Linear or Not?

Why is QM described as a linear theory and GR non-linear?

Because whoever is doing so is confusing completely different aspects of the two theories.

The linearity of QM has to do with the basic framework of the theory, in which the observables are represented by linear operators on a Hilbert space (as opposed to functions on some space of possible configurations); but the equations relating those observables (such as dA/dt=[H,A]) are generally not linear. However the quantum structure does allow us to predict the effects of non-linear evolution of observables by solving a linear (Schrodinger-type) equation for the quantum state.

The non-linearity of GR is about the relationships between observables, and is no more surprising than the non-linearity of Newtonian gravity or of the theory of fluids and elastic media (though it may be more difficult to find interesting situations with useful linear approximations in the case of the GR field equations than for some other classical theories). However this does not rule out the possibility of somehow eventually finding a Quantum version of GR in which observables correspond to linear operators and in which some linear equation satisfied by the quantum states is equivalent to the non-linear evolution of the observables. One reason this might be very difficult though (or even impossible) is due to the fact that, in GR, space and time coordinates are dependent on other observables in a way that may make them unusable as independent variables in some analogue of the Schrodinger equation.

Source: (1000) Alan Cooper’s answer to Why is QM described as a linear theory and GR non-linear? – Quora

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