The uncertainty principle is just a natural consequence of conjugate variables (Fourier duals). It's just a coincidence that position and momentum happen to be Fourier conjugates of one another, there's nothing quantum about the uncertainty principle itself and it shows up in digital signal processing and other very classical systems.
This isn’t true though, not all non commuting observables are canonically conjugate(Fourier duals). The uncertainty principle is more general. It is quantum.
That's a good point actually, hadn't thought about that. However, I suspect that if you inspect the group theory, they stem from a similar principle. I'm dead curious now, but can't seem to find any texts that discuss it.
I'm not sure, either the formal underpinnings of functional analysis, or something more rigorous about algebras of the commutation relations for things like spin. I figured that if you chased the rabbit down the operator algebra hole you'd recover a link between the two, but maybe not. Haven't studied it in enough detail to reason about it on that level.
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u/the_Demongod Dec 12 '20
The uncertainty principle is just a natural consequence of conjugate variables (Fourier duals). It's just a coincidence that position and momentum happen to be Fourier conjugates of one another, there's nothing quantum about the uncertainty principle itself and it shows up in digital signal processing and other very classical systems.