QFT with respect to general relativity

[Finbar]

No I am saying that the ratio of the gravitational coupling constant with other gauge coupling constant never exceeds one..

Please read the introduction of Birrel and Davies or alternatively section 3 of this introductory paper
arXiv:1011.0543

and the following gives the details of the energy expansion in slightly more detail, including a test case calculation of the change to the effective gravitational coupling constant where you see the effects arising from quantum corrections.

http://arxiv.org/abs/gr-qc/9712070v1

Alternatively the papers on asymptotic safety also seem show the same general pattern (Geff goes to zero)

Slightly more universal and highbrow statements can be found in this brilliant paper

http://arxiv.org/abs/hep-th/0601001

where they argue that the existence of incredibly small coupling constants arising from new Yang Mills like physics cannot occur in nature.

There is no solid evidence to support your claim that the ratio never exceeds one here. Only


http://arxiv.org/abs/hep-th/0601001

sugests that the mass/charge ratio never exceeds one and only for U(1).

If we take a yang-mills coupling it will go to zero as it is asymptotically free at high energies. The dimensionless gravitational coupling grows with energy and even in AS will reach a non-zero fixed point (note: you can't take the ratio of the dimensionful gravity coupling with the gauge one since this will carry dimensions and there would be no meaning to it being one). Now gravity might spoil asymptotic freedom but the current evidence suggests it might not. If we still have asymptotic freedom for YM theories then the ratio of the gravity to gauge coupling will exceed one.


Now I am in complete agreement that we can't neglect the running of any of the couplings once gravity is involved. But we have to do the calculation to see what the ratio of different couplings will do. I think you make a good point that pure gravity it is not a good enough model to tell us about QG and we need to include matter too.

 

[Haelfix]

If we take a yang-mills coupling it will go to zero as it is asymptotically free at high energies.

Yes.

The dimensionless gravitational coupling grows with energy and even in AS will reach a non-zero fixed point (note: you can't take the ratio of the dimensionful gravity coupling with the gauge one since this will carry dimensions and there would be no meaning to it being one).

Yes. I didn't want to get into this, b/c it gets away from the point and becomes technical, but what we actually compare are the couplings in the energy expansion (the Cn's in the derivative expansion of the effective lagrangian of gravity coupled to whatever matter survives up to the Planck scale) not the uncoupled SM gauge couplings by themselves (which probably are altered by strongly coupled GUT dynamics anyway). The former by construction has the same units, however, there are ambiguities in what one means by this, arising from renormalization scheme differences ... Again its not difficult, but its a bit of a chore to identify b/c there is a tremendous amount of mixing and field redefinitions taking place when you take the counter lagrangian etc.. Especially when you have a large amount of matter species. Suffice it to say, you can show that my statement will hold in some sense.

Anyway let's go over this again from the top. I'm sure you have seen the famous RG log graphs where you have the 3 SM couplings that unify as lines, and the gravitational constant that also unifies about 2 orders of magnitude later from underneath. We both agree that up until the Planck scale, gravity is certainly the weakest force (at least with just the SM + gravity). Good!

What happens when quantum mechanics becomes involved and the expansion becomes strongly coupled?

Well, the previous behaviour of the couplings alters, and a precise description would require calculating the new beta functions of the full Planckian theory (requiring knowledge of whatever physics and matter exists there). Of course a UV completion would completely alter the physics entirely, and it wouldn't make sense to talk about 'gravity' perse anymore. --So I will assume from here on out that we are not doing that and are not introducing new degrees of freedom--

So can we make some guesses as to the behavior? You bet! The first guess is to examine what happens to the effective gravitational coupling constant with the first quantum corrections arises at one loop in the case of pure gravity in the context of graviton graviton scattering. If you read page 13-14 of the Donoghue paper, you will see that he refers to a calculation that gives a correction that is *negative*. The first analytic albeit perturbative evidence that the behaviour actually turns around and becomes weaker again.

Second piece of evidence. In asymptotic safety papers, they find the same exact phenomenon.. EG the weakening of Newtons constant! This is slightly more powerful than the previous result, b/c they are probing some amount of strongly coupled behavior through the method of the exact renormalization group equations.

See Percacci's FAQ and the references he links too

http://www.percacci.it/roberto/physics/as/faq.html

Now the final piece of evidence comes from Nima's paper that I linked earlier.

There the conjecture is claimed to be universal for all BYSM physics that you might include before the Planck scale -its motivated in part with a U(1) but can be generalized to any Nonabelian group that can be higgsed down to the U(1)- They further motivate it with several no go arguments. The point is that if you include new YM forces with absurdly tiny couplings (in such a way that it would actually be weaker than gravity), you are secretely introducing a new intermediate scale in physics, and are conspiring for there to be an infinite tower of very light stable charged particles that are not protected by any symmetry. The nonobservation thereof and theoretical implausibility of such objects is then claimed to be evidence for the conjecture.

So I would say the case is very strong, that gravity always stays the weakest force in our world, or indeed any consistent world with physics like our own.