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I have looked at some of 't Hooft's recent papers and, unfortunately, they are well beyond my current level of comprehension. The same holds for the discussions that took place on this website. (See, for example, here.) I therefore tried to imagine what these papers might be about in my own terms. The following is one such imagination.

Is, possibly, the essence of his recent papers that 't Hooft forces probability amplitudes to be of the form ${\Bbb {Q}}e^{2\pi i{\Bbb Q}}$,* which, I presume, is dense in $\Bbb C$? That is, does 't Hooft provide an unfamiliar, and possibly cumbersome, interpretation of probability, which nonetheless might be considered appealing and/or insightful, especially since the set of allowed probability amplitudes is countable in such an interpretation?

Is this the gist of his models? Or am I a long way off?

*I'm using ${\Bbb {Q}}e^{2\pi i{\Bbb Q}}$ as a notational shortcut for $\left.\left\{re^{2\pi i \theta}\,\right|\, r,\theta\in\Bbb Q\right\}$.

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Sorry to be mean, but consider $z = (1+i)/\sqrt{2}.$ Then $r = 1$ and $\theta = 1/8$ (in your notation). But clearly $z$ is not of the form $p + q i$ with $p,q \in \Bbb{Q}.$ –  Vibert Mar 8 '13 at 23:06
    
@Gugg. mathworld.wolfram.com/NivensTheorem.html. Only a few values of theta would satisfy your condition. –  Mew Mar 8 '13 at 23:35
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However, more on-topic, both sets are most definitely dense in $\Bbb{C}.$ –  Vibert Mar 8 '13 at 23:58
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@Vibert and Chris: Thanks! I removed one of the sets from the question. I hope the one that I left is the right one. :) –  Glen The Udderboat Mar 9 '13 at 6:23
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't Hooft considers wavefunctions which, as they evolve, are in an eigenstate, then, after a certain period of time, another eigenstate, and which keep passing through eigenstates at a fixed rate. These "moments when the wavefunction is in an eigenstate" correspond to the discrete timesteps of the cellular automaton. –  Mitchell Porter Mar 10 '13 at 3:33

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(I'm copy-pasting from @MitchellPorter's comments into this community wiki.)

Off the mark. In QM you have states and you have "observables" and the states give the probabilities for the observables. The states (wavefunctions) evolve deterministically, the probabilities only pertain to how they manifest in observables. A state can imply a 100% probability for a particular outcome, then it's an "eigenstate" of that observable...

't Hooft considers wavefunctions which, as they evolve, are in an eigenstate, then, after a certain period of time, another eigenstate, and which keep passing through eigenstates at a fixed rate. These "moments when the wavefunction is in an eigenstate" correspond to the discrete timesteps of the cellular automaton.

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