How does abstract algebra relate to canonically conjugate in physics?

[nateHI]

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

 

[tom.stoer]

In QM this is related to the Heisenberg goup
http://en.wikipedia.org/wiki/Heisenberg_group

Classically this is related to the Poisson bracket on phase space carrying a symplectic structure
http://en.wikipedia.org/wiki/Poisson_bracket
http://en.wikipedia.org/wiki/Symplectic_manifold