I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).
Re:Quantum Physics Falls Apart Without Imaginary Numbers
by nathan_compton on Tuesday April 25, 2023 @11:37AM
(#35701966)
Exactly. But I want to point out that the field character of the complex numbers is not some incidental quality which happens to distinguish them from 2-vectors. Its absolutely essential to their mathematical character and usage and it also distinguishes them from other complexes we might want to form that behave in a real number like fashion. For instance, there is no way to form a field over the three vectors. In general, one has to give up more and more structure as the dimensions go up.
You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity.
You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1].
[0] https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra