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Mudassir Ali

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There are two classes of problems, one is easy and is more or less not important (though frequently trotted out as a problem), the second is the real problem.

The first problem that people usually say is that quantum gravity is not “renormalizable” or “violates unitarity at high energy”. This class of problems have to do with the low energy behavior of quantizing the graviton. More or less, these these problems are were solved by Ken Wilson and effective field theory. We know how gravitons scatter at energies below the Planck scale. We know how to formulate a theory gravitons and do most things that we do with quantum electrodynamics.

The second class of problems is much more problematic and are the real problems of quantum gravity. These problems show that there is an inherent mismatch between General Relativity and quantum field theory (which is how we formulate modern quantum theories). Hawking and Bekenstein showed that the entropy of a black hole is proportional to the area of the event horizon. The entropy is related to the number of states in a system Quantum field theory more or less treats space as a little lattice and every little bit of volume as a degree of freedom. That means that in quantum field theory, the entropy is proportional to the volume. So these two scalings don’t match up: General Relativity says that the number of degrees of freedom is proportional to the area of a region, while quantum field theory says it’s the volume. Volume is much bigger than area, so quantum field theory says that there are many more states than General Relativity. The issue arises from the fact that you can show that black holes have the most information/entropy of any state in the theory and therefore, there are only an area’s worth of degrees of freedom in a system. The full resolution of this mismatch isn’t known. ‘t Hooft made a conceptual proposal known as “holography,” and in limited circumstances, we see that some problems of quantum gravity can be resolved. Strominger and Vafa also showed that in some special circumstances of string theory, the number of states of a black hole constructed in string theory is precisely the Bekenstein-Hawking prediction. So there has been progress, but it is limited. There are still major problems in understanding the quantum mechanics of black holes.

The issues become even more severe when we start talking about quantum cosmology. There are straight up paradoxes when quantum mechanics and cosmology are combined. We don’t even know what the right variables are because quantum mechanics has its most straightforward interpretation as a probabilistic theory, however, no observer gets to see more than one cosmology.