Quote:
Originally Posted by Nereid
This thread is motivated by a number of posts in the ATM section, where the usual things we think we understand are blithely tossed about as being meaningful in the Planck regime (or era, or at the Planck scale, or ...) - length (or space), time, energy, mass, ...
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What are some good explanations - other than those using the math of each theory - that you've come across which show this incompatibility?
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Well, I'm not John Archibald Wheeler and I'm not an expert on QM, but do know a bit about GR and I think I can provide an explanation (I'm sure there are others here who can correct me where I'm wrong (cough Eta C, hint Celestial Mechanic). I want to point out that no matter how much one wants to separate the math from the explanation, you must realize that the reason QM and GR are incompatible is because of the math. There are two approaches to consider. Getting the math of QM to work with GR or using the math of GR to work with QM.
In the case of the first, extremely simplified, in Quantum Elelectrodynamics(QED) (EM force), when you calculate the force between two electrons, most of the time the you simply consider a photon being emitted from one of the electrons and being absorbed by the other electron. This is known as a coupling. However, an energetic enough photon, can produce a electron-positron pair while moving between the electrons (they would then annihilate each other and the photon produced from their annihilation would be absorbed by the second electron, another coupling. This second coupling is considered a "perturbation" and should be a small correction to the main "one photon between electrons" coupling. It turns out the when calculating the charge, all the possible types of couplings have to be considered. You would think that these small corrections shouldn't be a problem, however when those "perturbations" are considered, the equations diverge and you end up with infinities in the calculations.
A way around this was found by Shin'ichiro Tomonaga, Julian Schwinger, Richard Feynman(They shared the 1965 Nobel Physics prize for it) which has been dubbed
"Renormalization". Again, extremely simplified, renormalization basically subtracting out the infinities (I know, I know, that's not allowed I hear you screaming, but there is a lot more involved, I'm just trying, per Nereid's post to stay away from any advanced math and give a simplified version). A similar process was found to work with the color force on gluons for quark interaction and was named Quantum Chromodynamics (note the chromo to designate the color force).
You might think that it should be simple to just use the "Perturbative Theory" used in QM to find a gravitational theory. This would use the graviton as the mediator of gravitational force, much like the the photon is the mediator of the EM force. The problem is, the renormalization techiques using in QED and QCD, don't work for the graviton. The reason has to do with the properties the graviton has to have to be compatible with GR. And it has to be compatible to get the match that there is between GR predictions and observations. It's extremely technical (translation, I don't understand it enough to provide a good explaination) and math intensive. So throw out this approach(however, see near the bottom of this post)
If that approach doesn't work, how about trying to get QM to fit in with GR? Well, you run right in to the backround problem. In GR, the backround (space-time) is dynamic. Both time and space can change depending on the energy in a given volume (this is what we normally refer to as gravity). In QM, the backround used is Minkowski space, which is a flat space-time with no changes. Once you use a dynamic backround, problems arise in QM (such as the Unruh effect, which simply states that the vacumn, is dependent on the path through spacetime. In other words, the vacumn effects predicted by QM (virtual pair production for example) depend on the path the observer follows and two different observers may not see the same thing. QM at present can't explain this. Another problem comes about by trying to fit the Heisenberg uncertainty principle (HUP) into GR. To calculate the effects of the gravitational field in GR, you need to localize the amound of energy in a given volume. Under the HUP, we can't know the location percise enough, within that volume, to calculate the gravitational field generated by the energy.
Now, with all these problems, let me point out that while we don't have a full quantum gravitational theory, there have been some success in combining the two in limited ways. The Laws of Quantum Mechanics in Curved Spacetime were first developed by Hawking and the most notable result is Hawking radiation from black holes. In addition, there has been success in calculating the first order quantum mechanical corrections to the gravitational potential between two masses using QM perturbative methods. So far, this only works in the low energy regime, where the infinities don't materialize.
There are quite a few other techincal details to work out. Anyone interested in the techical details or the math can find these by searching the web.