Luke, please correct me if I’m misunderstanding something.
The rule follows directly if you require that the wavefunction behaves like a “vector probability”. Then you look for a measure that behaves like probability should (basically, nonnegative and adding up to 1). And you find that for this the wavefunction should be complex-valued and the probability should be its squared amplitude. You can also show that anything “larger” than complex numbers (e.g. quaternions) will not work.
But, as you said, the question is not how to derive the Born rule from “vector probability”, but rather why would we make the connection of wavefunction with probability in the first place (and why the former should be vector rather than scalar). And in this respect I find the exposition that starts from probability and gets to the wavefunction very valuable.
The two requirements are that it be on the domain of probabilities (reals on 0-1), and that they nest properly.
Quaternions would be OK as far as the Born rule is concerned—why not? They have a magnitude too. If we run into trouble with them, it’s with some other part of QM, not the Born rule (and I’m not entirely confident that we do—I have hazy recollection of a formulation of the Dirac equation using quaternions instead of complex numbers).
Luke, please correct me if I’m misunderstanding something.
The rule follows directly if you require that the wavefunction behaves like a “vector probability”. Then you look for a measure that behaves like probability should (basically, nonnegative and adding up to 1). And you find that for this the wavefunction should be complex-valued and the probability should be its squared amplitude. You can also show that anything “larger” than complex numbers (e.g. quaternions) will not work.
But, as you said, the question is not how to derive the Born rule from “vector probability”, but rather why would we make the connection of wavefunction with probability in the first place (and why the former should be vector rather than scalar). And in this respect I find the exposition that starts from probability and gets to the wavefunction very valuable.
The two requirements are that it be on the domain of probabilities (reals on 0-1), and that they nest properly.
Quaternions would be OK as far as the Born rule is concerned—why not? They have a magnitude too. If we run into trouble with them, it’s with some other part of QM, not the Born rule (and I’m not entirely confident that we do—I have hazy recollection of a formulation of the Dirac equation using quaternions instead of complex numbers).
Here are some nice arguments about different what-if/why-not scenarios, not fully rigorous but sometimes quite persuasive: http://www.scottaaronson.com/democritus/lec9.html