If you end up with complex probabilities, you won’t be able to plug them into an expected utility formula to get a preference ordering. This has always been the knockdown argument for quantitatively scaled real-number subjective probabilities in my book. Even if underlying physics turns out to use complex-numbered reality fluid, I don’t see how I can make choices if my degree of anticipation for something happening to me is not a real number—I don’t know of any complex analogue of the von Neumann-Morgenstern theorem which yields actual decision outputs.
To put it simply, complex numbers lack an order compatible with the algebraic structure: just that makes them unsuitable for decision theoretic criteria.
After the first dozen responses, I’m currently thinking of writing something along the lines: “While the unusual math of noncommutative probabilities allows for complex probabilities, which have applications in quantum superpositions and eigenstates, there is little likelihood of any practical application involving (the protocol). A (protocol) statement may be written with a complex number for its confidence, but a (protocol) reader or interpreter need only concern itself with the real portion of that number.”
Either that, or just stating ‘real numbers only’.
(PS: I’ve never written anything which has even a chance at being an Internet Draft, let alone an RFC; but if the tag: URI made it in, nym: just might pass muster, too—and I would welcome any and all advice.)
No, in accordance with whatchamacallit’s law.
If you end up with complex probabilities, you won’t be able to plug them into an expected utility formula to get a preference ordering. This has always been the knockdown argument for quantitatively scaled real-number subjective probabilities in my book. Even if underlying physics turns out to use complex-numbered reality fluid, I don’t see how I can make choices if my degree of anticipation for something happening to me is not a real number—I don’t know of any complex analogue of the von Neumann-Morgenstern theorem which yields actual decision outputs.
To put it simply, complex numbers lack an order compatible with the algebraic structure: just that makes them unsuitable for decision theoretic criteria.
After the first dozen responses, I’m currently thinking of writing something along the lines: “While the unusual math of noncommutative probabilities allows for complex probabilities, which have applications in quantum superpositions and eigenstates, there is little likelihood of any practical application involving (the protocol). A (protocol) statement may be written with a complex number for its confidence, but a (protocol) reader or interpreter need only concern itself with the real portion of that number.”
Either that, or just stating ‘real numbers only’.
(PS: I’ve never written anything which has even a chance at being an Internet Draft, let alone an RFC; but if the tag: URI made it in, nym: just might pass muster, too—and I would welcome any and all advice.)