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- Thread starter zetafunction
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tom.stoer

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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.

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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)

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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.

- #5

tom.stoer

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But is this reasonable?

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

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Bob.

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tom.stoer

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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!

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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.

- #9

tom.stoer

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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

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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=3;

exp(-3) ≈ 1/20=0.05;

1-x+x

1/(1+x+x

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

Bob_for_short.

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- #11

nrqed

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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.

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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.

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.

- #13

tom.stoer

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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.

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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

Bob_for_short.

http://arxiv.org/abs/0811.4416

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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.

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tom.stoer

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

I doThe theory is just physically wrong

What about

Tom

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I donotagree.

What aboutNaive quantization with perturbation theory in terms of weakly interacting gravitons on a flat spacetime is just physically wrongThe 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

Bob.

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- #18

tom.stoer

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And what is the electronium of QG?

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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.

- #20

tom.stoer

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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.

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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.

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