Why do the same Conjugate Pairs appear in Fourier Transforms, Commutators, The Uncertainty Principle, and Noether’s Theorem?

Question

When studying time series analysis, the answer is usually ‘the FT variables are defined such that they are duals of one another’ but this isn’t a very satisfying answer and doesn’t explain why the same pairs come up again and again.

Are the uncertainty principle pairs the same because the uncertainty principle relies on Fourier transforms in some way or vice versa?

Or is there some underlying mechanism or symmetry which causes then to pop up all over the place, and if so, how does it cause them to appear?

In: Mathematics

Answer 1

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