I am not a physicist.

For as long as I can remember, I've been told that the main problem in physics is to unify quantum mechanics and general relativity into a single theory.

I once heard someone say, that the reason we haven't combined general relativity with QM, is simply because in GR, quantum operators wouldn't be linear (due to the curvature of spacetime), so that the superposition principle fails.

In other words, that would mean we have no problem understanding how they would be unified, conceptually, we just don't know how to deal with the nonlinear mathematics that would result from this.

Is that understanding correct?

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    $\begingroup$ That's one approach to the solution. But non-linearity, while being mathematically difficult, is not the actual problem. The problem is a conceptual one. $\endgroup$ – Yuzuriha Inori Mar 5 '18 at 16:36
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    $\begingroup$ Possible duplicate of A list of inconveniences between quantum mechanics and (general) relativity? $\endgroup$ – AccidentalFourierTransform Mar 5 '18 at 16:55
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    $\begingroup$ Physics is made of mathematics. Not being able to write something down mathematically (at a physicist's level of rigor) is the same as not conceptually understanding it. $\endgroup$ – knzhou Mar 5 '18 at 17:24
  • $\begingroup$ I'm not an expert in this area, but it's my understanding that there are both linear QM theories on curved spacetime and nonlinear theories in flat spacetime. The nonlinearity is due to the form of the differential equation, not necessarily the spacetime. $\endgroup$ – D. Halsey Mar 6 '18 at 0:31

At the moment we have no idea how to construct a theory of quantum gravity, and I'd say that's as conceptual as it gets. We don't even know whether it makes sense to quantise gravity. There have been suggestions that general relativity is an emergent theory, much like hydrodynamics, and if so quantising it makes no more sense than trying to quantise the Navier-Stokes equations.

The many totally different approaches to quantum gravity should make it immediately obvious how little we know about the subject. The leading contender is of course string theory, but even there the interest appears to moving away from string theory as a description of quantum gravity.

If you are interested in knowing more I recommend the paper Conceptual Problems in Quantum Gravity and Quantum Cosmology by Claus Kiefer, ISRN Math.Phys. 2013 (2013) 509316. This gives a pretty good overview of the issues and it was published in 2014 so it is up to date.

  • $\begingroup$ The third line - "... general relativity is an emergent theory..." is bliss $\endgroup$ – Yuzuriha Inori Mar 5 '18 at 17:01

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