QFT with respect to general relativity

[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 wouldn't 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 altered

does 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 continuous 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.