# QFT with respect to general relativity

1. Dec 21, 2011

### 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

2. Dec 21, 2011

### Staff: Mentor

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.

3. Dec 22, 2011

### 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

4. Dec 22, 2011

### 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.

5. Dec 23, 2011

### 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

6. Dec 23, 2011

### Harv

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 doesnt 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 thats 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.

7. Dec 24, 2011

### tom.stoer

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)

8. Dec 25, 2011

### 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?

9. Dec 25, 2011

### atyy

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."

10. Dec 25, 2011

### 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.

11. Dec 25, 2011

### suprised

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.

12. Dec 25, 2011

### friend

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?

13. Dec 25, 2011

### tom.stoer

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.

14. Dec 25, 2011

### friend

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.

15. Dec 25, 2011

### tom.stoer

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

16. Dec 25, 2011

### martinbn

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

17. Dec 25, 2011

### tom.stoer

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?

18. Dec 25, 2011

### stglyde

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?

19. Dec 25, 2011

### 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)

20. Dec 26, 2011

### 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.