H.-W. Hammer (INT), Brian G. Swingle (Georgia Tech and INT)

 

[Demystifier]

Summary:: Why the physicists have troubles with infinities in many physical theories, such as quantum gravity? Why cannot they just use divergent integrals and regularize or renormalize them in the end so to obtain finite values?

Why the physicists have troubles with infinities in many physical theories, such as quantum gravity? Why cannot they just use divergent integrals and regularize or renormalize them in the end so to obtain finite values?
I mean, operations on divergent integrals are not a problem, and techniques for regularization are known.

To understand the answer to this question, it's important to distinguish regularization from renormalization. Any field theory can be regularized, which makes all quantities finite. However, the values of quantities obtained that way are typically very large, much larger than seen in experiments. Hence regularization is not enough. That's why we also need renormalization, which is a way to absorb those large quantities into not so large coupling constants obtained through measurements. The problem is that such absorbtion cannot always be made. Namely, for some theories it turns out that one would need an infinite number of such constants to be obtained through measurements, which of course cannot be done because nobody can measure an infinite number of coupling constants. Such theories are called non-renormalizable.

However, for practical purposes even such non-renormalizable theories make sense. Even though there is an infinite number of coupling constants to be dealt with, in practice most of them can be neglected. When the theory is applied to sufficiently low energies, only a few coupling constants are important, while the effect of the rest of them is negligible. Theories used only at low energies in that sense are called effective theories.

 

[vanhees71]

One has to distinguish between Dyson renormalizable QFTs and more general "effective" QFTs. A QFT is called Dyson renormalizable if it can be renormalized by fixing a finite number of constants (wave-function normalization, masses, and coupling constants), i.e., by a finite number of local counter terms in the effective action.

An effective QFT is renormalizable in the sense that you interpret as a low-energy theory defined by some finite energy scale, $\Lambda$. Then you consider expansions in powers of $p/\Lambda$. Then at any loop order of diagrams you have a finite number of "low-energy constants" to fix through renormalization, but you get more and more when going to higher orders in the loop expansion and the expansion in powers of $p/\Lambda$. The predictive power of such theories is usually in the symmetries, because also in this case you can show that the counter terms at any order of the expansion are compatible with the symmetries (e.g., the counter terms in chiral perturbation theory is consistent with chiral symmetry, taking the explicit symmetry breaking as a perturbation in the same sense, i.e., another perturbation expansion is also in terms of powers of $m_{\pi}/\Lambda$).

A very illuminating article about this "Wilsonian point of view" of renormalization is

https://arxiv.org/abs/hep-th/9702027

A more complete formal explanation is found in Weinberg, Quantum theory of Fields Vol. 1 (Sect. 12.3).

 

[Doctor Who]

They simply can not. Can't find the Theory of Everything.