Comment by sixo

I think the issue with "modeling" is really a human one, not a mathematical one.

It's helpful sometimes to think of our collective body of mathematical knowledge as like a "codebase", and our notations and concepts as the "interface" to the abstractions at play within. Any software engineer would immediately acknowledge that some interfaces are FAR better than others.

The complex numbers numbers are one interface to the thing they model, and as you say, in a certain sense, it may be the case that the thing is C. But other interfaces exist: 2x2 antisymmetric traceless matrices, or a certain bivector in the geometric-algebra sense.

Different interfaces: a) suggest different extensions, b) interface with other abstractions more or less naturally, c) lend themselves to different physical interpretations d) compress the "real" information of the abstraction to different degrees.

An example of (a): when we first learn about electric and magnetic fields we treat them both as the "same kind of thing"—vector fields—only to later find they are not (B is better thought of as bivector field, or better still, both are certain components of dA). The first hint is their different properties under reflections and rotations. "E and B are both vector fields" is certainly an abstraction you CAN use, but it is poorly-matched to the underlying abstraction and winds up with a bunch of extra epicycles.

Of (d): you could of course write all of quantum mechanics with `i` replaced by a 2x2 rotation matrix. (This might be "matrix mechanics", I'm not sure?) This gives you many more d.o.f. than you need, and a SWE-minded person would come in and say: ah, see, you should make invalid states unrepresentable. Here, use this: `i = (0 -1; 1 0)`. An improvement!

Of (b): the Pauli matrices, used for spin-1/2 two-state systems, represent the quaternions. Yet here we don't limit ourselves to `{1, i, j, k}`; we prefer a 2-state representation—why? Because (IIRC) the 2 states emerge intuitively from the physical problems which lead to 2-state systems; because the 2 states mix in other reference frames; things like that (I can't really remember). Who's to say something similar doesn't happen with the 2 states of the phase `i`, but that it's obscured by our abstraction? (Probably it isn't, but, prove it!)

I have not given it much more thought than this, but, I find that this line of thinking places the "discontent with the complex numbers in physics" a number of people in this thread attest to in a productive light. That dissatisfaction is with the interface of the abstraction: why? Where was the friction? In what way does it feel unnecessarily mystifying, or unparsimonious?

Of course, the hope is that something physical is obscured by the abstraction: that we learn something new by viewing the problem in another frame, and might realize, say, that the interface we supposed to be universally applicable actually ceases to work in some interesting case, and turns out to explain something new.