# Gravity as a non renormalizable theory

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## Main Question or Discussion Point

Dear All
I have a project on the topic: " Explaining Why gravity is a non renormalizable theory?". I have downloaded many papers but i need your advice about which paper you think would most benefit me.
Thank you

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Haelfix
The papers you want are:
One-loop divergencies in the theory of gravitation. G. 't HOOFT (*) and M. VELTMAN
and
The ultraviolet behavior of Einstein gravity. Goroff and Sagnotti

The heuristic proof of the power counting argument for the nonrenormalizability of the Einstein Hilbert action can be found in many textbooks. For instance Zee, quantum field theory in a nutshell.

A. Neumaier
2019 Award
MTd2
Gold Member
canonical quantum gravity is perfectly valid as an effective field theory:
Effective to what energy scales?

Also, does that effective means without other forces? Or just gravity? What is the importance of this:

"this constitutes hard evidence that the asymptotic safety program indeed can give rise to a consistent quantum theory of gravity within the framework of quantum field theory along the lines envisioned by Weinberg"

http://arxiv.org/pdf/1601.01800v1.pdf

How does this new work relate to the doubts posed by Distler?

A. Neumaier
2019 Award
Effective to what energy scales?
Such that the leading effects become noticeable. below that, quantum gravity is not needed; sufficiently above that, one would have to fit a few further constants to the experiments, the number of constants needed slowly increases with the energy.

In the paper just gravity. But nothing fundamental is in the way of treating in the same way gravity plus standard model, except that one probably cannot do the computations anymore.

Asymptotic safety is a possible way to raise canonical gravity from an effective field theory to a fundamental one. On first sight, the paper looks very promising.

I don't know the doubts posed by Distler. Note that I am not an expert in quantum gravity, though I know enough QFT to understand the paper I cited.

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MTd2
Gold Member
It's that you posted the links in your website to his criticism, so I thought you could help me understand.

A. Neumaier
2019 Award
you posted the links in your website to his criticism, so I thought you could help me understand.
Ah, I had written it so long ago that I didn't remember all the details. Here is Distler's criticism:

''The trouble is that all hell breaks loose for ε∼1. Then all of these infinite number of coupling become equally important, and we lose control, both computationally and conceptually.''

He writes $\varepsilon$ for the square of $E/M_p$, where $E$ is the energy and $M_p$ is the Planck mass. Yes, I agree with his criticism, but none of us will live till the time when this regime can be experimentally probed. Thus I am happy to leave the resolution of this difficulty to the future.

But there are others who are more impatient than I am. Since the trouble would be cured by giving a nonperturbative definition of gravity, asymptotic safety will do it, if it can be demonstrated nonperturbatively. The paper you cited shows that asymptotic safety is, or seems to be [I haven't checked the details] present on the perturbative level. Thus the prospects are good that it is also valid nonperturbatively.

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atyy
But there are others who are more impatient than I am. Since the trouble would be cured by giving a nonperturbative definition of gravity, asymptotic safety will do it, if it can be demonstrated nonperturbatively. The paper you cited shows that asymptotic safety is, or seems to be [I haven't checked the details] present on the perturbative level. Thus the prospects are good that it is also valid nonperturbatively.
Perturbatively the prospects have been good for a long time, eg. http://arxiv.org/abs/0805.2909 - but how can one go from perturbative to non-perturbative - I think that's where asymptotic safety is stuck.

A. Neumaier
2019 Award
non-perturbative - I think that's where asymptotic safety is stuck.
Just like any other approach. This doesn't count against it.

Nonperturbatively, even standard QED or QCD are stuck.

MTd2
Gold Member
Nonperturbatively, even standard QED or QCD are stuck.
But aren't those proved in principle, by counting the divergence of the terms?

A. Neumaier
2019 Award
counting the divergence of the terms?
This only proves the existence of the renormalized perturbative series.

But this doesn't define the theory since it gives only an asymptotic series for the physical quantities, which is not good enough since infinitely many functions have the same asymptotic series.

That's why people talk about nonperturbative construction. One can treat quantum field theories on the lattice, which counts as nonperturbative, but the lattice approach has other difficulties. In particular, it breaks all continuous symmetries. To recover these, one has to take a continuum limit, and there all problems resurface.

No interactive relativistic QFT in 1+3 dimensions is under full nonperturbative control - which would mean: has a sound mathematical basis. One can do it either with full control of all limits in lower dimension, or with uncontrolled approximations in 4D. To construct an interactive QFT in 4 dimensions in a fully sound way is one of the big unsolved problems. The simplest case to be constructed first is considered by many to be Yang-Mills theory (i.e., QCD without quarks, glueballs only). But even this is already deemed very hard - it is one of the 7 Clay Millennium problems.

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MTd2
Gold Member
BTW, has anyone tried to arrange diagrames in predictable, infinite patters, like fractals of perturbative expantion until a boundary which is continuous?

A. Neumaier
2019 Award
has anyone tried to arrange diagrames in predictable, infinite patters
Yes. This is called resumming the series. There are various ways of doing it. The most standard one is the use of the Schwinger-Dyson equation to get useful approximations to the renormalized propagators. The resulting propagators have poles at (approximations to) the physical masses, while the perturbation series itself doesn't show them. It is useful and necessary in practice.

But it is all black art without any real support in the math. It is known rigorously that for any arbitrary asymptotic series there exist infinitely many different functions that have this series as their Taylor expansion. Hence no amount of rearranging the series can supply the missing information needed to identify the right nonperturbative function.

There is another important resummation technique called Borel summation. This has a mathematical rigorous setting, and reconstructs under certain assumptions on the growth of the coefficients a unique function with the given asymptotic series. Thehe applicability of the math depends on assumptions on the analytic behavior of the limit function, which sometimes can be proved to work in lower dimension. But they haven't been verified in 4D QFT, and there are obstructions (renormalons) that are believed to invalidate this approach to 4D QFT.

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MTd2
Gold Member
But it seems some black art yields more confidence than others. So, how to increase the confidence?

A. Neumaier
2019 Award
how to increase the confidence?
If the results agree well with experiments, the black art is considered fully justified. This even justifies (with a little less force) the same black art applied to similar problems. (The amount of similarity is in the eyes of the beholder). With even less force, black art is considered justified if you can convince the referees of your research paper that what you do is plausible.

Because of this, theoretical physics is quite different from mathematics. This is why much of physics can be non-rigorous without undue harm.

MTd2
Gold Member
I would like a more technical answer...

A. Neumaier
2019 Award
I would like a more technical answer...
I don't think there is any. As I indicated, it is a social process by which theoretical physics is justified. Much of it leads to predictions that are in good agreement with experiment. Everything that uses the thus established techniques is considered good scientific practice. Even if it is not (or not yet) backed up by experiment - as long as there are no contradictory experimental findings.

MTd2
Gold Member
A. Neumaier
2019 Award
In order not to raise false beliefs, you should also explain in your project why, in spite of its nonrenormalizability, canonical quantum gravity is perfectly valid as an effective field theory:
C.P. Burgess, Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory, Living Reviews in Relativity 7 (2004), 5
I just found the following recent survey of canonical quantum gravity and its confrontation with exciting experimental data:

R.P. Woodard, Perturbative Quantum Gravity Comes of Age, Int. J. Modern Physics D 23 (2014), 1430020. http://arxiv.org/abs/1407.4748.

Woodard writes in the introduction:
All of the problems that had to be solved for flat space scattering theory in the mid 20th century are being re-examined, in particular, defining observables which are infrared finite, renormalizable (at least in the sense of low energy effective field theory) and in rough agreement with the way things are measured. [...] The transformation was forced upon us by the overwhelming data in support of inflationary cosmology.

This only proves the existence of the renormalized perturbative series.

But this doesn't define the theory since it gives only an asymptotic series for the physical quantities, which is not good enough since infinitely many functions have the same asymptotic series.

That's why people talk about nonperturbative construction. One can treat quantum field theories on the lattice, which counts as nonperturbative, but the lattice approach has other difficulties. In particular, it breaks all continuous symmetries. To recover these, one has to take a continuum limit, and there all problems resurface.

No interactive relativistic QFT in 1+3 dimensions is under full nonperturbative control - which would mean: has a sound mathematical basis. One can do it either with full control of all limits in lower dimension, or with uncontrolled approximations in 4D. To construct an interactive QFT in 4 dimensions in a fully sound way is one of the big unsolved problems. The simplest case to be constructed first is considered by many to be Yang-Mills theory (i.e., QCD without quarks, glueballs only). But even this is already deemed very hard - it is one of the 7 Clay Millennium problems.
This makes me wonder just how fundamental and valid QFT as a framework really is, in general terms. I have only just begun studying QFT in recent weeks, so I'm still largely ignorant of the finer points and issues, but even in the most simplistic of textbook cases ( e.g. quantisation of Klein-Gordon fields ) there already seem to be fundamental issues and quite a bit of handwaving going on ("let's just subtract that infinity to fix that other infinity..."). I do not for a minute doubt or dispute the successes of the Standard Model in describing real-world physics, but the foundation this all stands on appears to me to be shaky and ad-hoc at best. Something just does not feel right about QFT to me; I can't put my finger on it, it's just a matter of intuition. I will keep learning, but there remains an itch that feels as if it badly needs to be scratched Am I the only one who feels that way ?

Staff Emeritus
2019 Award
So, in just a few weeks of study you have concluded that everyone else is doing it wrong?

A. Neumaier
2019 Award
the foundation this all stands on appears to me to be shaky and ad-hoc at best. Something just does not feel right about QFT to me; I can't put my finger on it, it's just a matter of intuition. I will keep learning, but there remains an itch that feels as if it badly needs to be scratched Am I the only one who feels that way ?
The mathematical foundations of interacting relativistic quantum field theory in 4 dimensions are nonexistent but there is lots of evidence that there is a consistent foundation independent of all the handwaving: All approximate, ad hoc, and/or handwaving ways of doing things lead with proper care to exactly the same results where they are comparable, and no inconsistency has ever been found. (Quantum gravity being not renormalizable is not a counterexample.) That the foundations are one of the Clay Millennium Problems is indication that something very important is missing and worth the prize. Until someone solved it, we must be content with the established mess, no matter how badly one feels about it. You are definitely not an exception in this respect. I still have this feeling after nearly 30 years of studying quantum field theory.

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ShayanJ
Gold Member
I just came across these lectures notes on arxiv. From its title, I get the impression that it may help some people in this thread. But I haven't read it yet.

So, in just a few weeks of study you have concluded that everyone else is doing it wrong?
Absolutely not, and to be honest I am quite surprised that this is how you interpreted my post. QFT is manifestly a very successful theory, there can be no doubt or argument over that. It is just that there are elements to it that feel ad-hoc and "messy" to me; it's a matter of intuition, I can't help it. And judging by A. Neumeier's comment as well as the link that ShayanJ has provided, it seems I am not the only person feeling that way.
Perhaps things will become clearer and more logical to me as I progress onwards and keep learning - time will tell.

Allow me to quote one sentence in particular from the link in post #23 which captures perfectly how I feel at the moment ( bottom page 1 ) :

"Often, one is left with the impression that there is some blind faith required on the side of the physicists or at least that some black magic is helping to obtain numerical values that fit so impressively what is measured in experiments from very doubtful expressions."

I could not have put it more succinctly.