WHY dimensional regularization does not work for Gravity ?

• zetafunction
In summary: Second, if the terms in the perturbation series are small, then the corrections to the original Lagrangian are also small, and so the PT series will converge. However, if the terms in the PT series are large, then the corrections to the original Lagrangian will also be large, and the PT series will not converge.

zetafunction

i have been reading several introductory papers to 'dimensional regularization' they tell how it can be applied to QED and so on, but the problem is why this dimensional regularization technique can not be applied to get finite answer in Quantum Gravity ??

I think you can use dim. reg. for gravity theories, but you will immediately run into the non-renormalizibility problem.

With dim. reg. you will probably find the divergent terms as a 4-d expansion; but when you now start to calculate and fix the counter terms, e.g. by something like minimal subtraction, you will find counter terms that do not appear in the original lagrangian. That means that you have to add new counter terms for each new order of the perturbation series.

The problem is not the REGULARIZATION you are using but the fact that new counter terms appear; that's what is meant be non-RENORMALIZIBILITY; this should be independent from the details of the regularization scheme.

um, however i found another link saying that if you use zeta regularization instead of Dimensional regularization, you could obtaine finite results for the divergent integrals

is this true ? http://www.scribd.com/doc/9650858/Z...lem-of-renormalization-and-Riemann-Hypothesis

author uses zeta regularization to overcome the divergences, but i can not fully understand his purpose (see section 2)

zetafunction said:
i have been reading several introductory papers to 'dimensional regularization' they tell how it can be applied to QED and so on, but the problem is why this dimensional regularization technique can not be applied to get finite answer in Quantum Gravity ??

As I explained in my papers and in my posts, the renormalization prescription "success" is a pure luck.

Being lucky in QED, it fails in the quantum gravity. It is partially so because physicists rely too much on this luck. They build the interaction Lagrangians and/or Hamiltonians by "analogy" with QED - by multiplying fields involved into interaction. By chance it gives "renormalizable" theories, but not always. See my publications and posts on this subject.

Bob.

A question: In the last decades people believed in renormalizable theories. That means that most theories leading to non-renormalizable theories were abandoned. And people always looked for renormalizability for GUTs, SUSY, SUGRA, ...

But is this reasonable?

If a theory fails to be (perturbatively) renormalizable, what has failed? the theory - or only its perturbation expansion?

The theory, of course. It means that the theoretical entities (fields, particles) and/or their interaction are badly "guessed", figured out, understood, and implemented. I give a simple example in RiR.

Bob.

I don't think so.

1) GR is not perturbatively renormalizable, but seems to be mathematically well-posed in terms of LQG.
2) what does it help if the perturbation series is finite order-by-order but not summable?

If you can't repair your car using needle and stitch, don't bame the car!

tom.stoer said:
I don't think so.

1) GR is not perturbatively renormalizable, but seems to be mathematically well-posed in terms of LQG.
2) what does it help if the perturbation series is finite order-by-order but not summable?

If you can't repair your car using needle and stitch, don't bame the car!

In my opinion, a physically meaningful formulation of any theory starts from well defined entities. As in QED, in quantum gravity one makes an error. It is especially evident while applying the perturbation theory (PT).

The good physics consists in good initial approximation for PT. Then the perturbative corrections will be small. Then there is no problem with the PT series convergence: it will converge to the exact solution. Physically well-posed theory does not lead to mathematical or conceptual difficulties and bad surprises.

It seem to me I managed to repair the QED. The same ideas should be applied to QG to advance in calculations.

Bob.

I can't agree: I do not see how and why QED (or perhaps better: QCD) should have a convergent perturbation series. Where is the proof?

So if QCD is not finite in PT (it is order-by-order, but not the sum): what are the "true" dregrees of freedom if not quarks and gluons?

Look at lattice gauge theory: the same theory with the same degrees of freedom can be regularized and has a well-defined continuum limit. So I think PT has failed, not QCD (I use QCD as an example because it has nice UV properties :-)

I agree that you have to start with well-defined entities. In QCD these are perhaps not plane wave states as you use them in PT, but in the original Lagrangian or path integral you do not have plane wave states - they enter via PT.

In QG you already have a viable theory w/o PT, namely LQG (it may have different problems, but they are not related to PT).

Tom

2) what does it help if the perturbation series is finite order-by-order but not summable?

First, there are non-linear methods of summing asymptotic series, for example, Padé approximants. They often converge to the exact functions. Demonstration: a truncated series for e-x 1-x+x2/2!-x3/3! is not good at x=3 but its positive Padé approximant [0,3](x)=1/(1+x+x2/2!+x3/3!) is much better - it decreases when x → ∞ and it is finite:

x=3;
exp(-3) ≈ 1/20=0.05;
1-x+x2/2!-x3/3! ≈ -2;
1/(1+x+x2/2!+x3/3!)≈ 1/13 ≈ 0.077.

There also other methods of divergent and asymptotic series summation known since long ago. It is a whole science.

Second, after reformulation, the most "singular" function dependence f(x) may be included in the zeroth approximation f(0)(x) so the remaining series may be convergent in a regular sense.

Bob_for_short.

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tom.stoer said:
A question: In the last decades people believed in renormalizable theories. That means that most theories leading to non-renormalizable theories were abandoned. And people always looked for renormalizability for GUTs, SUSY, SUGRA, ...

But is this reasonable?

If a theory fails to be (perturbatively) renormalizable, what has failed? the theory - or only its perturbation expansion?

You are right that it's just a failure of the approach, not necessarily of the theory. But since perturbation theory was the only known way to do calculations, I think that the spirit was more that if we were to try to check a theory against experiments, we had to pick renormalizable theories.

Of course, in the spirit of effective field theories, even perturbatively nonrenormalizable theories are ok.

nrqed said:
You are right that it's just a failure of the approach, not necessarily of the theory. But since perturbation theory was the only known way to do calculations, I think that the spirit was more that if we were to try to check a theory against experiments, we had to pick renormalizable theories.

That is the main trap. A stop-gap made an icon.

nrqed said:
Of course, in the spirit of effective field theories, even perturbatively nonrenormalizable theories are ok.

And there is always a possibility to reformulate theory in better terms. That is the correct direction.

Bob_for_short.

Bob_for_short said:
And there is always a possibility to reformulate theory in better terms. That is the correct direction.Bob_for_short.

Have a look at QCD and chiral perturbation theory: imagine we would have startet with CPT not knowing QCD. Then reformulating CPT as QCD would have been the correct direction in terms - but you would no loner be able to calculate the soft amplitudes in QCD.

I agree, if I have a non-renormalizable theory I should look for improvements. But those improvements can point to different directions and perhaps only experiment can tell you what is right. For GR it could be SUGRA or LQG. Both should be finite (I think in SUGRA they are working hard on a proof - perhaps they are nearly there).

So the failure of PT is only a hint that you must improve the theory, it is not necessarily a hint HOW to do that. In the very end it could mean that only PT failed, but that the underlying theory (if correctly qunatized w/o PT) was right.

tom.stoer said:
I agree, if I have a non-renormalizable theory I should look for improvements. But those improvements can point to different directions and perhaps only experiment can tell you what is right. For GR it could be SUGRA or LQG. Both should be finite (I think in SUGRA they are working hard on a proof - perhaps they are nearly there).

So the failure of PT is only a hint that you must improve the theory, it is not necessarily a hint HOW to do that. In the very end it could mean that only PT failed, but that the underlying theory (if correctly qunatized w/o PT) was right.

If you take patience to read my article "Reformulation instead of Renormalizations", you will see how and why to do that. It is physically rather natural but completely unexplored direction. The article is made simple on purpose - I reduced the essential of the renormalization problem to a two-body CM problem in the mixed variables.

Bob_for_short.

http://arxiv.org/abs/0811.4416

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=== WHY dimensional regularization does not work for Gravity ? ===

None of regularizations work for Gravity, not only the dimensional one. The regularized expressions are useless because of non renormalizability of QG. The theory is just physically wrong, thus mathematical difficulties.

Bob_for_short.

Bob_for_short said:
The regularized expressions are useless because of non renormalizability of QG
I agree; that was my stratement as well.

Bob_for_short said:
The theory is just physically wrong
I do not agree

What about Naive quantization with perturbarion theory in terms of weakly interacting gravitons on a flat spacetime is just physically wrong The theory has to be reformulated - and LQG should be a reasonable way out.

Tom

tom.stoer said:
I do not agree.

What about Naive quantization with perturbation theory in terms of weakly interacting gravitons on a flat spacetime is just physically wrong The theory has to be reformulated - and LQG should be a reasonable way out.

Tom

I do not think so. The right way is a compound system approach, like my electronium in RiR and in "Atom...". This way excludes infinities and renormalizations and it gives the correct classical limit.

Bob.

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Why do you ignore the fact that LQG is already finite after regularization?
And what is the electronium of QG?

tom.stoer said:
Why do you ignore the fact that LQG is already finite after regularization?
And what is the electronium of QG?

I do not ignore LQG being finite after regularization. I just think, no, I am sure it is a wrong direction. Let us take QED. It is also finite after regularization and renormalizable. But to what extent it is ugly! It starts from free particles as an initial approximation whereas the photon-electron interaction is very strong - there is always soft radiation. The same happens in LQG.

I could write here what the electronium is but it is already written. So if you like to know, you may read "Reformulation instead of Renormalizations" and "Atom as a "dressed" nucleus" by Vladimir Kalitvianski. I take the electron-field coupling into account exactly rather than perturbatively.

Regards,

Bob_for_short.

I am just started reading some articles, but I do no really see how this should be related to LQG.

In QED you start with an approximation, namely weak coupling and PT of plane waves. QED is not UV safe, so you can be sure that PT fails to be summable.

But in LQG you do not make any approximation. Instead you start with and keep the full nonlinear theory. You do not make any truncation at all, but you decide for each calculation which approximation fits best. Of course the theory is not exactly solvable, so for every problem approximations are in order. But the formulation of theory does not force you to use PT, whereas in QED you are not free to chose the approximation.

tom.stoer said:
I am just started reading some articles, but I do no really see how this should be related to LQG.

In QED you start with an approximation, namely weak coupling and PT of plane waves. QED is not UV safe, so you can be sure that PT fails to be summable.

But in LQG you do not make any approximation. Instead you start with and keep the full nonlinear theory. You do not make any truncation at all, but you decide for each calculation which approximation fits best. Of course the theory is not exactly solvable, so for every problem approximations are in order. But the formulation of theory does not force you to use PT, whereas in QED you are not free to chose the approximation.

I could not conclude from your post whether you are reading my articles or somebody else's.
My Novel QED starts from coupled quantized EMF and electron (=electronium). The charge in electronium is quantum mechanically smeared, so no UV problem arise. Pushing such a coupled charge excites unavoidably the quantized EMF (photon emission), so no IR divergence appear. My approach is radically different from the standard QED.

I think, in QG we should also start from a compound system like an electronium - with the quantized gravitational field coupled to an elementary mass (?). But I think the quantum character of the gravitational field will never manifest itself. Rather, it will always be the inclusive picture.

If you are reading my articles, do not hurry to make conclusion. Try first to understand my idea of electronium.

Bob_for_short.

1. Why can't we use dimensional regularization for gravity?

Dimensional regularization is a mathematical technique that is commonly used in quantum field theory to deal with divergent integrals. However, it does not work for gravity because gravity is a non-renormalizable theory. This means that the divergences in gravity cannot be removed by adding counterterms, which is the standard procedure in dimensional regularization.

2. What is a non-renormalizable theory?

A non-renormalizable theory is a theory in which the divergences cannot be removed by adding counterterms. This means that the theory cannot be made finite by adjusting a finite number of parameters. In the case of gravity, this is because the theory becomes infinitely sensitive to high energies, making it impossible to predict physical quantities beyond a certain energy scale.

3. Can we use other regularization methods for gravity?

Yes, there are other regularization methods that can be used for gravity, such as cutoff regularization and zeta-function regularization. However, these methods also have their limitations and may not be applicable in all cases.

4. Why is gravity a non-renormalizable theory?

Gravity is a non-renormalizable theory because it is described by a non-renormalizable Lagrangian. This means that the terms in the Lagrangian become infinitely large as the energy of the system increases, making it impossible to remove the divergences through renormalization.

5. Are there any successful theories that use dimensional regularization for gravity?

No, there are currently no successful theories that use dimensional regularization for gravity. However, there are ongoing efforts to develop a theory of quantum gravity that can incorporate the principles of dimensional regularization and other regularization methods.