If you take the Pontryagin dual of the inverse system of cyclic groups $\mathbb{Z}/N\mathbb{Z}$ and projections, you get a system of cyclic groups $\frac{1}{N}\mathbb{Z}/\mathbb{Z}$ with inclusions into each other. The direct limit of this system of dual groups is $\mathbb{Q}/\mathbb{Z}$. By taking a complex exponential, you can identify this system of dual groups with the groups of $N$th roots of unity in the complex line $\mathbb{C}$. The direct limit is then the subgroup of $U(1,\mathbb{R})$ made of the elements of finite order. In either case, there is a metric completion of the limit that yields the circle group $\mathbb{R}/\mathbb{Z}$.

You can view elements of $\widehat{\mathbb{Z}}$ as homomorphisms from $\mathbb{Z} \to \widehat{\mathbb{Z}}$ by taking the image of 1, so by Pontryagin duality, they can be identified with homomorphisms from $\mathbb{Q}/\mathbb{Z}$ to $\mathbb{R}/\mathbb{Z}$. The elements coming from the subgroup $\mathbb{Z}$ are precisely those that extend to continuous maps $\mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$. This is one way to distinguish elements of $\mathbb{Z}$.

The subspace topology on $\mathbb{Z} \subset \widehat{\mathbb{Z}}$ is not the usual discrete topology on $\mathbb{Z}$. The open sets are unions of unbounded arithmetic progressions. In fact, this is the topology seen in Furstenburg's (recasting of Euclid's) proof of the infinitude of primes.

Knowing the topology, we can describe the representation theory. A representation of $\mathbb{Z}$ is determined by the image of $1$, so it amounts to a matrix. A continuous representation of $\widehat{\mathbb{Z}}$ is also determined by the image of $1$, but if the representation takes values in endomorphisms of a real or complex vector space, continuity forces the corresponding matrix to have finite order, i.e., the representation factors through a finite cyclic quotient. In particular, the spectrum is restricted to roots of unity.

Everything I just said can be found in standard texts, like Ramakrishnan, Valenza, *Fourier analysis on number fields*. What follows is baseless speculation.

I know next to nothing about thermodynamics on lattices, but I think the inverse limit approach to formalizing the large $N$ limit loses some essential information. My reason is that the process of passing to a larger lattice along an inclusion $\{ -N, -N+1,\ldots,N \} \to \{-N-1,-N,\ldots,N+1 \}$ should be meaningful in some weak sense (e.g., some part of the spectrum should experience minimal perturbation), and the inverse limit method only allows you to compare them after passing to high-degree covers. For example, the properties of a big chunk of crystal that are relevant to physical examination should not depend on whether or not you shave off an atom-thick layer from one side. My ill-informed guess is that people mainly use the cyclic group structure because periodic boundary conditions are the most easily computed - the fact that covers of groups give exact maps on spectra is a convenient accident. I would guess that a boundary condition more amenable to taking a limit over embeddings of increasingly large lattices might look something like a PML.

This post imported from StackExchange MathOverflow at 2014-06-15 16:33 (UCT), posted by SE-user S. Carnahan