# wouldn't a discretized space-time violate pontryagin duality?

While this question regards physics, it is more of a mathematical question, so here it is.

One often hears about attempts to model space time with tilings or some type of discretized structure. Physically, however, energy and momentum have discretized values, and so a red flag always goes up in my head regarding the former notions. If anything, don't quantized values of energy momentum indicate a closed continuous space time?

In more math-like speak, if the dual space is isomorphic to the integers, shouldn't the space be isomorphic to the circle (just considering one dimension for simplicity)? Does this line of thought make sense? FYI, I study physics, and sometimes they pass right over these principles without giving them due process.

Physics is based on lie groups, which in themselves respect Pontryagin duality. From this (admittedly simplistic) point of view it would seem feasible that the global geometry dictates quantization on a local level. Thoughts anyone? I'm working on something currently to this effect.

The extra compactified dimensions of string theory for example seem unnecessary if we just take the spacial dimensions we already have and consider them compact??

• A question and a comment: What evidence is there that energy and momentum are discrete? (That there are quantized energy levels in, for example, the hydrogen atom doesn't imply anything about physics as a whole. And the hydrogen atom itself has a continuous spectrum in addition to the discrete one.) A lattice on a torus would have a discrete momentum spectrum. You don't need continuous space for this. Jul 31, 2016 at 11:40
• @Will Orrick yes all bound states are discrete, while free particles have continuous energy momentum modes. However, when the larger system is considered there are no free particles. There will always be a weak interaction binding particles on some level. The spectra approach continuity as the mode goes to infinity, so we can safely approximate it as continuous. as for the latticeTorus, I hadn't thought of that, perhaps my question should read: Does quantum mechanics (via Pontryagin Duality) require the universe to be closed? Jul 31, 2016 at 19:11