How exactly are QM & GR incompatible?

[Thrice]

I've done college courses on both & it's not clear yet how they conflict. I'm looking for a more technical account. Have I done enough to understand it, or do I have to do wait till QFT?

 

[pervect]

Probably you need to wait till QFT, where you'll find out that quantum gravity isn't renormalizable. However, you'll also find out why this isn't really such a big deal if the energy is low enough.

 

[JesseM]

A similar question was asked on this thread, this was my post from there:

I don't know if there are any situations where they both make clear predictions that contradict each other, but see here for a discussion of one of the main problems in figuring out how to reconcile them, having to do with the fact that the uncertainty principle would seem to allow for huge uncertainty in energy at sufficiently small scales, but in GR big energies cause significant curvature of spacetime, and my understanding is that physicists only know how to make predictions in quantum field theory if they have a specific known background spacetime.

I guess another more general conflict is that quantum field theories treat the other set of forces using a common set of rules, but if you try to apply these rules to gravity you get infinities which can't be "renormalized" as in the case of the other forces.

 

[hellfire]

The basic problem for the incompatibility between GR and QM is that GR is a classical theory that does not include the superposition principle and the uncertainty principle.

If space-time is considered to be classical, the (instantaneous) collapse of the wavefunction of a particle would lead to superluminal propagation of the deformation of space-time. Otherwise, it is needed to incorporate the superposition principle to the definition of space-time states leaving the classical regime.

On the other hand, you can try save GR using expectation values of the energy-momentum tensor as input for the Einstein equations, but even this seams to be already disproved experimentally, as you can read here.

You can find a detailed explanation of this argumentation in chapter 13 of Wald's book "General Relativity".

Moreover, with the title or term "incompatibility between GR and QM" is often referred to a more wider problem, namely the possible incompatibility between diffeomorphism invariance and quantum principles. This means, whether a quantum theory of gravitation can be generally covariant if QM principles are saved, or whether QM principles can be saved mantaining general covariance.

 

[MeJennifer]

There is also the issue of background independence

 

[masudr]

The fundamental principle of QM is the Principle of Superposition on a Hilbert space.
The fundamental principle of GR is the Principle of General Covariance on a 4-dimensional Lorentizian differentiable manifold.

Really they are about two completely different things. How should one go about combining/relating them?

 

[Zelos]

one problem with gravity bieng quantized is that when you try to do it a infinite amount of different variables need to be calculated which will take a infinite time to determen

 

[JesseM]

The fundamental principle of QM is the Principle of Superposition on a Hilbert space.
The fundamental principle of GR is the Principle of General Covariance on a 4-dimensional Lorentizian differentiable manifold.

Really they are about two completely different things. How should one go about combining/relating them?

This argument seems a bit sketchy, since without knowing more there's no obvious reason they couldn't be combined--after all, Maxwell's laws of electromagnetism don't incorporate the principle of superposition on a Hilbert space, and yet they were successfully combined with QM to make the theory of quantum electrodynamics.

 

[pervect]

There are several ways of combining GR and QM at low energies. One is to treat gravitation as an "effective field theory".

http://arxiv.org/abs/gr-qc/9512024

Of course we do not have a way of combining GR & QM that works at all energies, i.e. we do not have a "theory of everything".

Another rather interesting paper is

http://xxx.lanl.gov/abs/astro-ph/0006423

which develops a non-geometrical approach to GR. This sort of development will have problems if we ever observe a non-trivial topology (a wormhole, for example). Of course, one might turn that argument around and suggest that the exclusion of non-trivial topologies is a "feature" rather than a "problem".

The geometrical approach to gravity is still a very natural approach, because gravity acts on everything. The above papers shows that there are alternatives which can put gravity into a different framework, so that while the goemetrical approach is a very natural one it's not necessarily the only one, or the one that's the most compatible with quantum mechanics.

My view is that the fundamental problem with a TOE is that it will of necessity be rather speculative, in that it will describe very high energy physics that we won't be able to actually observer.

 

[masudr]

This argument seems a bit sketchy, since without knowing more there's no obvious reason they couldn't be combined--after all, Maxwell's laws of electromagnetism don't incorporate the principle of superposition on a Hilbert space, and yet they were successfully combined with QM to make the theory of quantum electrodynamics.

The fundamental principle of EM is U(1) gauge invariance which is pretty easily reconcilable with superposition.

 

[pervect]

As one of the papers I quoted earlier mentions, you can envision a spin-2 field on a background flat space-time as a place to start developing a quantum version of gravity. It turns out with some analyhsis that the background flat-space time is unobservable. and that what can actually be observed by experiment is a dynamical curved space-time.

 

[Thrice]

Well QFT it is. Thanks, guys.

 

[Karlisbad]

A question not mentioned..and another problem.. in GR time is just a coordinate of the curve ..whereas in QM time is "absolute" (all the observers seem to have the same time) since for every observer you have that they share the same Hamiltonian and $$H\rightarrow i\hbar \partial _{t}$$ which is not the spirit of GR.

the most direct quantization method (in my opinion) would be using Poisson Bracket then:

$$\dot g_{ab}=[g_{ab} , H]$$ $$\dot \pi_{ab}=[\pi _{ab} , H]$$

where gab and pab are the metric and the conjugate momenta to the metric...however i believe this can be done since Poisson approach only works well whenever H=T+V and L=T-V (lagrangian), by the way...¡¡the Hamiltonians in SR and GR are H=0¡¡ then there's no possible quantization.

 

[Hurkyl]

in QM time is "absolute" (all the observers seem to have the same time)

Not true; it depends on the particular quantum theory under consideration.

 

[dextercioby]

A question not mentioned..and another problem.. in GR time is just a coordinate of the curve ..whereas in QM time is "absolute" (all the observers seem to have the same time) since for every observer you have that they share the same Hamiltonian and $$H\rightarrow i\hbar \partial _{t}$$ which is not the spirit of GR.

the most direct quantization method (in my opinion) would be using Poisson Bracket then:

$$\dot g_{ab}=[g_{ab} , H]$$ $$\dot \pi_{ab}=[\pi _{ab} , H]$$

where gab and pab are the metric and the conjugate momenta to the metric...however i believe this can be done since Poisson approach only works well whenever H=T+V and L=T-V (lagrangian), by the way...¡¡the Hamiltonians in SR and GR are H=0¡¡ then there's no possible quantization.

Where did you get that idea from ?

Daniel.