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Comment by kevinventullo | Hacker News Reader

I think the question itself doesn’t make sense. It’s like asking if quantum mechanics “needs” linear algebra, or if general relativity “needs” differential geometry.

Like there’s probably some roundabout way to do the exact same thing but calling it something different (e.g. matrices or commutative algebra), but then this calls into question what exactly one means by “using complex numbers”.

I guess if you thought there was a simplification to be had, that would be a good reason to search for a fundamentally different way of framing things, but in my experience as a mathematician, the whole point of fancy theoretical frameworks is to arrive at conceptual simplicity. You prove something very general and then you can take it as a conceptual axiom.

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This is literally addressed verbatim by the article, and I quote:

"One can always devise new mathematical constructs that behave in all the same ways as complex numbers even though they’re called something else."

It's honestly as if people are just jumping to comment on the title of the article without actually reading it. The article is fundamentally about how trying to replace complex numbers by some real valued representation of them, for example using a vector of real numbers, or a matrix of real numbers, doesn't work due to the compounding of dimensions. The two examples they give where using pairs of real numbers fails is multiparticle states and a spin 1/2 qubit.

At any rate, there's no point rehashing the article here, since it's all written in a fairly straight forward manner. It's just frustrating reading so many people write the same trivial comment dismissing the article when the article is written precisely to demonstrate why these simplistic dismissals fail.

As I commented elsewhere, in the multi particle state example you can fix the problem by tensoring with respect to the vector or matrix algebra, so it works just fine. And like of course it does because C itself is isomorphic to these algebras.

My point about calling into question what one means by “using complex numbers” is that the article says it is “necessary to establish some ground rules that exclude real-valued quantum theories that restate the standard complex-valued theory by other names” but then handwaves the rules. Presumably you’re only allowed to tensor over R. But more importantly, it provides no motivation for why these ground rules wouldn’t be somewhat arbitrary. I guess I just don’t see the value of removing complex numbers from the formulation.

I’m not convinced. Complex numbers are a representation of a system. Yes, you can express it somehow else, but at its core it’s the same. As I understand it, they are saying that using complex numbers is the simplest (pun not intended) representation necessary and there’s no further simplification possible. That is meaningful because an ancient Greek would say that reals or even integers are sufficient to describe the world. They are not. Just like Euclides was not enough.

I guess I don’t find it that meaningful. The Greeks also didn’t have the differential geometry needed to describe relativity, but no one’s trying to remove that from physics.

For the record, as I mentioned elsewhere, I think further simplifications will only come from higher levels of abstraction, not less abstraction.

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