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It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them?

E.g. relativity theory uses a space-time continuum, while quantum theory uses discrete states.

I am not merely looking for a solution or rebuttal of such opposites, more for a survey of the field out of interest.

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"E.g. relativity theory uses a space-time continuum, while quantum theory uses discrete states." Not really ... –  Cedric H. Nov 8 '10 at 22:57
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Well clearly special relativity and quantum mechanics are quite compatible, in that QFT represents their unification in some way. QFT and GR is a much harder task, and I'll let someone else elaborate on the complications... –  Noldorin Nov 8 '10 at 23:05
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@Noldorin: my impression is that renormalization, rather than upsetting mathematicians, is really something that they would love to understand better; in particular, how it manages to give us physicists a handle on "universality". Recently, Brydges and colleagues have been using a rigorous version of renormalization to prove things about lattice models (see his lectures in the IAS/Park City book on Stat. Mech.). In another direction, note Costello's recent work on foundations of perturbative field theory math.northwestern.edu/~costello/renormalization –  j.c. Nov 18 '10 at 2:51
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@Noldorin: indeed, renormalization not only hints but in fact implies that something is wrong with QFT. But wrong in very specific sense, namely that it is only an effective theory of some better theory. But we knew this all along (there is no gravitation in it), so renormalization actually had to occur. By the way, renormalization is actually very beautiful thing (as opposed to horrible) that cures many conceptual problems of older theories. Also, renormalization doesn't really belong to QFT. In fact, it even originated elsewhere: in the work of Wilson and Kadanoff on critical phenomena. –  Marek Nov 18 '10 at 4:31
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@Noldorin: I wonder about that. In my opinion renormalization is great. Anyway, we've just officially left the realm of physics and moved into arts :-) –  Marek Nov 18 '10 at 18:56
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5 Answers 5

There are zero contradictions between quantum mechanics and special relativity; quantum field theory is the framework that unifies them.

General relativity also works perfectly well as a low-energy effective quantum field theory. For questions like the low-energy scattering of photons and gravitons, for instance, the Standard Model coupled to general relativity is a perfectly good theory. It only breaks down when you ask questions involving invariants of order the Planck scale, where it fails to be predictive; this is the problem of "nonrenormalizability."

Nonrenormalizability itself is no big deal; the Fermi theory of weak interactions was nonrenormalizable, but now we know how to complete it into a quantum theory involving W and Z bosons that is consistent at higher energies. So nonrenormalizability doesn't necessarily point to a contradiction in the theory; it merely means the theory is incomplete.

Gravity is more subtle, though: the real problem is not so much nonrenormalizability as high-energy behavior inconsistent with local quantum field theory. In quantum mechanics, if you want to probe physics at short distances, you can scatter particles at high energies. (You can think of this as being due to Heisenberg's uncertainty principle, if you like, or just about properties of Fourier transforms where making localized wave packets requires the use of high frequencies.) By doing ever-higher-energy scattering experiments, you learn about physics at ever-shorter-length scales. (This is why we build the LHC to study physics at the attometer length scale.)

With gravity, this high-energy/short-distance correspondence breaks down. If you could collide two particles with center-of-mass energy much larger than the Planck scale, then when they collide their wave packets would contain more than the Planck energy localized in a Planck-length-sized region. This creates a black hole. If you scatter them at even higher energy, you would make an even bigger black hole, because the Schwarzschild radius grows with mass. So the harder you try to study shorter distances, the worse off you are: you make black holes that are bigger and bigger and swallow up ever-larger distances. No matter what completes general relativity to solve the renormalizability problem, the physics of large black holes will be dominated by the Einstein action, so we can make this statement even without knowing the full details of quantum gravity.

This tells us that quantum gravity, at very high energies, is not a quantum field theory in the traditional sense. It's a stranger theory, which probably involves a subtle sort of nonlocality that is relevant for situations like black hole horizons.

None of this is really a contradiction between general relativity and quantum mechanics. For instance, string theory is a quantum mechanical theory that includes general relativity as a low-energy limit. What it does mean is that quantum field theory, the framework we use to understand all non-gravitational forces, is not sufficient for understanding gravity. Black holes lead to subtle issues that are still not fully understood.

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+1 This looks like a pretty thorough and accurate overview. –  Noldorin Nov 9 '10 at 19:17
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Excellent answer. The only thing I might add is a mention of string theory having GR as a low energy limit, and thus being a candidate quantum gravity theory. And here's a more technical article going more into the details of nonrenormalizability of gravity: arxiv.org/abs/0709.3555 –  j.c. Nov 11 '10 at 23:13
    
Isn't the CGHS model a renormalizable model of quantum gravity? –  QGR Jan 21 '11 at 22:28
    
This answer is completely wrong. I am sorry but I cannot find something that can be saved. I will comment only on what is relevant for the OP question. First, as stated in textbooks, QFT is not QM+SR. In fact QFT and QM are disjoint theories. Second, QFT does not solve the problems of the RQM (QM+SR) but avoids them. –  juanrga Dec 31 '12 at 21:52
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@juanrga: QFT is QM + SR applied to various situations with such an interpretation that all interactions occur locally. –  Dimensio1n0 Jun 27 '13 at 4:20
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Matt Reece gives a good answer, but one additional area of tension that seems worth mentioning is the problem of time. The role of time in quantum theory is quite different from general relativity.

For a review of some of the issues involved, see

Canonical Quantum Gravity and the Problem of Time. C. J. Isham. "Recent Problems in Mathematical Physics", NATO Advanced Study Institute, Salamanca, June 15-27, 1992. arXiv:gr-qc/9210011.

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Links to time operator and Pauli's objection physics.stackexchange.com/q/6584/2451 and physics.stackexchange.com/q/5268/2451 –  Qmechanic Mar 30 '11 at 18:44
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A superposition of causal structures. More precisely, given two events A and B, they could be in a superposition of being spacelike, null and timelike separated. Quantum field theory is built upon a sharp distinction between localized operators which are spacelike separated from those which aren't. With a superposition of causal structures, such distinctions break down.

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I'm not sure this should be an answer - it is really an anti-answer.

In Quantum Relativity, David Finkelstein has a list of analogies between QM and relativity, detailing "an extended parallel between the structures and developments of relativity and quantum theory". (Section 1.4.2)

Yes, he does have GR in mind when he speaks of relativity.

While he points out some deep similarities, the rest of the book explores in depth, in ways that would thrill only a theorist, the underlying nature of each, analogies and differences.

Anyone interested in the relation of QM and GR would benefit from a browse through this book, though it is hardly the only one that should be read.

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I myself overlooked it too but wikipedia actually happens to have a great such list at https://en.wikipedia.org/wiki/Quantum_gravity#Points_of_tension

There are other points of tension between quantum mechanics and general relativity.

  • First, classical general relativity breaks down at singularities, and quantum mechanics becomes inconsistent with general relativity in the neighborhood of singularities (however, no one is certain that classical general relativity applies near singularities in the first place).

  • Second, it is not clear how to determine the gravitational field of a particle, since under the Heisenberg uncertainty principle of quantum mechanics its location and velocity cannot be known with certainty. The resolution of these points may come from a better understanding of general relativity.

  • Third, there is the Problem of Time in quantum gravity. Time has a different meaning in quantum mechanics and general relativity and hence there are subtle issues to resolve when trying to formulate a theory which combines the two.

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