A question about graviton self-interactions in QFT

[ensabah6]

Gravity when expressed as QFT suffers from renormalization problems.

From the point of view of the techniques of (four-dimensional) quantum field theory, and as the numerous and heroic efforts to formulate a consistent quantum gravity theory by some very able minds attests, gravitational quantization was, and is still, the reigning champion for bad behavior. There are problems and frustrations stemming from the fact that the gravitational coupling constant has dimensions involving inverse powers of mass, and as a simple consequence, it is plagued by badly behaved (in the sense of perturbation theory) non-linear and violent self-interactions. Gravity, basically, gravitates, which in turn...gravitates...and so on, (i.e., gravity is itself a source of gravity,...,) thus creating a nightmare at all orders of perturbation theory. Also, gravity couples to all energy equally strongly, as per the equivalence principle, so this makes the notion of ever really "switching-off", "cutting-off" or separating, the gravitational interaction from other interactions ambiguous and impossible since, with gravitation, we are dealing with the very structure of space-time itself.​

Suppose one were to introduce as a hypothetical physical principle, the idea of "quantum minimal energy" -- simply put, there is a minimal "quanta" of energy that results in graviton interactions, and below this threshold, there is no graviton interactions. Suppose that one were to conjecture, as a physical principle, the graviton carries too little energy (or for some unknown physical principle, spin-2 particles do not self-interact below a certain threshold)

would this make gravitons perturbation in non-SUSY 4D QFT re-normalizable? While this might seem ad-hoc, if it works, it might be worth researching. Also, there is no technology to explore such incredibly weak effects.

there might not been a need for strings, SUSY, or loops. Just QFT with some additional physical principles, like a cut-off.

 

[tom.stoer]

I am not sure if I get your idea correctly, but I guess that vanishing self-intercation of gravitons below a certain threshold would affect the IR sector only, whereas the divergencies are in the UV.

In addition I do not know how to use this as input. Instead it should be a prediction of the theory (look at QCD: asymptotic freedom = "zero coupling strength in the UV" is a prediction).

Do you know about the "asymptotic safety" research program? It goes back to an idea of Weinberg which is a generalization of asymptotic freedom. It basically says that only a finite number of couplings survive in the UV and that their values remain finite (whereas all other couplings can be neglected). There are some promising results indicating that this could indeed be true in quantum gravity.

Then there is a result of the LQC program which removes singularities like black holes and the big bang entirely. It basically says that gravity coupled to scalar particles becomes repulsive near the Planck scale.

None of these results has been derived rigorously, neither are the fundamental theories widely accepted theoretically or tested experimentally. I mention these results mainly to indicate that there are theories making similar predictions instead of using such an idea as input.

 

[CHIKO-2010]

Gravity when expressed as QFT suffers from renormalization problems.

From the point of view of the techniques of (four-dimensional) quantum field theory, and as the numerous and heroic efforts to formulate a consistent quantum gravity theory by some very able minds attests, gravitational quantization was, and is still, the reigning champion for bad behavior. There are problems and frustrations stemming from the fact that the gravitational coupling constant has dimensions involving inverse powers of mass, and as a simple consequence, it is plagued by badly behaved (in the sense of perturbation theory) non-linear and violent self-interactions. Gravity, basically, gravitates, which in turn...gravitates...and so on, (i.e., gravity is itself a source of gravity,...,) thus creating a nightmare at all orders of perturbation theory. Also, gravity couples to all energy equally strongly, as per the equivalence principle, so this makes the notion of ever really "switching-off", "cutting-off" or separating, the gravitational interaction from other interactions ambiguous and impossible since, with gravitation, we are dealing with the very structure of space-time itself.​

Suppose one were to introduce as a hypothetical physical principle, the idea of "quantum minimal energy" -- simply put, there is a minimal "quanta" of energy that results in graviton interactions, and below this threshold, there is no graviton interactions. Suppose that one were to conjecture, as a physical principle, the graviton carries too little energy (or for some unknown physical principle, spin-2 particles do not self-interact below a certain threshold)

would this make gravitons perturbation in non-SUSY 4D QFT re-normalizable? While this might seem ad-hoc, if it works, it might be worth researching. Also, there is no technology to explore such incredibly weak effects.

there might not been a need for strings, SUSY, or loops. Just QFT with some additional physical principles, like a cut-off.

If I understand correctly you are suggesting a theory where below certain energy scale gravity becomes a linear theory. I guess this may happen if the effective self-interaction couplings vanishes in the IR. This does not happen in GR. However, if there is a theory where this indeed happens you still won't be able to avoid UV divergences. BTW, the asymptotic safety scenario is based on the (non-Gaussian) UV fixed point for the gravitational constant. if this is correct than gravity can be viewed as non-perturbatively renormalizable theory.

P.S. I didn't noticed the previous post. Mine essentially repeats it, sorry

 

[tom.stoer]

P.S. I didn't noticed the previous post. Mine essentially repeats it, sorry

No problem.

 

[ensabah6]

As I understand it, the reason QED is renormalizable in QFT, is that while photons carry EM force, photons themselves are uncharged, so it is straightforward to eliminate those infinities.

The reason gravitons is not renormalizable, is that gravitons carry energy and gravitons interact with themselves and everything else, so the infinities in the perturbation series are nonrenormalizable. SUSY attempts to tame these infinities by positing a fermionic SUSY-partner to cancel some of these infinities, string theory attempts to do so via extended interactions and higher dimensions.

What if instead, gravitons do not interact with other gravitons, the way photons in QED do not interact with other photons, b/c gravitons either do not have enough energy, or graviton-spacetime interaction causes them to not interact. Would such gravitons in 4D non-SUSY QFT spin-2 bosons have a perturbation series that is renormalizable?

 

[CHIKO-2010]

As I understand it, the reason QED is renormalizable in QFT, is that while photons carry EM force, photons themselves are uncharged, so it is straightforward to eliminate those infinities.

The reason gravitons is not renormalizable, is that gravitons carry energy and gravitons interact with themselves and everything else, so the infinities in the perturbation series are nonrenormalizable. SUSY attempts to tame these infinities by positing a fermionic SUSY-partner to cancel some of these infinities, string theory attempts to do so via extended interactions and higher dimensions.

What if instead, gravitons do not interact with other gravitons, the way photons in QED do not interact with other photons, b/c gravitons either do not have enough energy, or graviton-spacetime interaction causes them to not interact. Would such gravitons in 4D non-SUSY QFT spin-2 bosons have a perturbation series that is renormalizable?

It is not entirely true that photons do not interact with each other in QED. QED is a nonlinear theory where quantum corrections through the matter loops can induce radiative processes such as e.g., light-by-light scattering. Similar non-linear interactions will be induced if you couple gravitons to matter, in such a way that diff invariance is preserved. Indeed, the linearized gravity coupled to matter is known to be non-renormalizable. Matter loop contributions will be divergent and since the Newton constant has a dimension mass^{-2}, the theory will be non-renormalizable anyway. Diff non-invariant theory of spin-2 field, on the other hand, cannot be a consistent theory. There is no much choice indeed :-)

Renormalizability of QED is essentially due to the fact that the fine structure constant is dimensionless + other stuff (non-anomalous gauge invariance, etc). There are renormalizable non-linear theories also, such as non-Abelian gauge theories: QCD and the electroweak theory. Gluons (QCD cousins of photons) are charged under the color group, but that does not preclude QCD from being a renormalizable theory.

 

[ensabah6]

It is not entirely true that photons do not interact with each other in QED. QED is a nonlinear theory where quantum corrections through the matter loops can induce radiative processes such as e.g., light-by-light scattering. Similar non-linear interactions will be induced if you couple gravitons to matter, in such a way that diff invariance is preserved. Indeed, the linearized gravity coupled to matter is known to be non-renormalizable. Matter loop contributions will be divergent and since the Newton constant has a dimension mass^{-2}, the theory will be non-renormalizable anyway. Diff non-invariant theory of spin-2 field, on the other hand, cannot be a consistent theory. There is no much choice indeed :-)

Renormalizability of QED is essentially due to the fact that the fine structure constant is dimensionless + other stuff (non-anomalous gauge invariance, etc). There are renormalizable non-linear theories also, such as non-Abelian gauge theories: QCD and the electroweak theory. Gluons (QCD cousins of photons) are charged under the color group, but that does not preclude QCD from being a renormalizable theory.

What about developing a theory of gravity that has a fine structure constant that is dimensionless, but reproduces some of the predictions of GR in the classical limit.

 

[CHIKO-2010]

What about developing a theory of gravity that has a fine structure constant that is dimensionless, but reproduces some of the predictions of GR in the classical limit.

Well, the only consistent theory of that sort known up to now is the string theory

 

[ensabah6]

Well, the only consistent theory of that sort known up to now is the string theory

If QED can have a dimensionless fine structure constant, why can't a gravity be modeled after this, one whose mathematical form exactly duplicates QED?

 

[CHIKO-2010]

If QED can have a dimensionless fine structure constant, why can't a gravity be modeled after this, one whose mathematical form exactly duplicates QED?

No gravity does not duplicate QED. Simply compare Coulomb force with the Newton force. "Charges" for gravity are masses therefore the coupling constant for gravity (the Newton constant) has a dimension mass^{-2} (in natural units).

In the string theory the basic objects are not point-like particles but extended strings, therefore there is inevitably new scale in the theory, the string scale L. The dimensionless string constant than is made up from L and the Newton constant, g_s~ L^-1G_N^1/2. this is crude, but I believe correct explanation. The string theory reproduces then Einstein's general relativity for distances > L.

 

[ensabah6]

No gravity does not duplicate QED. Simply compare Coulomb force with the Newton force. "Charges" for gravity are masses therefore the coupling constant for gravity (the Newton constant) has a dimension mass^{-2} (in natural units).

In the string theory the basic objects are not point-like particles but extended strings, therefore there is inevitably new scale in the theory, the string scale L. The dimensionless string constant than is made up from L and the Newton constant, g_s~ L^-1G_N^1/2. this is crude, but I believe correct explanation. The string theory reproduces then Einstein's general relativity for distances > L.

the coloumb force is coupled to electric charge. shouldn't gravity really couple to energy rather than mass?

 

[CHIKO-2010]

the coloumb force is coupled to electric charge. shouldn't gravity really couple to energy rather than mass?

Yes, in GR gravity couples to energy and pressure (energy-momentum tensor). I just gave you simpler example of non-relativistic Newton's law, which is obtained from the GR in the corresponding limit. The constant of interactions (the Newton constant) is dimensionfull both in GR as well as in its Newtonian limit. This is the very basics of the theory, I would suggest to consult some textbook, if you are really interested in the subject.

 

[ensabah6]

Yes, in GR gravity couples to energy and pressure (energy-momentum tensor). I just gave you simpler example of non-relativistic Newton's law, which is obtained from the GR in the corresponding limit. The constant of interactions (the Newton constant) is dimensionfull both in GR as well as in its Newtonian limit. This is the very basics of the theory, I would suggest to consult some textbook, if you are really interested in the subject.

ok, but since QED couples to electric charge, why is it "dimensionless"

 

[CHIKO-2010]

ok, but since QED couples to electric charge, why is it "dimensionless"

Dear ensabah6, In the relativistic QFT it is customary to use the units where c=hbar=1, called natural units. In this units the electric charge is dimensionless while Newton's constant has dimension mass^-2. The use of these units is also convenient because you can discuss renormalizability of the theory by power counting...Of course you can use whatever units you would like this does not change the conclusion about renormalizability of a given theory.

My sincere suggestion: before attacking such a complicated problems as renormalization of quantum gravitation, it is useful at least to understand very basics of QFT. Sorry, I would not reply anymore to your subsequent post on this matter. Cheers.