Incorrect. You can construct an isomorphism between the even subalgebra of the 2D geometric algebra Cl(2) and the complex numbers that maps 1 to the unit scalar and i to the pseudoscalar: https://link.springer.com/article/10.1007/BF01883676
Are you interested in proving me wrong, or figuring out the right answer? If you actually read that article instead of just the title, you would have noticed at the end it says
This leads us to say a few words about the widely held opinion that, because complex numbers are fundamental to quantum mechanics, it is
desirable to “complexify” every bit of physics, including spacetime itself. It will be apparent that we disagree with this view, and hope earnestly that it is quite wrong, and that complex numbers (as mystical uninterpreted scalars) will prove to be unnecessary even in quantum mechanics
They literally say that “complex numbers are fundamental to quantum mechanics”. In other fields of physics complex numbers are just a convenient tool, but in quantum mechanics they are(as far as we know) fundamental, even if the author hopes that to be proved wrong at some point.
You seem like you know a bit about alternatives to complex numbers in other areas of physics, so it would be interesting to have a further conversation, as long as you stop being so defensive.
Complex numbers seem to be used either as 2d vectors or as representation of waves/circles in exponentials, is there an alternative that combines both of those uses?
nope, they’re just one mathematical construct out of many (e.g. 2D vector calculus or geometric algebra), and they just happened to stick
Nope, you’re just wrong. Quantum mechanics without complex numbers(real quantum theory) is less predictive than complex(regular) quantum theory. https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/
Incorrect. You can construct an isomorphism between the even subalgebra of the 2D geometric algebra Cl(2) and the complex numbers that maps 1 to the unit scalar and i to the pseudoscalar: https://link.springer.com/article/10.1007/BF01883676
Are you interested in proving me wrong, or figuring out the right answer? If you actually read that article instead of just the title, you would have noticed at the end it says
They literally say that “complex numbers are fundamental to quantum mechanics”. In other fields of physics complex numbers are just a convenient tool, but in quantum mechanics they are(as far as we know) fundamental, even if the author hopes that to be proved wrong at some point.
You seem like you know a bit about alternatives to complex numbers in other areas of physics, so it would be interesting to have a further conversation, as long as you stop being so defensive.
Complex numbers seem to be used either as 2d vectors or as representation of waves/circles in exponentials, is there an alternative that combines both of those uses?