(I’ve also been wondering for a few years whether it is possible to make a version of quantum mechanics which works with the wavefunctions taking on values from a (very large) finite field instead of the complex numbers. My impression so far is “probably not” because you need exponentials, and the multiplication group of a finite field doesn’t have a nontrivial group homomorphism to/from the additive group. Also, for the simple way of defining differentiation for functions over finite fields, has 0 as the only eigenvalue of differentiation, and I’m thinking that the other definition I made up for it also does. I think I saw some definition of differentiation for finite fields which maybe does have a non-zero eigenvalue, Yeah found it, https://arxiv.org/abs/1501.07502 but, this doesn’t have the “derivative” of a constant be 0, so I’m not sure that it would be suitable either.

So, my guess is that probably one can’t make quantum mechanics work with wave functions that have their values all from a finite field. Still going to keep looking a bit more though. )