Why is GR and QM not compatible?

[dipole]

You always hear that GR and QM aren't compatible, but I've never seen an explanation as to why. If I had to guess I would say it's more than quantum mechanics isn't compatible with GR than the other way around, but I really don't know where the conflicts are.

 

[Simon Bridge]

Put simply:

QM treats the time axis as something that events happen against while GR treats time as another dimension of space. So which is it?

Philosophically GR produces a deterministic Universe where everything is set in space-time while QM produces a statistical Universe where everything is indeterminate.

There is very strong evidence for both, so reconciling these pictures is an important task. There are several approaches tried so far - which give us the Standard Model. Quite a lot has been written about this, not least of which by Einstein and Bohr, so you have a lot of reading to do.

 

[kloptok]

QM treats the time axis as something that events happen against while GR treats time as another dimension of space. So which is it?

But this can be resolved in Quantum Field Theory, which is basically relativistic quantum mechanics, so I wouldn't say that this is an issue for quantizing GR. However, QFT only deals with Special Relativity and not General Relativity.

One problem with quantizing gravity is that GR is not renormalizable, which means that there will be problems with predictions as the calculations will lead to infinities. Put in another way, GR can be formulated as a classical field theory but when attempting to quantize this theory you will run into trouble since GR is not renormalizable. In QED, renormalizability means that it is possible to redefine parameters such as the electron charge in a way as to avoid the infinities that otherwise appear. This is not possible in GR.

Of course, it might be the case that gravity is not possible to formulate as a renormalizable QFT, maybe QFT breaks down at the high energy scale of gravity (quantified by the Planck mass at ~10¹⁹ GeV). Weinberg has made a suggestion that gravity obeys a generalized form of renormalizability called asymptotic safety. String theory is another suggestion, which as far as I understand (I have no real knowledge about this though) is not really QFT in the original sense.

I am not an expert but it seems like the Wikipedia page on "Quantum Gravity" is a decent reference to learn more. http://en.wikipedia.org/wiki/Quantum_gravity

 

[Simon Bridge]

But this can be resolved in Quantum Field Theory, which is basically relativistic quantum mechanics, so I wouldn't say that this is an issue for quantizing GR. However, QFT only deals with Special Relativity and not General Relativity.

That's right - field theory attempts to reconcile SR with QM - not GR, which is what was asked.

I didn't mention it because (a) renormaliziability is difficult to describe without math and (b) field theory involves reformulating relativity to make it fit with quantum mechanics ... if we are to count a similar reformulation in GR as being GR then the original question becomes meaningless: there is no conflict.

When various books talk about a conflict between GR and QM, they are not normally talking about any of the formulations that attempt to reconcile the conflict. Mind you, some may do ;)

 

[Ken G]

I would like to echo the original question in a different way, which is, why do we feel it necessary to unify gravity and quantum fields? Perhaps the "contradiction" between them is simply an essential tension between two fundamentally different things that we are trying to shove into the same box. I realize that the goal of physics is always to unify, why would we not try to unify if we think it might succeed, but here I'm not convinced unity is a good thing. After all, GR and QFT have very different goals, it seems to me: the goal of GR is to describe the geometry of inertial motion, and the goal of QFT is to describe noninertial motion, playing out against the geometry of inertial motion. Why would we want to unify those things, when their interplay might be the very thing we are trying to understand?

 

[dipole]

Thanks, that answers my question.

 

[Simon Bridge]

QFT and GR are indeed intended to describe different types of thing - however, they also each claim to be general models of nature. Their scope overlaps - and, where they do we would expect them to provide a different, but consistent, understanding of the same thing. Studying the interplay is exactly what unifying the theories is all about.

 

[mathman]

QFT and GR are indeed intended to describe different types of thing - however, they also each claim to be general models of nature. Their scope overlaps - and, where they do we would expect them to provide a different, but consistent, understanding of the same thing. Studying the interplay is exactly what unifying the theories is all about.

Inside black holes where both have to apply, the results are nonsense (mathematically). One or both theories have to be modified.

 

[Ken G]

Inside black holes where both have to apply, the results are nonsense (mathematically). One or both theories have to be modified.

How can that be? If I'm doing quantum mechanical experiments as I fall through an event horizon, I expect QM to completely describe what I see, yet I also expect GR to tell me how long I have before I hit the center, and so forth. I can see that if tidal forces get stronger than the atomic effects I'm studying with quantum mechanics, that's a problem, but that doesn't necessarily happen as I cross an event horizon. What's more, if I'm in a realm of very strong tidal forces, I expect new physics anyway, so I'm not troubled if QM doesn't agree with GR in that domain-- I could just look for modifications of either GR or QM or both, without quantizing gravity or seeking any other type of unification. And, I'm probably going to have a hard time doing that, without experimental results from that regime.

 

[Nabeshin]

the goal of GR is to describe the geometry of inertial motion

That's a really naive way to view it. If this were the case, all of GR would be contained in the geodesic equation,
$$\frac{d^2 x^{\alpha}}{d\tau ^2} + \Gamma ^{\alpha}_{\beta \gamma} \frac{dx^\beta}{d \tau} \frac{ dx^\gamma}{d \tau}=0.$$

But as I'm sure you know GR is much more, it's about the dynamics of spacetime itself:
$$G_{\mu \nu} = 8 \pi T_{\mu \nu}.$$

In comparison, the geodesic equation is rather boring, since it's the Einstein equation which really makes the theory something interesting, and it's here, not with the geodesic equation, that you run into trouble with quantization.

 

[K^2]

The biggest contradiction between the theories is that GR is non-linear, which violates superposition postulate of QM. In other words, as soon as you add GR into your Hamiltonian, QM breaks down.

Everything else is perfectly reconcilable.

 

[Ken G]

That's a really naive way to view it. If this were the case, all of GR would be contained in the geodesic equation,
$$\frac{d^2 x^{\alpha}}{d\tau ^2} + \Gamma ^{\alpha}_{\beta \gamma} \frac{dx^\beta}{d \tau} \frac{ dx^\gamma}{d \tau}=0.$$

But as I'm sure you know GR is much more, it's about the dynamics of spacetime itself:
$$G_{\mu \nu} = 8 \pi T_{\mu \nu}.$$

Good point, even inertial motion has to be understood in the context of noninertial dynamics. The feedback between the two is crucial. So one cannot support a demarcation between the two.

In comparison, the geodesic equation is rather boring, since it's the Einstein equation which really makes the theory something interesting, and it's here, not with the geodesic equation, that you run into trouble with quantization.

Actually, there is still a problem with the geodesic equation, at the Planck scale. But I take your meaning that quantization of gravity runs into trouble with the meaning of "stress-energy" well before we get to the Planck scale, and so the difficulty is fundamentally dynamical.

 

[Dickfore]

I think that it had been shown that a QFT with a Hilbert action (the action of space-time, which upon equating its variation to zero gives the Einstein field equations) is non-renormalizable.

According to Ken Wilson's ideas, non-renormalizable theories should be regarded as effective field theories that give the correct low-energy physics, but, eventually need to be replaced by a more fundamental theory.

 

[laserblue]

See website and writings of Dr. Mendel Sachs. http://mendelsachs.com/the-future-of-physics/

 

[cosmik debris]

As well as the problem of non-renormalizability there is the question of the background spacetime. One theory (GR) is the generator, the other (QFT) works quite comfortably in an a priori curved spacetime.