QFT with respect to general relativity

[jacksonb62]

After recently researching about Quantum Field Theory and more specifically gravitons, I am slightly confused with how this theory of the gravitational force fits in with general relativity. I know it hasn't disproved it so there must be some connection. Do gravitons in 11 dimensions cause curvature in 4 dimensional space-time that we observe as gravity? I've been thinking hard about this one and its been stumping me

[jtbell]

We don't yet have a generally-accepted theory that combines general relativity and quantum field theory. People are working on different approaches (e.g. string theory, loop quantum gravity) but none of them has won out.

[jacksonb62]

So string theory is postulating that gravitons are closed loops that can move between branes correct? Would this possibly explain why they would cause curvature in the 4 dimensions that we can observe? If the particles travel between extra dimensions it seems to me that the effects in our 4 dimensions would then be what we observe as general relativity. Just a thought

[mitchell porter]

A few comments in lieu of a comprehensive explanation...

If you have read about general relativity, you may be aware that the curvature of space is described by the metric, and the metric is described by a tensor field.

In quantum field theory, particles (like the graviton) are associated with fields; they arise by applying the laws of quantum mechanics, such as the uncertainty principle, to the field.

The original way to get a quantum theory of gravitons, as pioneered e.g. by Feynman, is as follows: You take the dynamical metrical field of general relativity. You express it as a deviation from the constant metric of flat space (Minkowski space). Then you treat this deviation itself as the graviton field.

From this perspective, the graviton is a quantized deviation from flat space.

You mention 11 dimensions and string theory. Well, before we get to string theory, let's talk about 11 dimensions. The original 11-dimensional theory was the 11-dimensional form of "supergravity" (which can also be defined for a lower number of dimensions). In supergravity, you have an 11-dimensional metric, an extra "3-form" field that is a generalized version of the electromagnetic field, and then a "gravitino" field which is a matter (fermion) field. So at the quantum level, you have the 11-dimensional graviton (which can be defined in the way I mentioned above), an 11-dimensional photon-like gauge boson, and an 11-dimensional fermion.

If you were trying to get the real world out of 11-dimensional supergravity, you would probably treat 7 of the dimensions as "compact" or "closed", with a radius much less than that of an atomic nucleus. Fundamentally, you still only have the graviton, the 3-form field, and the gravitino. However, the way that e.g. the graviton manifests itself depends on whether it's traveling in one of the extra, compact, closed directions, or whether it's traveling in one of the 3 "large" directions of space. Gravitons traveling in the large directions show up as gravity in 3 dimensions, while gravitons circulating in the compact directions can show up as other forces. This was part of the agenda of pre-string "Kaluza-Klein" unification efforts - the other forces would be explained as resulting from higher-dimensional gravity. (That idea goes back to about 1921.)

In M-theory, along with the fields I've described, you have "M-branes" (of 2 and 5 dimensions) which interact with the graviton, the 3-form, and the gravitino fields. A string is really an M2-brane with one of its internal directions wrapped around the compact dimensions. Anyway, these complexities aside, if we go right back to where we started, the key point is that quantum fields have particles, whose presence indicates a deviation from the ground state of the field, and the graviton is the particle of the metric field, indicating a deviation from flat space.

[tom.stoer]

some people think the the artificial split into a static background metric + quantized fluctuations on top of it cause severe problems for the whole program, and that no such background must be introduced

[Harv]

some people think the the artificial split into a static background metric + quantized fluctuations on top of it cause severe problems for the whole program, and that no such background must be introduced


As discovered long ago, this naïve perturbative approach to obtaining a quantum theory of gravity that reduces to General Relativity in the low-energy limit by simply quantizing the linearized gravitational field doesn`t work because General Relativity cannot be fully understood as just a theory of a self-interacting massless spin-2 field. There is only one known consistent perturbative approach to quantum gravity that does have the proper low-energy limit, and that`s string theory. In fact, at this point there is no nonperturbative approach (e.g., LQG etc) to quantum gravity that is known to achieve this.

[tom.stoer]

There is only one known consistent perturbative approach to quantum gravity that does have the proper low-energy limit, and that`s string theory.
The problem is that even with string theory you do not get fully dynamical quantized spacetime b/c spacetime (the classical background) is frozen in this approach. So even if perturbative string theory is consistent, it misses an essential feature of GR. Other non-perturbative and background independent approaches like LQG or AS seem to do a better job regarding fully dynamical spacetime and background independence, even if they fall short w.r.t. to the overall picture (but I know that I will never reach consensus here, neither with the loop nor with the string community)

[stglyde]

Even without going to planck scale. I think the search for physics of wave functions of the metric is a separate thing, isn't it? Or how to quantize the metric.. this is not related to Planck scale, correct?

[atyy]

Even without going to planck scale. I think the search for physics of wave functions of the metric is a separate thing, isn't it? Or how to quantize the metric.. this is not related to Planck scale, correct?

No. It is only near the Planck scale and above that it is uncertain what a consistent theory of quantum gravity is. See http://arxiv.org/abs/gr-qc/0108040 p17, the discussion starting from "Note that even though the perturbation theory described here does not provide an ultimate quantum theory of gravity, it can still provide a good effective theory for the low energy behavior of quantum gravity."

[Dickfore]

There is a well developed theory called Quantum Field Theory in curved space-time. It treats the dynamics of "matter fields" on a background metric caused by massive bodies. Then, you can go ahead and calculate the stress-energy tensor due to these fields and use it in the Einstein's field equations.

In this respect, the "gravitational field" is treated classically, i.e. it develops according to Einstein's equations which minimize the action of the gravitational field. However, the sources of the gravitational field, namely, the stress-energy tensor of various particles is treated in a fully quantum fashion.

This partial theory predicts emission of particle-antiparticle pairs from the exterior of an event horizon of a black hole. The emitted spectrum looks just like a blackbody spectrum, with the temperature of the black hole being inversely proportional to its Schwarzschild radius (smaller black holes emit more). This causes evaporation of black holes.

It is interesting to notice that what was a static, or stationary, problem in General Relativity (we were solving for a metric that does not depend explicitly in time. As a necessary condition, the total mass-energy enclosed inside the Schwarzschild radius remains fixed, and the radius remains constant), has become an explicitly time-dependent problem, because as the black hole evaporates and looses energy, its radius shrinks.

To me, this is very similar to the failure of Classical Electrodynamics when applied to the atomic system, or simply by its own predictions. Namely, in the Rutherford model, the electron used to be in a dynamical balance because the attractive Coulomb force caused centripetal acceleration keeping it in a stable orbit around the nucleus. However, when we apply the laws of Classical Electrodynamics to the model, the accelerated electron, being a charged particle, should emit electromagnetic radiation, and spiral down to the nucleus in a very short time (of the order of 10-8 s). Nevertheless, this never happens. It took the genius of Niels Bohr to postulate that there are particular orbits on which the electron does not emit electromagnetic radiation. Thus, he essentially modified Classical Electrodynamics. The criterion by which these orbits were chosen was the quantization of the angular momentum of the electron around the nucleus, which also modified the laws of Classical Mechanics. Of course, it was later shown that the latter corresponds to so called semi-classical quantization conditions of the Quantum Mechanics. It took the development of Quantum Electrodynamics to resolve the mystery of the former prediction. QED also solves the absurdity of the prediction of classical electrodynamics that a charged particle should exponentially accelerate once it was accelerated in some external electric field due to its own radiation reaction force.

Up to now, there has been no conclusive evidence that Hawking radiation exists.

[suprised]

No. It is only near the Planck scale and above that it is uncertain what a consistent theory of quantum gravity is.

Not necessarily. Quantum gravity effects are expected to be relevant at much larger distances than the Planck scale. Relatedly, non-perturbative, non-local/long-distance effects are likely to be relevant at the horizon of black holes, which can be very far away from the singularity at the origin.

See eg. here for a readable exposition: http://arXiv.org/pdf/1105.2036

Citation:

These notes have given sharpened statements that this unitarity crisis is a long-distance issue, and there is no clear path to its resolution in short-distance alterations of the theory.....

While specific frameworks for quantum gravity have been proposed, they do not yet satisfactorily resolve these problems. Loop quantum gravity is still grappling with the problem of approximating flat space and producing an S-matrix. Despite initial promise, string theory has not yet advanced to the stage where it directly addresses the tension between the asymptotic and local approaches, or is able to compute a unitary S-matrix in the relevant strong gravity regime. Because of the long-distance and non-perturbative nature of the problem, it is also not clear how it would be addressed if other problems of quantum gravity were resolved, for example if supergravity indeed yields perturbatively finite amplitudes.

[friend]

No. It is only near the Planck scale and above that it is uncertain what a consistent theory of quantum gravity is. See http://arxiv.org/abs/gr-qc/0108040 p17, the discussion starting from "Note that even though the perturbation theory described here does not provide an ultimate quantum theory of gravity, it can still provide a good effective theory for the low energy behavior of quantum gravity."

Forgive me if this seems ignorant, but why should it be necessay to quantize the gravitational field? I mean aren't we really only interested in how the two fit together, where one comes from in terms of the other? I don't see that as necessarily requiring quantizing the gravitational field. Perhaps gravity is a emergent property. Or perhaps the metric is continuous, though curved, all the way down to the particle level. What phenomina or logic necessitates quantizing the gravitational field?

[tom.stoer]

Forgive me if this seems ignorant, but why should it be necessay to quantize the gravitational field?
First reason: the Einstein equations read "metric-dependent terms = matter-dependent term"; b/c the r.h.s. is quantized, the l.h.s. should be quantized, too.

[friend]

First reason: the Einstein equations read "metric-dependent terms = matter-dependent term"; b/c the r.h.s. is quantized, the l.h.s. should be quantized, too.

The quantization procesure of the matter-dependent term relies on a specific, continuous space-time background metric which is not quantized. This argues that there is no quantized gravity.

[tom.stoer]

The quantization procesure of the matter-dependent term relies on a specific, continuous space-time background metric which is not quantized. This argues that there is no quantized gravity.
No, it means that quantization is incomplete. There MUST be quantized gravity, otherwise the equation is ill-definied.

[martinbn]

No, it means that quantization is incomplete. There MUST be quantized gravity, otherwise the equation is ill-definied.

So, if the equation is modified there may be no need for quantization of gravity.

[tom.stoer]

So, if the equation is modified there may be no need for quantization of gravity.
Einstein equations couple gravity to matter - and we know that matter is described by QFT. So how do you want to change the equation and couple gravity to non-quantized matter?

[stglyde]

No. It is only near the Planck scale and above that it is uncertain what a consistent theory of quantum gravity is. See http://arxiv.org/abs/gr-qc/0108040 p17, the discussion starting from "Note that even though the perturbation theory described here does not provide an ultimate quantum theory of gravity, it can still provide a good effective theory for the low energy behavior of quantum gravity."

I meant.. for low energy limit far from the planck scale.. should the metric be quantized.. or should it only be quantized near the planck scale, and why?

[tom.stoer]

As I said, we expect the geometry to be quantized for several reasons - mainly consistency reasons. Quantum effects would then be small far away from the Planck scale, i.e. quantum gravity would be the UV completion of an effective QFT on smooth classical spacetime (however there are proposals for so-called fuzzball blackholes in string theory which indicate deviations from classical metric even far away from the Planck sale)

[suprised]

Actually to be fair, there are some proposals out that challenge the conventional wisdom. One is classicalization and self-completeness. This posits that if one tries to probe the Planck scale, eg by an energetic scattering process, then one creates black holes before one ever enters into the quantum gravity regime. These are classical objects, so in this sense one never would be able to probe quantum gravity near the Planck scale: the theory protects itself. Pumping in more energy just makes the black holes larger and even more classical.

This is not undisputed, however, but some version of this may be true, perhaps only in particular kinematical regimes; see the ref. in my previous post. The key point is unitarity, not renormalizeability.

Nevertheless, for consistency, the whole theory needs to be quantum mechanical. This is independent of whether one can probe the Planck scale by scattering experiments or not.

[martinbn]

Einstein equations couple gravity to matter - and we know that matter is described by QFT. So how do you want to change the equation and couple gravity to non-quantized matter?

I don't want that, just saying that may be the equation can be changed so that gravity need not be quantized. You, yourself, say that the equation has to be changed (the whole theory), but you use the equation (that has to be changed) as the reason why gravity should be quantized. I am only saying that that is not very convincing.

[Dickfore]

I think you guys are running in circles with your discussion. Would you please define what you mean when you say an equation is "quantized", and similar terms. What is "classical" then?

[martinbn]

Would you please define what you mean when you say an equation is "quantized", and similar terms.

Where was that said?

[Dickfore]

Where was that said?

First reason: the Einstein equations read "metric-dependent terms = matter-dependent term"; b/c the r.h.s. is quantized, the l.h.s. should be quantized, too.

Also, if you do a search of this thread for "quantized", you will see it is applied very liberally for various concepts. Could you define what you mean by "quantized" before you start discussing?

[martinbn]

First reason: the Einstein equations read "metric-dependent terms = matter-dependent term"; b/c the r.h.s. is quantized, the l.h.s. should be quantized, too.

Ah, but it does NOT say anything about an equation being quantized, right?

Also, if you do a search of this thread for "quantized", you will see it is applied very liberally for various concepts. Could you define what you mean by "quantized" before you start discussing?

I could.

[Dickfore]

Ah, but it does NOT say anything about an equation being quantized, right?

What does r.h.s or l.h.s. stand for?!

I could.
Please do.

[suprised]

There seems a lot of confusion. So let's do a little thought experiment. Just scatter two electrons - one from the left, the other coming from the right, in some rest frame.

Quantum mechanics is used to describe the scattering matrix. This is like a black box which tells you what comes out from this scattering process, given the incoming particles. And you want to have unitary scattering, so that probabilities do not exceed one. So far so good, I guess nobody objects that QM is the right concept here.

To make things easier, the electrons have an offset, or impact parameter, which is large, say 1km. Ordinarily one wouldnt expect that something would be peculiar or problematic.

But I didnt tell you that the kinetic energy of the electrons equals to the mass of a large star. A star with such a mass would form a black hole. So what's going to happen is that when the electrons are still, say 2km apart, a large black hole forms. But you dont really want to know the details now; all that matters is the "black box", or S-Matrix, and the question is, without caring about the details of what happens in the black box, what are the final states? Is the scattering unitary? This is obviously a quantum mechanical question. And if the scattering is unitary, this implies that the black hole must be able to decay. So Hawking radiation must necessarily occur, if quantum mechanics is supposed to be valid.

Note that this involves quantum mechanics and gravity, and ultra-plankian energies, but still these questions are insensitive to the Planck scale: small distances are not relevant here. So we talk about highly non-perturbative non-local effects.

Related problems occur when considering loops of virtual black holes; do these induce non-unitary scattering for low-energy particle physics? Better not!

Obviously one needs to describe gravity and quantum mechanics in one single coherent framework, in order to address this kind of questions. AFAIK a suitable framework to describe this quantitatively is still lacking. Although I know of some attempts using AdS/CFT.

[friend]

As I said, we expect the geometry to be quantized for several reasons - mainly consistency reasons. Quantum effects would then be small far away from the Planck scale, ...

Planck scale this and Planck scale that... How can we be sure that any of the constants of nature and thus the planck scale should remain the same as we approach ever more tightly curled up spacetimes? I mean, if we cannot see inside a black hole or cannot see the big bang, then it seems we are just guessing.

[Fra]

There seems a lot of confusion. So let's do a little thought experiment. Just scatter two electrons - one from the left, the other coming from the right, in some rest frame.

Quantum mechanics is used to describe the scattering matrix. This is like a black box which tells you what comes out from this scattering process, given the incoming particles. And you want to have unitary scattering, so that probabilities do not exceed one. So far so good, I guess nobody objects that QM is the right concept here.

I see a big problem with this closed black box view - it is valid ONLY when the scattering picture is which is when you have an inert observer that can make observations as well as preparations from a distance where the coupling to the black box is is weak/controlled in the sense that the observer itself (which is a generalized "background") does not severly deform during the interaction.

The other problem is that it also only makes sense when ensembles can be realized.

In cosmological pictures, where the observer is strongly coupled, the observers entire ENVIRONMENT (ie remainder of the universe) is the effective "black box", and here most of the premises in the scattering picture fails. Also an inside observer can hardly encode arbitrary amounts of inforamtion - someting that is usually not cared about in a good way in the scattering pictures as I see it.

It's no news that my own view is that QM formalism as it stands is unlikely to be sufficient here. That's not to say the scattering matrix is interesting, it is. But I think it's a good abstraction of observed reality only in limiting/special case.

/Fredrik

[suprised]

In cosmological pictures, where the observer is strongly coupled, the observers entire ENVIRONMENT (ie remainder of the universe) is the effective "black box", and here most of the premises in the scattering picture fails. Also an inside observer can hardly encode arbitrary amounts of inforamtion - someting that is usually not cared about in a good way in the scattering pictures as I see it.

It's no news that my own view is that QM formalism as it stands is unlikely to be sufficient here. That's not to say the scattering matrix is interesting, it is. But I think it's a good abstraction of observed reality only in limiting/special case.


Well I am not talking about cosmological pictures, but just a transplanckian scattering experiment, if you wish with asymptotic oberservers. So what is the S-Matrix for this scattering? It should have a concrete answer, and better be unitary.

If you dispute the valitidy of QM and the S-Matrix - well QM has been proven to be extremely robust against deformations and so far no one, AFIAK, was able to replace it by something else. It is very common (because cheap) to say "according to my opinion QM needs somehow be modified", but very difficult to actually do it ...

[suprised]

I think you guys are running in circles with your discussion. Would you please define what you mean when you say an equation is "quantized", and similar terms. What is "classical" then?

Why circles? This just means it's an equation involving operators. And this makes sense only if the complete equation, and not just part of it, becomes operator valued.

[Dickfore]

Why circles? This just means it's an equation involving operators. And this makes sense only if the complete equation, and not just part of it, becomes operator valued.

:uhh:

So, what is the meaning of the operators g_{\mu \nu}, and R_{\mu \nu}?

[friend]

:uhh:

So, what is the meaning of the operators g_{\mu \nu}, and R_{\mu \nu}?

It turns them into probability distributions instead of absolute values.

[Dickfore]

It turns them into probability distributions instead of absolute values.

Please elaborate. Are you saying the metric tensor becomes a probability distribution? If yes, whose random variable it is a distribution of? Or, is the metric tensor a (multivariate) random variable. In this case, what determines its distribution?

[tom.stoer]

In canonically quantized GR g and R are field operators with a huge gauge symmetry and therefore w/o a direct physical meaning.

[Dickfore]

In canonically quantized GR g and R are field operators with a huge gauge symmetry and therefore w/o a direct physical meaning.

What do you mean by "gauge symmetry" of GR?

Also, if you canonically quantize the gravitational field, what are the canonical commutation relations?

[tom.stoer]

Have a lokk at the ADM formulation of GR

http://arxiv.org/abs/gr-qc/0405109
The Dynamics of General Relativity
R. Arnowitt (Syracuse Univ.), S. Deser (Brandeis Univ.), C. W. Misner (Princeton Univ.)
(Submitted on 19 May 2004)
Abstract: This article--summarizing the authors' then novel formulation of General Relativity--appeared as Chapter 7 of an often cited compendium edited by L. Witten in 1962, which is now long out of print. Intentionally unretouched, this posting is intended to provide contemporary accessibility to the flavor of the original ideas. Some typographical corrections have been made: footnote and page numbering have changed--but not section nor equation numbering etc.

The 'gauge symmetry' is related to the diffeomorphism invariance

[Finbar]

To make things easier, the electrons have an offset, or impact parameter, which is large, say 1km. Ordinarily one wouldnt expect that something would be peculiar or problematic.

But I didnt tell you that the kinetic energy of the electrons equals to the mass of a large star. A star with such a mass would form a black hole. So what's going to happen is that when the electrons are still, say 2km apart, a large black hole forms.

Suprised,

How can you be certain a black hole forms for the two electrons? This is not a kinematical regime that we have any experimental knowledge of and there is not an established theory at transplanckian energies. You cannot simply apply general relativity to two electrons with these energies.

It is a fact that one can only state when a black hole is formed with knowledge of the complete dynamical history of the spacetime. Yes it is true that initially when the two electrons are 2km apart that they should begin to collapse. But since they are transplanckian as they get closer to each other the quantum gravity effects will become important and it is possible that the collapse will cease to continue. So although an apparent horizon will form it is possible that once the electrons reach Planckian distances their coupling to the gravitational field will be vastly altered and a classical spacetime is unlikely to be a valid assumption.


To make rash statements about the formation of black holes one must at least take three quantities into account:

1) The total energy

2) The impact parameter

3) The number of degrees of freedom

The important thing in your example is the number of degrees of freedom is very small, just those of two electrons. Roughly speaking GR is only valid when the number of degrees of freedom is very large. So the normal hoop conjecture rational is good when we assume that there is a large number of degrees of freedom and so we only concern ourselves with 1) and 2). For a star this is fine but in your example it is clearly not.

[tom.stoer]

How can you be certain a black hole forms for the two electrons? This is not a kinematical regime that we have any experimental knowledge of and there is not an established theory at transplanckian energies. You cannot simply apply general relativity to two electrons with these energies.
I think this is what he wants to show: the usual reasoning of GR and even perturbative QG do no longer apply b/c what you mean by
coupling to the gravitational field will be vastly altereddoes not emerge from this ansatz.

What does the asymptotic safety program say about transplanckian scattering?

[suprised]

How can you be certain a black hole forms for the two electrons? This is not a kinematical regime that we have any experimental knowledge of and there is not an established theory at transplanckian energies. You cannot simply apply general relativity to two electrons with these energies.


Actually one cannot be certain and I should perhaps have said: “So, _according to standard expectations_, what's going to happen is that when the electrons are still, say 2km apart, a large black hole forms.”

There was a paper by t' Hooft in the 80's supporting this idea.
Indeed, this is also the viewpoint of the more recent “classicalization” or “UV-self-completeness” approach to gravity by Dvali & Co, see eg:

arXiv:1006.0984v1:

Physics of Trans-Planckian Gravity
Authors: Gia Dvali, Sarah Folkerts, Cristiano Germani
(Submitted on 4 Jun 2010)

But this is by no means undisputed, and AFAIK no one really knows what is going to happen under these circumstances. So the question about the S-Matrix is a very important one.



… Yes it is true that initially when the two electrons are 2km apart that they should begin to collapse. But since they are transplanckian as they get closer to each other the quantum gravity effects will become important and it is possible that the collapse will cease to continue. So although an apparent horizon will form it is possible that once the electrons reach Planckian distances ….

With the large impact parameter they will never reach Planckian distances, that was the whole point. I presented this, in the context of the thread, as an example where quantum gravity effects may become important, despite one is _not_ probing distances close to the Planck scale; so this has little to do with the UV completion of gravity.

There are indications that inside of black holes macroscopic quantum effects occur (horizonless “fuzzball states”), that are extremely non-local. So what could happen in the scattering process, roughly speaking, is that one huge extended fuzzball state is created, which decays afterwards in a perfectly unitary way; and no classical black hole is ever formed.


To make rash statements about the formation of black holes one must at least take three quantities into account:

1) The total energy

2) The impact parameter

3) The number of degrees of freedom

The important thing in your example is the number of degrees of freedom is very small, just those of two electrons. Roughly speaking GR is only valid when the number of degrees of freedom is very large.

Indeed so, classical GR may not be relevant at all here. This what I would tend to believe. But again, the classicalization approach tries to argue otherwise. Note (tom) that this approach vehemently denies asymptotic safety.

[tom.stoer]

Indeed so, classical GR may not be relevant at all here. This what I would tend to believe. But again, the classicalization approach tries to argue otherwise. Note (tom) that this approach vehemently denies asymptotic safety.
Do you have a good reference about classicalization?

Isn't AS somthing like "classicalization" as well? It's an effective action (but as such a 'classical' expression) taking into account quantum effects via renormalized couplings - but no new structures or interactions (at least if the usual truncation remains valid).

[suprised]

Do you have a good reference about classicalization?

Isn't AS somthing like "classicalization" as well? It's an effective action (but as such a 'classical' expression) taking into account quantum effects via renormalized couplings - but no new structures or interactions (at least if the usual truncation remains valid).

I guess the paper cited above and refs. therein, eg. ref.3, is a good start.

No, these authors claim that the regime where AS would take place can never be probed; nothing can ever become weaker coupled than standard gravity.

[atyy]

Suprised, from the renormalization point of view, unless there is asymptotic safety, new degrees of freedom are expected at high enough energies (and small impact parameter).

But from the unitarity point of view, from the Giddings paper you linked, there seems to be a problem at high energies and large impact parameter, so he says unitarity is really the problem. But shouldn't the two problems somehow be linked, ie. if the new degrees of freedom are properly incorporated, the problem should go away?

[suprised]

Suprised, from the renormalization point of view, unless there is asymptotic safety, new degrees of freedom are expected at high enough energies (and small impact parameter).

But from the unitarity point of view, from the Giddings paper you linked, there seems to be a problem at high energies and large impact parameter, so he says unitarity is really the problem. But shouldn't the two problems somehow be linked, ie. if the new degrees of freedom are properly incorporated, the problem should go away?

Yes this is likely related and the expectation is of course that the problem goes away in a proper formulation of quantum gravity, but how does this work precisely? There were some attempts from AdS/CFT, but I don't quite recall now as to how far this could be pushed.

On the other hand, the classicalization people claim that new degrees of freedom are not required, since the ultra-high energy regime maps back to classical physics.

[Finbar]

With the large impact parameter they will never reach Planckian distances, that was the whole point. I presented this, in the context of the thread, as an example where quantum gravity effects may become important, despite one is _not_ probing distances close to the Planck scale; so this has little to do with the UV completion of gravity.


If we assume that a black hole does form then they do reach Planckian distances when they collapse towards the singularity. The two electrons are attracted to each other my gravity so they will not remain 2km apart. So we can only say that the UV effects can be ignored if they are hidden behind the horizon. But the existence of the horizon really depends on the whole dynamical history of the electrons. So we can't really assume that a black hole does form. So I would say that the idea that we can ignore the UV is actually circular logic.

[friend]

I just had a thought, perhaps this is the best place for it.

Considering the nature of spacetime and QFT, as I understand it, virtual particles pop into existence, travel about, and then come back together such that the uncertainty principle is not violated. But how much space do the virtual particles travel through before coming back together? And how can you define space without events in the form of particle trajectories that establish the concept of relative distances? It may be that we cannot define one without the other. And the ultimate equations will have to account for both in a single equation.

[Dickfore]

Have a lokk at the ADM formulation of GR

http://arxiv.org/abs/gr-qc/0405109
The Dynamics of General Relativity
R. Arnowitt (Syracuse Univ.), S. Deser (Brandeis Univ.), C. W. Misner (Princeton Univ.)
(Submitted on 19 May 2004)
Abstract: This article--summarizing the authors' then novel formulation of General Relativity--appeared as Chapter 7 of an often cited compendium edited by L. Witten in 1962, which is now long out of print. Intentionally unretouched, this posting is intended to provide contemporary accessibility to the flavor of the original ideas. Some typographical corrections have been made: footnote and page numbering have changed--but not section nor equation numbering etc.

The 'gauge symmetry' is related to the diffeomorphism invariance

What specifically should I look for? I don't feel like going through a whole chapter of a textbook.

[tom.stoer]

The canonical variables and the constraints are defined in section 3-2 and chapter 4

[friend]

Considering the nature of spacetime and QFT, as I understand it, virtual particles pop into existence, travel about, and then come back together such that the uncertainty principle is not violated. But how much space do the virtual particles travel through before coming back together? And how can you define space without events in the form of particle trajectories that establish the concept of relative distances? It may be that we cannot define one without the other. And the ultimate equations will have to account for both in a single equation.

So I'm having trouble with how some quantum gravity programmes make an effort to quantize gravity without matter or other particles of any kind. I guess they expect to couple matter into the equations at a later time. But gravity is the geometry of spacetime, and it seems the only thing that established distance in reality is the relative distance between particles. So what relavance is there to quantizing geometry without respect to particles. Even virtual particles would at least give us a a source of particles between which there is distance, right? So it seems we have to quantize gravity with respect to QFT or we're just quantizing geometry as an exercise.

[tom.stoer]

So I'm having trouble with how some quantum gravity programmes make an effort to quantize gravity without matter ... gravity is the geometry of spacetime ... the only thing that established distance in reality is the relative distance between particles
Good point.

But of course we know that there are vacuum solutions in GR with non-trivial dynamics (dS spacetime, black holes, brill waves, ...), so it's not totally unreasonable.

[friend]

Good point.

But of course we know that there are vacuum solutions in GR with non-trivial dynamics (dS spacetime, black holes, brill waves, ...), so it's not totally unreasonable.

That's why I'm thinking that this is where virtual particles come in. At least there are virtual particles in the vacuum to establish relative distances with respect to them. But if geometric quantities are justifiably quantized irrespective of particle quantization, then there is nothing to specify when to quantize and when not to quantize any geometric quantities simply because it is there. If you take away the context of QFT of particles away from the quantization of geometry, then it seems arbitrary to quantize geometry. So I think we need to formulate the problem of quantum gravity by finding geometric quantities as dynamic variables in a more diffeomorphic generalization of the usual QFT.

[tom.stoer]

I don't think that virtual particles are of any relevance as they are purely perturbative artefacts; OK, perhaps matter will play a key role, but not via virtual particles

[Fra]

Well I am not talking about cosmological pictures, but just a transplanckian scattering experiment, if you wish with asymptotic oberservers. So what is the S-Matrix for this scattering? It should have a concrete answer, and better be unitary.

If you dispute the valitidy of QM and the S-Matrix - well QM has been proven to be extremely robust against deformations and so far no one, AFIAK, was able to replace it by something else. It is very common (because cheap) to say "according to my opinion QM needs somehow be modified", but very difficult to actually do it ...

Sorry for the slow response, haven't had much time lately :( a short comment.

First of all, I agree it's cheap to say you need something better, without actually showing what this better thing is. But it's still of importance to be able to distinguish problem if there is one. To cure it is hard, but it gets even harder if you don't see it first.

My objection to your transplanckian scattering picture is this: I agree with you that given a FIXED given observer, the scattering must be unitary due to consistency. The problem is that I think it's still ambigous, becaseu there is not unique "asymptotic observer", there is rather a sort of landscape of them. (I'm not talking just about string landscape, i'm talking generally).

And I do not think there exists a conventional "renormalization" picture where you can define observer invariants here.

Cheap as it may be, I think one needs to consider the backreaction on the observing context, and then referring to the "asymptotic limiting observer" sort of misses the point.

I sure don't have anything better at the moment, but I see problems with current approach, that for myself I'm not letting pass.

/Fredrik

[ohwilleke]

@Dickfore

1. lhs means left hand side, rhs means right hand side. The reference in this case is to the conventional formulation of general relativity which on the left hand side has a tensor that represents the geometrical manifestation of gravity, and on the right hand side has a tensor called the stress energy tensor.

2. A tensor is another name for a matrix, usually in the context of a tensor that is used in the physical sciences to represent quantities analogous to vectors but that contain more data points. In the matrix that is the stress-energy tensor there is an element for each possible source of matter or energy in a given matter-energy field (e.g. rest mass, kinetic energy in three directions, pressure in three directions, electromagnetic flux in three directions, etc.) in each direction. The magnitude of each element of mass and energy in each direction contributes individually to the distortion in space-time geometry that we call gravity, so in general relativity, rather than merely the total amount of mass in Newtonian gravity mattering we care about its character and dynamics (and of course, since it is geometric, even massless stuff like photons are affected by it until Newtonian gravity) - hence we get gravitomagnetic effects (those arising from the motion of matter), etc.

3. A good quick introduction to the equations of general relativity and what each part stands for can be found in the relevant wikipedia articles, but it is all a little opaque if you don't have a good grasp of tensor mathematics (usually taught at the upper division undergraduate level to math and physics and engineering majors) and both the notion that general relativity is based on a continuous matter-energy field rather than a point sources that emit forces like Maxwell's equations, Newtonian gravity and to Standard Model interactions do and the notion that it is geometric while observing special relativity makes it all rather mind numbing and hard to process.

4. When we talk about quantitizing general relativity, several concepts are implicated: (a) general relativity is formulated with regard to continous matter-energy fields while a quantum gravity theory would operate at the level of individual particles of matter, energy; (b) general relativity envisions space-time as continuous, while quantum gravity could have a discrete space-time; (c) most theories of quantum gravity would suggest a mechanism such as force carrying by a graviton which is a spin-2 boson, rather than a mere equation that says that two continuous quantities are related in a particular way which would give gravity particle-like as well as wave-like properties; amd (d) a quantum gravity theory would probably have a mechanism that propogates these bosons via a probabilistic rather than deterministic set of rules.

5. A quantum gravity theory could have little elaboration of general relativity at all, but provide a way to incorporate gravitational effects into quantum field theory, for example, in strong gravitational fields at the boundaries of black holes and in the Big Bang. In weak gravitational fields, the corrections would be on orders of magnitude so trivial relative to Standard Model forces that we don't care. In hyperprecise applications for fairly strong fields, we do care. Point-like particles are inherently inconsistent with general relativity, in ways that don't matter in Standard Model equations, but create general relativity singularties. The QFT on curved space-time approach you mention is basically an ad hoc, non-rigorous way of pushing classical GR and Standard Model physics as far as they will go and estimating on that basis with a fair bit of artfulness what their mutual natural extensions would suggest.

6. There are two practical reasons to do this, aside from the joy of having a fully rigorous and consistent set of equations of everything. One is that there are some extreme situations where no amount of artful extensions of each is enough to resolve how GR and the Standard Model forces interact because there are multiple, mutually inconsistent ways of going about doing it and we don't have enough guidance to say. The other one is that lots of people think that while a quantum gravity theory that is true must reduce to GR in ordinary consideration that there may be new physics predicted in very weak gravitational fields in deep space (the IR limit) and in very strong fields like those of the Big Bang and black holes (the UV limit). Many people also think that by putting them together in a way that explains how matter and energy took the forms that it did after the Big Bang (baryogenesis, leptogenesis, dark matter formation, inflation, cosmic background radiation, etc.) that we might be able to discern UV behavior of not only gravity but all Standard Model forces in a way that would make clear how to unify them into a fundamental theory of everything at high energies that naturally segments into the distinct low energy four forces and x many particles we observe governed by the equations we use in a low energy limit.

[friend]

I don't think that virtual particles are of any relevance as they are purely perturbative artefacts; OK, perhaps matter will play a key role, but not via virtual particles

Well let's see, what about the graviton? Even the trajectories of gravitons are at least particles which can be used to establish relative distances, ... as opposed to quantizing geometry for the sake of geometry just to see what happens.

Here's a question: Suppose we have a complete diffeomorphic invariant generalization of QFT. Could that formulation be separated into a purely geometric component coupled to a particle component? Then it would make sense to try to quantize the geometric component first and add matter latter. Is this decoupling of geometry from particles the thing that we are assuming? Do we have a diffeomorphic invariant generalization of QFT?

[suprised]

Is this decoupling of geometry from particles the thing that we are assuming?

No, at least not in string theory. There the particles arise from geometry, and are unified with gravity in one coherent framework in higher dimensions. Quantum consistency actually requires extra matter fields, so if string theory is any right, pure gravity cannot be consistently quantized.

[tom.stoer]

Is this decoupling of geometry from particles the thing that we are assuming? No, at least not in string theory. There the particles arise from geometry, and are unified with gravity in one coherent framework in higher dimensions. Quantum consistency actually requires extra matter fields, so if string theory is any right, pure gravity cannot be consistently quantized.
Yes, at least in LQG, AS, ... ;-) these guys assume that a stepwise approach is fine and that unification of matter and geometry is not required for a consistent quantization of gravity. So 'stepwise' means that one can indeed quantize gravity but postpone unification.

LQG as of today is by no means complete, i.e. the quantization itself is still not fully understood (especially not in the celebrated spin foam models which I would describe as preliminary). But the technical problems in LQG are - as far as I can see - not related to the missing matter degrees of freedom. Another approach without relation between quantum gravity and unification is the asymptotic safety program which again points towards a consistent theory of quantum gravity to which matter degrees of fredom can be added later. So it seems that there are promising proposals for consistent theories of quantum gravity w/o unification. Whether they are physically right is a different question!

Nevertheless my feeling is that matter d.o.f. have a geometric origin and that a consistent and correct theory of quantum gravity along the lines of LQG or AS may not be the final word. Perhaps there is some potential in q-deformed / framed spin networks and 'braided matter", but this is more an idea than a theory. Another very interesting approach (with a very small community) seems to be the "exotic smoothness program".

[suprised]

But the technical problems in LQG are - as far as I can see - not related to the missing matter degrees of freedom....

Indeed so, they have problems already at a much more basic level. The issues I mean have to do eg with consistent graviton scattering at higher order (where, as seen from string theory, matter is needed to render the amplitudes finite and unitary). LQG is still far away to even see and address this question.

[tom.stoer]

The question in LQG is if you need something like perturbative gravitons, if it's the right question to be asked. If it changes the fundamental structure of spacetime its fundamental d.o.f. will not be gravitons at all.

(In lattice gauge theory there is neither the possibility nor the need to ask for perturbative quark-gluon scattering; in full QCD this is a valid question, but only b/c there is the regime of asymptotic freedom; so you need something but lattice gauge theory, too; it may very well be that there is no regime in gravity where gravitons and graviton scattering are required)

So the problem is NOT that LQG is not able to provide tools for calculating graviton scattering, the problem is that LQG is not able to tell you if this is a reasonable question. It all boils down to the incomplete quantization regarding dynamics, constraints, hamiltonian, PI measure, anomalies etc.

As soon as these issues are solved you are able address all these physical questions.

[friend]

No, at least not in string theory. There the particles arise from geometry, and are unified with gravity in one coherent framework in higher dimensions. Quantum consistency actually requires extra matter fields, so if string theory is any right, pure gravity cannot be consistently quantized.

Yes, at least in LQG, AS, ... ;-) these guys assume that a stepwise approach is fine and that unification of matter and geometry is not required for a consistent quantization of gravity. So 'stepwise' means that one can indeed quantize gravity but postpone unification.

What does it tell us about this decoupling between geometry and matter to know that kinetic energy associated with travel through spacetime is converted to particles and visa versa? What does it tell us that event horizons associated with accerated frames radiate particles? Here it seems spacetime features equate to particles and so cannot be decoupled.

[Haelfix]

"it may very well be that there is no regime in gravity where gravitons and graviton scattering are required"

I don't think so.

There is always a regime where the latter is important, and is basically a consequence of various non-decoupling theorems and renormalization group arguments. One can show that b/c the gravitational coupling constant is small (and in fact it always stays small relative to the other forces at any scale), there will always be an effective semiclassical description that must exist at some scale.

As a consequence you can't ignore the divergences that occur there, which is why all quantum gravity theories implicitly require the existence of either a nontrivial fixed point set or alternatively a UV completion.

The former of course hits a big problem (amongst many), which is that even if such a thing existed for pure gravity, b/c gravity couples to everything one must understand the dynamics of all the other forces to perfect precision and hope that they do not alter the UV critical surface. Which is of course rather silly, since the other forces are manifestly more important to gravitational dynamics than gravity is to itself at those energies.

So I mean its perfectly valid to look for novel top down approaches, but you do eventually have to answer the above foundational questions.

[Finbar]

"it may very well be that there is no regime in gravity where gravitons and graviton scattering are required"

I don't think so.

There is always a regime where the latter is important, and is basically a consequence of various non-decoupling theorems and renormalization group arguments. One can show that b/c the gravitational coupling constant is small (and in fact it always stays small relative to the other forces at any scale), there will always be an effective semiclassical description that must exist at some scale.

As a consequence you can't ignore the divergences that occur there, which is why all quantum gravity theories implicitly require the existence of either a nontrivial fixed point set or alternatively a UV completion.

The former of course hits a big problem (amongst many), which is that even if such a thing existed for pure gravity, b/c gravity couples to everything one must understand the dynamics of all the other forces to perfect precision and hope that they do not alter the UV critical surface. Which is of course rather silly, since the other forces are manifestly more important to gravitational dynamics than gravity is to itself at those energies.

So I mean its perfectly valid to look for novel top down approaches, but you do eventually have to answer the above foundational questions.

I do agree that there is a regime where gravitons are valid but...

The gravitational coupling increases with energy where as yang-mills couplings decrease with energy. So at high energies we would expect that gravity becomes stronger than the other forces. Of coarse we could only know for sure by preforming the calculation. But your claim that "one can show" that the other forces remain stronger than gravity at all scales is certainly not the case.

You're right that one must couple matter gravity to check that the UV surface remains finite dimensional. In the end if someone can do the calculation and show that there is some asymptotically safe theory that reproduces GR coupled to the SM at low energies then that is that.

[friend]

I just had a thought, perhaps this is the best place for it.

Considering the nature of spacetime and QFT, as I understand it, virtual particles pop into existence, travel about, and then come back together such that the uncertainty principle is not violated. But how much space do the virtual particles travel through before coming back together? And how can you define space without events in the form of particle trajectories that establish the concept of relative distances? It may be that we cannot define one without the other. And the ultimate equations will have to account for both in a single equation.

Other questions would be what mechanism causes the virtual particles to come back together. They were created, go out in opposite directions, and then somehow curl around and come back together. Is this because spacetime is curved at that level to cause the particles to come back together? That suggests that the uncertainty principle is somehow connected to the curvature of spacetime. Any thoughts?

[tom.stoer]

Other questions would be what mechanism causes the virtual particles to come back together. They were created, go out in opposite directions, and then somehow curl around and come back together. Is this because spacetime is curved at that level to cause the particles to come back together? That suggests that the uncertainty principle is somehow connected to the curvature of spacetime. Any thoughts?
Yes, some thoughts:

This is over-interpreting virtual particles; they do not have position, direction, they do not got here or there, they don't come back and neither do they follow spacetime curvature.

It's is hard to describe or explain what particles are in QFT; it's even harder to describe virtual particles. They are just mathematical entities - Forget about them!

[friend]

Yes, some thoughts:

This is over-interpreting virtual particles; they do not have position, direction, they do not got here or there, they don't come back and neither do they follow spacetime curvature.

It's is hard to describe or explain what particles are in QFT; it's even harder to describe virtual particles. They are just mathematical entities - Forget about them!

What about the Casimir effect. That seems to prove the zero point energy made up of virtual particles, right? What about black hole radiation, where the negative energy virtual particles fall into the black hole near the horizon, but the positive energy virtual particles get stripped away from their partners and float off from the horizon as radiation. That seems to prove the existence of virtual particles too, right? And like the Casimir effect, aren't these virtual particles of the zero point energy the very thing causing the accelerated expansion pressure of the cosmological constant?

[Haelfix]

"So at high energies we would expect that gravity becomes stronger than the other forces."

Gravity does increase in strength and become strongly coupled at the planck scale. And exactly there, all other couplings are of order unity in the effective lagrangian. So there is no regime where gravity ever becomes stronger (indeed the first major divergences occur in the matter couplings), and so for N particle species one quickly see's that they end up dominating the dynamics. In other words, there is no regime where you can ever safely integrate out the other forces. I don't have time to track down references, but this is essentially a non-decoupling theorem.

You can make this relatively tight for beyond the standard model physics by analyzing bounds on black hole states and so forth and it goes by the name of the 'Weak gravity conjecture'. Arkani Hamed and collaboraters have done a lot of work on this.

[tom.stoer]

What about the Casimir effect. That seems to prove the zero point energy made up of virtual particles, right? What about black hole radiation, where the negative energy virtual particles fall into the black hole near the horizon, but the positive energy virtual particles get stripped away from their partners and float off from the horizon as radiation. That seems to prove the existence of virtual particles too, right? And like the Casimir effect, aren't these virtual particles of the zero point energy the very thing causing the accelerated expansion pressure of the cosmological constant?
Sometimes interpreting virtual particles is nice, sometimes it's nonsense.

Even Hawking radiation explained in terms of virtual particles is over-interpretation; it's funny, Hawking provides such an explanation, but in his original calculation there are no virtual particles at all ;-) For the Casimir effect there are two different calcutions, one using 'vacuum fluctuations' and another one using 1st order 'virtual particles'.

[Finbar]

"So at high energies we would expect that gravity becomes stronger than the other forces."

Gravity does increase in strength and become strongly coupled at the planck scale. And exactly there, all other couplings are of order unity in the effective lagrangian. However there is no regime where gravity ever becomes stronger (indeed the first major divergences occur in the matter couplings). In other words, there is no regime where you can ever safely integrate out the other forces. I don't have time to track down references, but this is essentially a non-decoupling theorem.

You can make this relatively tight for beyond the standard model physics and it goes by the name of the 'Weak gravity conjecture'. Arkani Hamed and collaboraters have done a lot of work on this.

You make some claims but I see nothing to back these up. If we take the ratio of some gauge coupling with the gravity coupling. Are there calculations that show that the gauge coupling diverges with respect to gravity?? You seem to be claiming that when gravity couples to other forces that it spoils asymptotic freedom? Please do provide me with the references to support your claims.

[Haelfix]

You make some claims but I see nothing to back these up. If we take the ratio of some gauge coupling with the gravity coupling. Are there calculations that show that the gauge coupling diverges with respect to gravity?? You seem to be claiming that when gravity couples to other forces that it spoils asymptotic freedom? Please do provide me with the references to support your claims.

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.

[Federation 2005]

We don't actually know that there is a consistent approach to quantizing General Relativity. What we do know is the following:

(1) in 2+1 dimensions, one can consistently formulate a quantum theory of "gravity". The reason for the scare-quotes is that no quantum theory in less than 4 dimensions can lead to the quantization of independent gravitational degrees of freedom -- because there are none. More precisely: the Weyl tensor (which contains the gravitational degrees of freedom) is 0 in less than 4 dimensions. Or, to put it another way: all quantum theories of gravity in 3 or fewer dimensions have c-number Weyl tensors (since the 0 tensor is a c-number).

(2) In 3+1 dimensions, the only known approach that has led to a quantization of Einstein's law of gravity was that devised by Carmelli in the 1980's. The most important feature of the formalism is that it is not cast in Riemannian geometry, but Riemann-Cartan geometry. The distinction is crucial because in it, the metric remains classical, while the connection is quantized as the connection of a gauge field (namely, an SL(2,C) gauge field).

The reason this has not been heralded as the Final Definitive Solution to the Problem is that it only works for purely gravitational fields. That is, if all you're interested in is the exterior solutions in a matter-free vacuum, Carmelli does the job. Unfortunately, Carmelli never found a way to even couple the classical theory with matter, much less the quantum theory.

The most notable feature of the theory is that the Weyl tensor is a c-number.

(3) The approach "Quantum Field Theory in Curved Spacetime" succeeds in formulating quantum theory in a general relativistic context. However, there are two main features that are both regarded as drawbacks (whether regarded rightly so as drawbacks, on the other hand, is itself a question for contention). First, there is no reaction of matter on geometry. Rather, the curved background serves to condition the propagation (the Greens functions) and the wave equation. Second, one needs to make restrictive assumptions that, themselves, cannot be framed in operator form in any theory that has the metric as a quantized dynamic variable -- namely, that the underlying spacetime be globally hyperbolic.

The global hyperbolicity assumption is not expressible in operator form. So in a prospective quantized theory of gravity, one could literally have a superposition of a globally hyperbolic state with one that is not. Unfortunately, since nobody's ever found a consistent way to do quantum theory in a globally non-hyperbolic setting (this is much of what the 1990's papers about time travel and closed time loops was about) then the situation could be likened with the worst form of a Schroedinger Cat: a superposition of a (globally hyperbolic) universe in which quantum theory can be defined, with a (globally non-hyperbolic) universe where it can't be.

The more basic problem is that even metric signature is not something that can be expressed in operator form. So, one could even have a superposition of a state that is a 3+1 spacetime with a state that corresponds to a 4 dimensional timeless space. Given how central the notion of time is to quantum theory, this seems to entail some serious consistency problems.

(4) Loop Quantum Gravity takes place in the setting of Riemann-Cartan geometry. It tries to adopt a "background free" approach, placing the diffeomorphism group at the center. Unfortunately, not all things relevant to physics are acted on by the diffeomorphism group, so that the very notion of background freeness itself can't be consistently defined. More precisely: the kinds of objects acted on by the diffeomorphism group are what mathematicians call "natural objects", and the corresponding operations are called "natural", while the underlying geometries are referred to as "natural bundles". The kind of geometry required for a theory of gravity that satisfies the equivalence principle is a subset of the tangent bundle known as an "orthonormal frame subbundle". I don't believe the orthonormal frame bundles are natural. This is closely tied to the problems raised at the end of (3). In addition, fermions require spinors. These reside on spinor bundles and I don't believe these are natural bundles either.

One way to approach this may be to relax (or redefine) the requirement so as to only require "gauge naturalness". In place of the diffeomorphism group Diff(M) on a spacetime manifold M, this broadens to the gauge group Gau(P) and automorphism group Aut(P) on a principal bundle P that has M as a base space. Gauge natural objects over P need not be reduce to natural objects on M. In addition, I don't believe gauge natural objects on P are natural on P (i.e. that they are not acted on by the diffeomorphism group Diff(P)). So, one has background structure by virtue of the confinement of focus to Aut(P) and Gau(P).

(5) Sardanashvily, Mangiarotti, et. al work off of a long-lasting strand that originated with Heisenberg and Ivanenko. They (rightly) point out that the reduction of the manifold M's tangent bundle TM to the orthonormal frame bundle F_g(M) associated with a metric g is a form of symmetry breaking. All this, of necessity, takes place in the broader setting of Riemann-Cartan geometries. This puts the spotlight on either the metric g or the frame fields as being the corresponding Goldstone-Higgs fields. In general, when you have symmetry breaking, the Goldstone-Higgs fields are essentially classical. Each different configuration corresponds to a different vacuum state and different coherent subspace; and between any two coherent subspaces no quantum superpositions can occur.

They highlight the issue of the fermions, noting that the very process of quantizing it, itself, critically depending on which subbundle F_g(M) of TM you choose; so that two quantizations corresponding to inequivalent g's must lead to different quantum state spaces; i.e. no quantum superpositions of the usual kind between different states can exist if the states disagree on whose motions are free fall/inertial.

Among other things, when symmetry breaking is present, the vacuum is no longer a unique state. So the premise of a unique state |0><0| underlying an equation such as <0|T|0> = kG (which the approach in (3) may use) is false, because the equation can't even be written down.

(6) String theory. I don't know enough to say anything about it.

(7) Arguments against classical/quantum hybrids, such as the famous Feynman argument tend to be premised on Riemannian geometry. When the arguments pass into folklore this tends to muddy the issue and lead to fallacious claims. In fact, the Feynman argument employs fermions, which require a Riemann-Cartan geometry. In a Riemann-Cartan geometry one necessarily make a distinction between a metrical extreme curve and a geodesic curve. The former is a "geodesic" for the Levi-Civita connection, while the latter (the *actual* geodesics) are geodesic for the connection native to the Riemann-Cartan geometry.

The Riemann-Cartan geodesics for two electrons in different spin states CAN diverge from one another, in the way that the Feynman argument visualizes.

Another issue, whose premise on Riemannian geometry is often forgotten, is the question of what equation would govern the determination of geometry from quantized matter. Here, the problem is that (in a Riemannian geometry) if the metric is classical, then so is the Levi-Civita connection. Therefore, the field equation would have a (classical) Einstein tensor G on one side and a (quantized) stress tensor T on the other. So, one tends to use a proxy, like <0| T |0> = kG, where |0><0| is the (assumed unique) vacuum state.

Given the issues previously raised about vacuum degeneracy, the premise of the equation is questionable.

If, on the other hand, the vacuum is degenerate with (say) one state |g><g| for each orthonormal frame bundle F_g(M) then it is quite feasible to write down an equation such as <g|T|g> = k G(g), where the right hand side is the Einstein tensor G for a given metric configuration g. The same question regarding how the matter is to be coupled to gravity still arises, but is partially negated in a Riemann-Cartan geometry since in such a geometry the connection is an independent object. So one could quantize the connection, while keeping the metric classical.

(8) Closely linked to this is the idea of gravity NOT being a fundamental force at all, but effective. This is advanced by Padmanabhan, Verlinde, et al.; and should also be linked to Sardanashvily, Mangiarotti et al. as well as to Jacobson's Gravity-as-Thermodynamics idea, which Verlinde descends from.

This is the approach I think is the right one. It can be deepened if one is able to derive an equation such as the one I posed <g|T|g>/h-bar = 8 pi A G(g) (A = Planck area) as something arising from an anomaly associated with a breakdown of classical symmetry. In particular, this could be something that comes about as a result of diffeomorphism symmetry being spoiled upon quantization.

(8) Classical/quantum hybrids.
All successful approaches I've described entail the same thing: the Weyl tensor is a c-number and the metric is classical -- even if the connection is quantized. This is not consistent in a Riemannian geometry, but is perfectly well feasible in a Riemann-Cartan geomtetry. The main observation is that the equations governing matter (particularly fermions) only see gravity through the connection, and only see it as just another gauge field. So, as long as the connection can be quantized, this part of the consistency problem is resolved.

The main issue with approaches to quantum field theory that adopt microcausality as a postulate is this. Since the axiom is posed at the operator level, as a defining condition on the field algebra itself, this has the effect of building in the light cone structure of WHATEVER geometry emerges from the algebra. But that sets into motion a chain of consequences:
(a) once you have the light cones, you have the conformal geometry
(b) once you have the conformal geometry, you have the Weyl tensor -- a unique tensor for each field algebra; i.e. a c-number tensor.
(c) once you have a c-number Weyl tensor, you have no quantized gravitational modes, since it's the Weyl tensor that defines the gravitational degrees of freedom.

So, eventually the correct approach to the problem will lead to an IMPOSSIBILITY THEOREM for Quantum Gravity (when "Quantum Gravity" is meant in the sense of "quantized metric" or "quantized Weyl tensor"), and the establishment of a quantum theory in which gravity emerges as an effective force, rather than as a fundamental force.

In such an approach, the item (3) "quantum theory in a curved background" IS all you have and all you need. The back-reaction of matter on geometry is embodied in an effective dynamics, such as the one I posed: <g|T|g> = kG(g).

Last, but not least, bearing on this question are the works of Penrose, Diosi et al., who have been seeking ways to hybridize classico-quantum forms of gravity.

[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 lets 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.