TITLE: What are details of this claim of Penrose about gravity and QFT being at odds with each other?
QUESTION [0 upvotes]: Question
Can someone flesh out the details of the argument Roger Penrose makes in this video of a lecture he gave on twistors (starting around 1:25:15) or recommend me the appropriate literature (preferably not behind a paywall)?

REPLY [2 votes]: The relevant paper is I believe On the Gravitization of Quantum Mechanics 1: Quantum State Reduction (Penrose, 2014). This is open access.
Penrose actually talks about two types of incompatibility between GR and quantum theory. 
One of them is his explicit point that two different vacua can not "legally" be superposed, which I think is mostly a limitation QFT would put, on a conceptual level, upon the validity of using QM to describe a superposed system with a gravitational component. It is discussed in the above paper, and also in chapter 4.2 of Penrose's book 'Fashion, Faith and Fantasy in the New Physics of the Universe' (2016). 
The second one is more implicit in the video (and perhaps not even present), and does not involve QFT but plain QM, and it is the fact that when gravity is to be taken into account in QM from the Einsteinian perspective, a quantum superposition actually has to be a superposition of spacetimes, which is not easily handled by the status of time in the QM formalism. While this point is not exposed in the lecture itself, it is developed by Penrose in chapters 30.10 and 30.11 of his book 'The road to reality' (2004) so I believe it is an important part of his thinking about the incompatibilities between GR and QM.
Keep in mind that the overall perspective of Penrose is to keep the GR conceptual framework as is, and 'gravitize quantum mechanics' instead of 'quantizing gravity' (which is by far the most popular approach).
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