The Euclidean Lattice

The object we will call the Euclidean lattice consists of a finite set of points with a certain relation of neighborhood established among them. The type and structure of this relation of neighborhood will determine the dimension of the lattice, a whole number that, for us, will always be between $1$ and $5$. The points will be called sites and the neighborhood relations will be represented by connections among the points, which we will call links. A simple example of a lattice could be

We have here four sites and five links in sequence, connecting sites which are neighbors to each other. Another lattice similar to this one could be obtained interconnecting the two loose ends of the outer links, resulting in the lattice

Lattices have the property that the number of links connected to each site is constant, the same for all sites. This number will always be even and equal to $2d$, where $d$ is the dimension of the lattice. In this second example we have $d=1$, the lattice has dimension $1$, and it is said to be one-dimensional. The lattice in the previous example is also one-dimensional, but in that case there are two links that are connected to only one site. We say that the lattice in the first example, unlike the one in the second example, has a boundary.

The existence or not of a boundary will have, later on, an important role to play in the development of the theory, relating to the different types of boundary conditions that may or may not be adopted under various circumstances. Since the second example can be obtained from the first, by the interconnection of the two links with loose ends, resulting in a cyclic structure, we say that the second example adopts periodic boundary conditions.

Observe that, for the time being, there is no additional structure in this object, besides the connectivity of sites and links. In particular, there is no geometric structure or notion of distance. Our second example could be represented as

without any change in the structure of the object as defined so far. We will usually employ symmetrical pictorial representations of the lattices, for simplicity of the drawings, but it is important to keep in mind that there are no predetermined notions regarding the geometry of the lattice or the length of the links.

Also for simplicity of the drawings that we will use to illustrate the ideas, we will usually use as examples lattices of dimension $1$ or $2$. However, our fundamental interest will be in lattices of dimension $3$ and $4$. Occasionally we may make use of lattices with $d=5$, but never with dimensions larger than this. The case $d=1$ is very different from the others, within the scope of the quantum theory to be developed, and will be used as counterpoint, in order to contrast its results with the corresponding results of the lattices with larger dimensions. The case $d=2$ is also significantly different from the others, and will be used only for illustration. A typical two-dimensional lattice could be represented as shown in figure 1.2.1.

Figure 1.2.1: The basic elements of a typical two-dimensional lattice.

In the case of the adoption of periodical boundary conditions the links identified in this drawing with the same numbers or letters would be interconnected. Since it is unpractical to draw the toroidal structure of a lattice of dimension $2$ or larger with periodical boundary conditions, sometimes we may simply state that such boundary conditions have been adopted and represent the lattice as shown in figure 1.2.2, leaving as implicitly understood the connections of the links at the boundary with those on the opposite side.

If we somehow associate physical lengths to the links, we may understand these lattices as rough representations of a finite volume of space (in the case $d=3$) or of space-time (in the case $d=4$), in this case in an Euclideanized version. What we mean by this is that this is not really the usual space-time, since there is here no temporal direction that differs fundamentally from the other three. However, it is possible to establish a relation between the Euclidean space of dimension $4$ and the space-time of physics. Further along we will come back to the issue of the relation between real space-time and its Euclideanized version.

It is clear that, in order to obtain a finer representation of the space, whatever its dimension, we must increase the number of sites in our lattice of that same dimension. The purely mathematical issue of the representation of a continuous space by a scheme based on lattices of increasing size is quite complex and, as it turns out, not relevant for our purposes here, so it will not be further discussed and will be left at this intuitive level. Anyway, it is clear that we will be interested in the properties of lattice systems when the number of sites they contain increases without bound. The examples we drew in the figures of this section have $N=4$, where $N$ is the number of vertices, that is, the number of consecutive sites, according to the relations of neighborhood established by the links, in each one of the $d$ directions of the lattice. The total number of sites of the lattice is given by $N^{d}$ where $d$ is its dimension.

Figure 1.2.2: A simpler representation of a two-dimensional lattice with periodical boundary conditions.

Our strategy is, then, to study the properties of lattices of finite but arbitrary size, with the purpose of eventually discovering to what these properties tend when $N$ tends to infinity. We will refer to this limit, using its traditional name, as the continuum limit. As we shall see, this limit contains the central mathematical difficulty of the theory. It is within it that we find the main problems and the deepest questions of the theory, and it is from it that arise all the fundamental difficulties we will find during the development of the theory.

Still for simplicity, we will usually consider only hypercubical lattices, with a symmetrical structure containing the same number $N$ of sites in all the $d$ directions, as in the examples given in figures 1.2.1 and 1.2.2. When it becomes necessary, for the discussion of some point of foundation of the theory, as will happen later on, we will lift this restriction. Also, we will usually employ periodical boundary conditions, in all directions of the lattice, except when we come to the specific discussion of the issue of the choice of boundary conditions.

It is clear that the structure of the lattices can be generalized and made more complex in many different ways. We are taking here a small subset of the set of all possible lattices, which is particularly simple and symmetrical. Our strategy here is to focus attention only on the simplest cases, until some fundamental reason appears to lead us to more complex and sophisticated representations of the structure of the physical models we are dealing with.

Finally, let us note once more that the limit $N\rightarrow \infty$ by itself means nothing in terms of the physical geometry of the lattice. In particular, nothing happens to the physical lengths of the links in this limit, since there is as yet no notion of physical length established within the structure we are building. Later on, when the notion of metric distance appears, it will be discussed in detail in regard to its nature, origin and role in the structure and in the physical interpretation of the quantum theory.