Saturday, December 24, 2016

Some brief thoughts on mathematics and physics

I'm really glad that my education was in mathematical physics, rather than just physics. I feel like I understand parts of physics at a much deeper level than a lot of my classmates. Aside from one classmate who told me how jealous he was that maths majors get to study analysis (the foundations of calculus), quantum mechanics in particular gets thrown in a completely different light when you're a mathematician.

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.