Fourier analysis can be expressed in terms of complex analysis (the standard "most elegant field of mathematics"). Basically, you can think of a periodic function as a function defined on the unit circle, and the corresponding Fourier series as being a Laurent series (think of a Taylor series, but where you can also have terms with negative exponents) on the whole complex plane. This can be generalized to hyperfunctions, where you can treat Fourier analysis of non-periodic functions this way as well. Hyperfunctions, in turn, can be further generalized to a very elegant piece of complex analysis called sheaf cohomology.

Since quantum
mechanics uses Fourier analysis a lot, you can start to see how the
mathematics of quantum mechanics can be recast in terms of complex
analysis. For example, in QFT the requirement that fields have positive
frequency can be reinterpreted as requiring that the Laurent series be
capable of being split into two Taylor series on different parts of the
complex plane. Likewise, the spin states of quantum fields can be
reinterpreted in terms of Riemann surfaces.

Actually, the only parts of quantum mechanics that don't mesh well with complex analysis are the requirement that observables be Hermitian, and the related Born rule for what happens during wavefunction collapse. Though, I believe that if you switch to a gravitationally-induced collapse theory, even the collapse can be described holomorphically.

Anyway, that's one example of how cool physics becomes from a mathematician's point of view. I hope to expand on this stuff more in the future.

Actually, the only parts of quantum mechanics that don't mesh well with complex analysis are the requirement that observables be Hermitian, and the related Born rule for what happens during wavefunction collapse. Though, I believe that if you switch to a gravitationally-induced collapse theory, even the collapse can be described holomorphically.

Anyway, that's one example of how cool physics becomes from a mathematician's point of view. I hope to expand on this stuff more in the future.

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