The Pontryagin dual of a discrete (locally compact abelian) group is a compact group, and the Pontryagin dual of a compact abelian group is a discrete (locally

The Pontryagin dual of a discrete (locally compact abelian) group is a compact group, and the Pontryagin dual of a compact abelian group is a discrete (locally compact abelian) group… Hm. Momentum space being compact does seem weird.. Of course, if rather than a discrete group for space, you just have a discrete uh, co-compact(? Unsure of term. Meaning, there is a finite radius such that the balls of that radius at each of the sites, covers the entire space [edit: “Delone set” is the term I wanted.]), uh, if you take a Fourier transform of that lattice… Err… wait, but if the lattice is a subgroup, how does the Fourier transform relate to… I think the Fourier transform of a Dirac comb is also a Dirac comb (with the spacings being inversely proportional) If you multiply the Dirac comb by something first… Well, if you multiply it pointwise by e^(i x p_0 /hbar) , then the Fourier transform will have whole thing shifted by p_0 , and this is periodic in (width of the spacing of the comb in momentum space) So, if you consider all the pointwise multiples of a Dirac comb in position space (multiplying it by arbitrary functions), then I guess the image of that space under the Fourier transform, is going to in some way correspond to functions on S^1, I guess it would be functions periodic in the width of the comb in momentum space. So, if instead of a regular comb, you jostle each of the Dirac deltas in the position space comb by a bit first (a different random amount for each)… I’m not sure quite what one would get…