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- Summary
- Which properties of ##\mathbb{C}## are actually necessary?

The following is speculative as well as a honestly meant question about the way QM is modelled. I don't want to create a new theory, just understand the necessities of the old one.

Physicists use complex numbers for QM. But why are they necessary? I understand that the existence of eigenvalues must be granted, but any algebraically closed field does it. I understand that characteristic two is problematic, but what if the characteristic is a high prime, and I assume something like ##99989## or maybe even ##89## already counts as high. I understand that a topology is necessary to perform differentiation, but does it have to contain ##\mathbb{R}##? What's used from the symmetry groups looks as if finite groups could do, i.e. mainly discrete calculations. Why isn't discreteness an intrinsic demand from the start? The Lie algebra representations wouldn't look much different on the algebraic closure of ##\mathbb{F}_{89}##.

And finally, if complex numbers are necessary, what could be achieved by transcendental extensions like ##\mathbb{C}(t)\,?## This way the entire spacetime could simply be part of the scalar field. Isn't this more natural to do than to look for a way to attach it afterwards?

Physicists use complex numbers for QM. But why are they necessary? I understand that the existence of eigenvalues must be granted, but any algebraically closed field does it. I understand that characteristic two is problematic, but what if the characteristic is a high prime, and I assume something like ##99989## or maybe even ##89## already counts as high. I understand that a topology is necessary to perform differentiation, but does it have to contain ##\mathbb{R}##? What's used from the symmetry groups looks as if finite groups could do, i.e. mainly discrete calculations. Why isn't discreteness an intrinsic demand from the start? The Lie algebra representations wouldn't look much different on the algebraic closure of ##\mathbb{F}_{89}##.

And finally, if complex numbers are necessary, what could be achieved by transcendental extensions like ##\mathbb{C}(t)\,?## This way the entire spacetime could simply be part of the scalar field. Isn't this more natural to do than to look for a way to attach it afterwards?