An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?

Question

In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the study of many-body systems, namely when one looks at understanding physical properties of materials: in this case $N$ represents the number of particles, molecules, etc. of your system; the limit behavior $N\rightarrow\infty$ is of interest when this number is vast.

My question is very related to this concept. Say I am a bit of a theoretical physicist and I would like to ``formalize" the following procedure. Imagine you have a finite one-dimensional lattice modelled by the group $\mathbb{Z}_N$. Then, it is common (in physics) to invoke the trick $N\rightarrow\infty$ and voilà the lattice becomes infinite. Of course, this trick might be the right thing to do in certain problems, but I am presently trying to learn more about some aspects of lattice theory that depend on the (mathematical) group that describes your lattice (such as Wigner functions). For these reasons, I would like to understand whether and how the group of integers can be obtained via limit procedure from modular groups. In particular I would like to understand if I can construct the integers in such way using some algebraic transformations (such isomorphisms, homomorphisms, but not only) so that you I can make statements about group theoretical properties of the integers from what I know about modular groups using my (perhaps convoluted) limit procedure/map.

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A more concrete idea.

I have been reading about the inverse limit procedure used in category theory and profinite completions, (I myself do not know category theory). I am aware that you can construct a profinite completion of the integers $\widehat{\mathbb{Z}}$ that is obtained via an inverse limit over modular groups and naturally contains $\mathbb{Z}$ (via an injective group homomorphism). This looks somewhat like what I was looking for. Still, $\widehat{\mathbb{Z}}$ is stricly larger than $\mathbb{Z}$. Still, these topics are quite alien to me and I do not know whether this construction might be something interesting to look at. I have some concrete questions about this inverse limit construction that I would like to ask here:

1. Might the inverse limit procedure be a rigorous possible approach to obtain $\mathbb{Z}$ from modular groups in the lattice example I mention?
2. How can I characterize which elements of $\widehat{\mathbb{Z}}$ are in $\mathbb{Z}$ (since the former is larger)? Is there a simple algorithm to do this?
3. Can you use the inverse limit construction to make meaningful statements about concepts like the topology of the integers, based on your knowledge of modular groups? For instance, is it possible to induce on the integers its usual LCA topology using this limit?
4. (I am particularly interested in this one) Can you derive the representation theory (irreps, regular representation, etc.) of the integers (or its Pontryagin-dual group, the torus), and its Fourier analysis using the inverse limit procedure? (Again, from what you know about modular groups.) For instance, can you characterize the irreps of $\mathbb{Z}$ in this way.

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Both answers and references are greatly welcomed!

Answer 1

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.

Answer 2

1. You won't obtain $\mathbb Z$ as inverse (also called projective)limit, since a projective limit of compact groups is compact. $\mathbb Z$ is dense in $\hat{\mathbb Z}$, though.

2. The projective limit $\hat{\mathbb Z}$ is also expressible as the product $\prod_p \mathbb Z_p$ of all $p$-adic integers, as $p$ runs over primes. I don't know what kind of algorithm you want to distinguish $\mathbb Z$ in there... but this gives a finer level of detail.

3. The topology of the projective limit is uniquely determined (for cliched categorical reasons) from the topologies of the limitands. Probably we should give each $\mathbb Z/N$ the discrete topology, since that's the unique Hausdorff topology on a finite set. In any case, it is compact, so the projective limit, being a closed subset of compact Hausdorff space, is compact.

3b. But/and the continuous map of $\mathbb Z\rightarrow \hat{\mathbb Z}$, because it has dense image, is not discrete.

For 4. depending on your ulterior motivations, this situation might suggest that the representation theory of $\mathbb Z$ is not what you want, if these limits really do reflect the situation, but that of $\hat{\mathbb Z}$. In any case, we can identify the one-dimensional repns of a totally disconnected group such as $\hat{\mathbb Z}$, namely, they all have finite images in the complex unit circle, or, equivalently, they have open kernels, equivalently, of finite index. That is, every such character factors through some $\mathbb Z/N$.

Thus, the only characters of $\mathbb Z$ you'd get from those of $\hat{\mathbb Z}$ are finite-order ones.

Answer 3

I doubt the following has anything to do with the thermodynamic limit. However, it has been pointed out that neither the inverse nor the projective limit of a sequence of $\mathbb{Z}/N$'s gives you $\mathbb{Z}$, so I thought I'd mention a different way of taking limits of groups which does work.

A marked group is a pair $(\Gamma,S)$ where $\Gamma$ is a group and $S$ is a finite generating set for $\Gamma$; alternatively it's an epimorphism $F_r\to\Gamma$, where $F_r$ is the free group of rank $r=|S|$. There's an obvious notion of isomorphism, and an isomorphism class can be identified with the kernel of the map $F_r\to\Gamma$. Thus, we can identify the isomorphism class of $(\Gamma,S)$ with an element of the power set $2^{F_r}$. The set of all normal subgroups of $2^{F_r}$ is called the space of marked groups of rank $r$, was introduced by Grigorchuk, and is denoted by $\mathcal{G}_r$.

We endow $2^{F_r}$ with the product topology, and $\mathcal{G}_r$ with the subspace topology. The power set $2^{F_r}$ is compact, by Tychonoff's theorem, and totally disconnected. It's easy to check that being a normal subgroup is a closed condition, and so $\mathcal{G}_r$ is also compact and totally disconnected.

It's worth spending a little while working out what the open sets look like. A system of neighbourhoods for $(\Gamma,S)$ is given by the set $U_n$ of marked groups with the same (marked) ball of radius $n$ about the identity in the Cayley graph. So convergence here is very like the intuitive picture presented in the question.

With this in hand, it's easy to see that in $\mathcal{G}_1$, if we identify $F_1=\mathbb{Z}$ and consider the obvious epimorphisms $\mathbb{Z}\to\mathbb{Z}/N$, then we obtain a sequence that converges to the identity $\mathbb{Z}\to\mathbb{Z}$. And of course one has the same phenomenon in any $\mathcal{G}_r$, for suitable choices of generating set.

A google search for 'space of marked groups' will give you lots of references, some written by Yves Cornulier, who is active on MO.