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??