21–30 of 38 posts
Earlier quoted context omitted.
Negative, irrational and transcendental numbers are all on the same number line. Getting out into a plane (and losing something as important as the order relation) is radically different from figuring out what other numbers are on a line.
Order relations don't appear in QM.
Earlier quoted context omitted.
> QM can be well explained with conventional probability theory already The subjective experience of a person performing QM experiments, sure, but not the actual universe, that's what Bell's theorem was about.
That's not what I'm saying. Bell or any CHSH-like experiment can be equivalently described using random variables and quasi-stochastic processes instead of quantum states, unitaries, and measurements. It would still involve non-local correlations and inequality violations, but without mentioning the Born rule, phases, and interference with imaginary numbers. It is just an equivalent mathematical framework.
Earlier quoted context omitted.
That's not what I'm saying. Bell or any CHSH-like experiment can be equivalently described using random variables and quasi-stochastic processes instead of quantum states, unitaries, and measurements. It would still involve non-local correlations and inequality violations, but without mentioning the Born rule, phases, and interference with imaginary numbers. It is just an equivalent mathematical framework.
This paper [1] doesn't stoop to providing an example, but isn't that just the thing where you can write 1 and i as 2x2 matrices? I don't think that's what Scott is talking about. Requiring that the elements of the density matrix be real (or allowing them to be quaternion) creates a non-equivalent theory. [1] https://arxiv.org/pdf/1704.08525.pdf
Basically, either you have unsigned numbers (conjugated magnitude), with growth/shrink operations, or you have numbers with a polarity, with displacement operations . It is incomplete to have the notion of "negative numbers" without also including the imaginary parts.
It's only incomplete if you require particular operations, you're perfectly fine with +-*/ and real numbers.
Earlier quoted context omitted.
This paper [1] doesn't stoop to providing an example, but isn't that just the thing where you can write 1 and i as 2x2 matrices? I don't think that's what Scott is talking about. Requiring that the elements of the density matrix be real (or allowing them to be quaternion) creates a non-equivalent theory. [1] https://arxiv.org/pdf/1704.08525.pdf
Right, these are different questions indeed. Scott wonders what happens to amplitudes as they already appear in the theory but with numbers being no longer complex. But those lifted representations effectively change to a specific basis in higher dimensions (think qubit's 2x2 density matrix becoming a 4-dimensional distribution vector, with the same 3 real degrees of freedom) where everything is real and interpreted…
I would very much enjoy reading the pre-reading for this post.
And the pre-pre, and probably pre-pre-pre. Much respect to those who can follow this.
Basically, either you have unsigned numbers (conjugated magnitude), with growth/shrink operations, or you have numbers with a polarity, with displacement operations . It is incomplete to have the notion of "negative numbers" without also including the imaginary parts.
It's only incomplete if you require particular operations, you're perfectly fine with +-*/ and real numbers.
Earlier quoted context omitted.
It's only incomplete if you require particular operations, you're perfectly fine with +-*/ and real numbers.
If you have real numbers instead of integers, then it implies fractional applications of the arithmetic operators. Fractional negation is complex rotation, so if you have negative numbers and they are non-integers then it necessarily must also include complex numbers.
Does it? Where does this implication come from?
Earlier quoted context omitted.
If you have real numbers instead of integers, then it implies fractional applications of the arithmetic operators. Fractional negation is complex rotation, so if you have negative numbers and they are non-integers then it necessarily must also include complex numbers.
> implies fractional applications of the arithmetic operators Does it? Where does this implication come from?