I think it’s a fascinating puzzle. There is a certain amount of philosophy involved, which I find frustrating: namely, do you really need to provide an argument that each term that one *could* add to the Lagrangian, consistent with renormalizability and the gauge symmetries, is not actually there (or has an extremely low coupling constant)? Endless ink has been spilt on this subject, so let’s not talk about it more. But if someone can use this sort of reasoning to find new physics—like an axion—then more power to them!

But experiments say the magnitude of the θ angle is less than Is this angle zero or not? Nobody knows. Why is it so small?

It seems to be one of the examples of fine-tuning which, as far as I know, has no convincing weak-anthropic explanation.

]]>The status of CPT in string theory is unclear. But one of Hawking’s last papers (1401.5761) used CPT and AdS/CFT to argue that there are no true event horizons, only temporary apparent horizons.

]]>CPT violation would break quantum field theory on Minkowski spacetime as we know it, since CPT symmetry is a theorem in all decent axiomatizations of that sort of quantum field theory.

Quantum gravity also probably breaks quantum field theory on Minkowski spacetime. So it’s perhaps not completely surprising that Hawking gave an argument that quantum gravity could violate CPT invariance, as well as the unitarity of time evolution:

• Stephen Hawking, The unpredictability of quantum gravity., *Communications in Mathematical Physics* **87** (1982): 395–415.

I don’t know what the AdS-CFT crowd thinks, but since they believe quantum gravity can be described by a quantum field theory “on the boundary”, they probably also believe it’s CPT invariant.

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