How does abstract algebra relate to canonically conjugate in physics?

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Momentum and position are canonically conjugate variables in physics, represented as Fourier transforms of each other. In abstract algebra, they belong to a group G, with a natural homomorphism ψ: G -> G/H, where H is the kernel of ψ. The discussion raises the question of whether momentum and position reside in the same coset within the set of cosets G/H. Additionally, the relationship between canonically conjugate variables and Fourier transforms is linked to the Heisenberg group in quantum mechanics and the Poisson bracket in classical mechanics.

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Momentum and position are canonically conjugate in physics because they are the Fourier transforms of each other.

In the context of abstract algebra what would that mean. More precisely, Let G be the group they both (p and x) belong to and let ψ:G->G/H be the natural homomorphism where H is the kernel of ψ. Would p and x be in the same coset in the set of cosets G/H?

Dang, I lost my train of thought and I'm not sure where I'm going with this now. I guess my question now is, please relate canonically conjugate in group theory to Fourier transforms.

Thanks, Nate
 
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