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Kiefer (2014) claims that,

"It is, in fact, the superposition principle that points towards the need for quantizing gravity."

Moreover, Kiefer (2009) stresses that,

"The only assumptions are the experimentally supported universality of the linear structure of quantum theory and the recovery of general relativity in the classical limit."

But why is the linear structure of quantum mechanics so important? Don't we already know that nature is non-linear (for instance, interacting field theories)?

If we do accept that the superposition principle is not really universal, how do the above arguments for quantum gravity hold?

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You've confused whether interactions are linear (i.e. the resulting equations of motion are linear, which as you noted they in general aren't) with a very different "linearity" question, namely whether the states of a system belong to a Hilbert space and can therefore be superposed in the usual quantum-mechanical way. Some approaches to quantum field theory in curved spacetime (whether or not $g_{\mu\nu}$ is quantised properly to some operator $\hat{g}_{\mu\nu}$) use a non-linear generalisation of Hilbert spaces, but neither of Kiefer's papers discuss that.

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