[mst_moonshine]

Agreed. So what you need is the 'complex structure' behind rather than just 'complex numbers'. Any form of representations (numbers, matrices, and so on) should correspond to a unique structure. The question why the complex structure emerges in quantum mechanics is more interesting.

[zarzavat]

Complex numbers have two roles in mathematics. The first is as a number system based upon SO(2) the group of rotations in 2D, the second is as the algebraic closure of the reals. That these two are the same thing is somewhat of a fluke (it doesn't work in higher dimensions).

Physics uses complex numbers in the first sense. There's really nothing too special about SO(2), there's an SO(n) for all n.

Whereas mathematics uses complex numbers in both senses. There is something rather special about complex numbers as the algebraic closure of the reals and it's what makes a lot of modern math tick.

[amai]

"That these two are the same thing is somewhat of a fluke (it doesn't work in higher dimensions)."

Can you elaborate on this? What is an algebraic closure of the reals in higher dimensions?

[zarzavat]

The complex numbers are the closure regardless of dimension. When I was writing that I was thinking of the Quaternions, which are the 4 dimensional analog of the complex numbers, in 2^N dimensions this is the Cayley Dickson construction.

The fluke is this: Euclidean space of dimension N has N(N-1)/2 rotational dimensions. If you plug 2 into that you get 2x1/2 which is 1 dimension. i.e. the rotations in 2D space look like a circle. If you add an extra dimension (the radius) you get the polar form of complex numbers.

In other dimensions this doesn't always work. In 3 dimensions we have 3x2/2 = 3 rotational dimensions, so we need a space with dimension 4 (the quaternions). In 4 dimensions we need a 6 dimensional rotation space. We just established that Cayley Dickson algebras only come in powers of 2, so it doesn't fit at all.

[amai]

But couldn't one use two unit quaternions to describe rotations in 4d space? An antisymmetric matrix in 4 dimensions has N(N-1)/2 = 6 independent variables. But each unit quaternion has three independent variables, so two of them would be enough to describe rotations in 4d.

[kgwgk]

Skilling and Knuth have some interesting papers on the subject:

The Symmetrical Foundation of Measure, Probability, and Quantum Theories https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800...

The ABC of Physics https://www.mdpi.com/2673-9984/3/1/9/htm