For accurate predictions, are there formulations of physical processes that require math objects not in the reals, besides Sqrt[-1] and [infinity]?

Question

I believe from [Renau, et al., 2021][ arxiv.org/pdf/2101.10873.pdf] and [Li, et al., 2022][ arxiv.org/pdf/2111.15128.pdf ] that the formulation of quantum mechanics requires complex numbers. Another example in the omitted category is any process evolving momentum. The proofs of the fundamental theorems of calculus require the use of infinity which is not in the reals and many processes require calculus.

Are there other physical processes that require math objects not in the reals, besides Sqrt[-1] and [infinity]?

Answer 1

Complex numbers are basically the same as 2D rotations. Anything involving rotations (pretty much all of physics) can therefore be interpreted as 'requiring complex numbers'.

2D and 3D rotations got packaged up into abstract algebraic number systems (complex numbers and quaternions respectively) and then their geometric origins were forgotten about. When quantum mechanics was developed, they proved the perfect tools for the job, and were adopted in their abstract, unintuitive forms. (Quaternions are closely related to the Pauli matrices.) Many assumed that there was no deeper geometric intuition behind them - that quantum physics requires quantities that are somehow 'unreal'. But it's just spacetime geometry.

In other bits of physics we go straight to the intuitive geometrical interpretation, and don't even realise we're still using complex/quaternionic algebras. But they're mathematically equivalent.

In 2D, if you apply a $90^\circ$ rotation twice, you get the scalar transformation that multiplies by $-1$. Rotations are thus 'complex' in the sense of being quantities that can square to $-1$, but there is obviously nothing 'unreal' about plain old rotations.

Geometric Algebra uses this perspective to model quantum mechanics entirely without using any 'unreal' quantities. The $i$ that appears in the Schrodinger and Dirac equations is given a geometric interpretation as a bivector: a geometric entity that represents plane areas in the same way a vector represents lengths. The algebra incorporates reals, complex numbers, quaternions, spinors, polar vectors, axial vectors, rotations, and reflections. It's more intuitive, but somewhat unconventional.

Answer 2

Any system undergoing harmonic motion can be conveniently represented using complex numbers. So they pop up in mechanical engineering (dynamical systems) and all over the place in electrical engineering, where time-varying AC signals get represented as vectors in the complex plane that rotate about the origin with a certain angular frequency. The complex number plane furnishes a tool that makes operations on those rotating vectors (called phasors) far more straightforward than if they were instead represented in the cartesian (x,y) plane.

In this example, I admit that it is not impossible to describe those systems using only real numbers in the cartesian plane- it is just so much easier to use complex numbers instead.

Answer 3

Putting user Nullius in Verba's answer a different way:

Complex numbers are not required for quantum mechanics.

You can think of complex numbers as multiplication and addition of 2D matrices, which means that you simply need to rewrite any equations in these terms, and voila a theory with only real numbers.