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From the mathematical point of view, where is the idea of the incompatibility between functional analysis, at the basis of quantum mechanics, and differential geometry, at the basis of general relativity?

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Smooth functions on a manifold commute, because they are valued in numbers. Operators (in functional analysis) might not commute, even when they are used to represent quantum measurements of physical observables. You might want to read Connes's book about noncommutative geometry. –  Ben McKay Mar 21 '13 at 13:49
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The kind of incompatibility that you seem to have in mind is an urban myth. It is often mentioned, but rarely explained accurately outside specialized literature. The question of combining quantum mechanics with general relativity has been discussed here before. Best to approach it only after you know what a quantum field theory is. See for instance my answer here: mathoverflow.net/questions/13205/13261#13261 –  Igor Khavkine Mar 21 '13 at 14:46
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What about the incompatibility between functional analysis, at the basis of differential equations, and differential gometry, also at the basis of differential equations? I think this question is based on misinterpretation, see Igor Khavkine's comment –  Yemon Choi Mar 21 '13 at 18:37
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