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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]?

L. Gorham
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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.

Nullius in Verba
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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.

niels nielsen
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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.

Steven Sagona
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  • You are correct that some quantum experiments, especially using one particle can be predicted using only real numbers as seen in the Bell [experiment][ https://arxiv.org/ftp/quant-ph/papers/0402/0402001.pdf].

    However, complex numbers are required in some multiparticle experiments as in this “Bell like” [prediction][ https://arxiv.org/pdf/2101.10873.pdf ] and [experiment][ https://arxiv.org/pdf/2111.15128.pdf].

     – L. Gorham Aug 16 '23 at 16:06
  • No I am still correct, even when considering the linked papers. So what they are saying is different than what I am saying. They are considering systems that simply remove the imaginary numbers without replacing them with their identical counterpart, which are these specific 2D matrices of real numbers. In just removing the imaginary numbers, doing so changes the physical model and is falsiable. – Steven Sagona Aug 16 '23 at 16:51
  • You can think of these linked papers as saying that there's no way that you can do quantum mechanics without working with these specific 2D matrices of real numbers. – Steven Sagona Aug 16 '23 at 16:55
  • Unless and/or until there is a published rebuttal, I will accept the authors’ conclusions in the references. – L. Gorham Aug 18 '23 at 13:16
  • @L.Gorham, the issue is your lack of understanding of the published papers. The author's conclusions are different than your understanding of their conclusions. – Steven Sagona Aug 18 '23 at 13:40
  • As I have linked in the answer, there is a fact so basic that it doesn't need to be "published": Imaginary numbers can always be represented 2D matrices of real numbers. Nonexperts who first learn imaginary numbers think that there is something particularly special about them, especially because of their name "imaginary." But mathematicians now don't find imaginary numbers to be anything but a useful shorthand way of doing linear algebra, and are often a matter of convience for things that oscillate sinusoidally. – Steven Sagona Aug 18 '23 at 13:45
  • Yes, you are correct, I don’t understand Renau, et al., 2021. I was accepting their conclusions without reservation. I now follow how the 2D matrix approach works. I think you are not suggesting a quantum mechanical formulation using real number Hilbert spaces but are saying that any equation in the complex formulation can be evaluated with 2D matrices. Is this equivalent to saying, “Let the computer do the complex arithmetic.” You have made a R/C distinction that I earnestly seek. I appreciate your help making this distinction. Meanwhile, I’ll continue Renau, et al. – L. Gorham Aug 21 '23 at 14:43
  • “Let the computer do the complex arithmetic.” Right so the math of QM suggests you have to pay attention to "two real numbers" instead of "one real number." The paper you link is trying to modify quantum mechanics to only have to work with "one real number" instead of two. But of course doing that totally changes how everything works, because it was invented paying attention two "two real numbers." – Steven Sagona Aug 22 '23 at 14:01
  • And yes, as you describe, you can completely write code that does all of the math of QM while only working giving real numbers. In this case, it works because you're telling the computer what are the rules for adding and multiplying these 2D pairs of real numbers. – Steven Sagona Aug 22 '23 at 14:02