Fields on the Lattice

The fundamental variables of our structure will be fields, that is, functions defined on the lattice, for example with real values. The notion of particle will not be an elementary notion in our theory. Instead, it will appear later, as a notion derived from the structure of the theory. We will make, on the other hand, extensive use of the notion of wave. However, the physical interpretation of these waves will still be kept at an imprecise and intuitive level, serving mostly to guide our intuition regarding the identification, for closer examination, of the most important elements of the structure.

A scalar field $\varphi$ is a function that associates, to each one of the $N^{d}$ sites of the lattice, a real number. Since a single real value is associated to each site, we say that this is a field with a single component. A particular one-component real field in our first example lattice could be represented as

It is usual to represent this field by means of a set of arrows located at the sites, pointing along a real axis $\mathbb{R}$, with lengths given by the values associated by the field to each site,

Note that the values of the field at each site are not discrete, they exist within a continuous set, the real line. Occasionally one may be interested in fields with discrete values, but this is not the case in general. In this book we will consider only real fields with continuous values. Note also that this field is dimensionless, its values are real numbers without units, because so far there is no physical dimension associated to the fields.

In later parts of the development of the theory one may be interested in scalar fields with values in spaces which are larger and more complex than the real line $\mathbb{R}$. For example, fields which $n$ components may have values in $\mathbb{R}^{n}$, and one may have fields with values in other spaces, not necessarily flat, such as the circle and the sphere. A field with values on the circle may be represented by arrows oriented along it,

Note that all such fields are scalars, not vectors. In further parts of the development one would see that vector fields in space-time are associated to the links and not to the sites of the lattice. The vectors drawn above are vectors in the internal space of the fields, not in space-time. These internal spaces are simple vector spaces in which act symmetry transformations among the fields. Usually these are continuous objects even in finite lattices, while space-time is represented in a discrete form by the lattice. In this book we will not discuss such possibilities any further, and will limit ourselves to scalar fields with a single real component. However, many of the results we will arrive at will apply to multi-component fields as well.

In part of what follows we will be interested in single configurations, or particular possibilities for the field function $\varphi$, specially when we discuss the so-called classical version of the theory. However, our greater objective is the exploration of what we will refer to as the quantum theory, for which we will be interested, not in particular cases for the function $\varphi$, but rather in the set of all possible functions $\varphi$, a set which we will call the space of configurations of $\varphi$. On finite lattices this usually is a continuous space with a large but finite dimension. For example, for a single-component real field in a lattice with $N^{d}$ sites the space of configurations is $\mathbb{R}^{(N^{d})}$. Only in the limit $N\rightarrow \infty$ we will have an infinite-dimension space as part of our structure.

We have, then, the structure of the lattice and the fields defined over it, which are the basic elements for the construction of the theory. The theory will be about functionals of the fields, that is, functions that associate a real value to each one of the possible configurations $\varphi$ of the field. We will be interested in several functionals of this type. A very simple example of such a functional is one which associates to each configuration $\varphi$ the value that it assumes at a given site $s_{0}$, namely the value $\varphi(s_{0})$. Usually we will be interested in more complex functionals than these, which involve sums over all the sites of the lattice. An example of such a functional, still quite simple, would be the one which associates to $\varphi$ the sum of its values at all the sites,

$$F_{0}[\varphi]=\sum_{s}\varphi(s),$$

where the dependence of the functional on the configuration will always be denoted by square brackets and the sum symbolized by the subscript $s$ extends over all the sites of the lattice. If we describe the lattice by a set of integer variables $n_{\mu}$, $\mu=1,\ldots d$, where each one of the $d$ components $n_{\mu}$ of the vector $\vec{n}$ is an integer which numbers the sites along one of the directions of the lattice, we may write this sum explicitly as

$$\sum_{s}\equiv\sum_{\vec{n}} \equiv\sum_{n_{1}=1}^{N}\ldots\sum_{n_{d}=1}^{N}.$$

Figure 1.3.1: A system of integer coordinates in a typical two-dimensional lattice.

Hence, we may denote $\varphi(s)$, equivalently, by $\varphi(\vec{n})$, the first notation being more conceptual while the second makes more explicit reference to the system of integer coordinates. In the two-dimensional case, for example, we will usually employ the system of integer coordinates which indexes the sites in the way indicated in figure 1.3.1.

A particularly important functional, which will be used to define completely each particular model to be considered in the theory, is the one we will call the action, usually denoted by $S[\varphi]$. From the physical standpoint, we will say that this functional determines the dynamics of a given model within the scope of the theory. In order for this to be possible this functional must satisfy some basic properties. First, it must be bounded from below, that is, there must exist a real number $S_{m}$ such that $S[\varphi]\geq S_{m}$ for any configuration $\varphi$ and for any lattice size $N$. Second, it must involve only sums of functions of the field at each site and sums of functions of products of fields at neighboring sites, that is, there is a veto against any dependence on products of fields at sites which are not neighbors, according to the relations of neighborhood established by the links of the lattice.

The first of these two conditions we call the stability condition of the model, for reasons to be made clear later. The second one we call the locality condition, for it means that this fundamental functional cannot depend on the product of values of the field which are associated to sites which are mutually distant from each other, in terms of the number of links that it is necessary to cross to go from one site to the other, in terms of the neighborhood relations of the lattice. This second condition may also be called the next-neighbor condition. It corresponds, in the continuum limit, to the inclusion in the action of terms containing derivatives of at most second order on the fields.

A possible example of an action $S$ would be given by

$$S[\varphi]=\sum_{s}\varphi^{2}(s).$$

This functional satisfies the two conditions, because it depends only on a function (the square) of the field at each site, and because there is a number $S_{m}=0$ such that $S[\varphi]\geq S_{m}$ for any configuration $\varphi$ and for any value of $N$. However, this kind of action, which depends only on the fields at individual sites, and which we may call ultra-local, is too simple to be of much interest to us. For the theory of quantum fields it is essential that the action depend also on products of the fields at neighboring sites. This is what happens when we include in the action, in the continuum limit, dependencies on the derivatives of the fields. In finite lattices we define as the objects that play the role of derivatives certain finite differences between values of the fields.

If $\varphi_{+}$ and $\varphi_{-}$ are the two values of a real field $\varphi$ on the sites at the two ends of a link in the direction $\mu$, $\varphi_{+}$ being on the positive side of the link and $\varphi_{-}$ on the negative side, according to the orientation defined by a given system of integer coordinates, the finite derivative of the field is defined as

$$\Delta_{\ell}\varphi=\Delta_{\mu}\varphi(s)=\varphi_{+}-\varphi_{-}.$$

The positive and negative orientations of the direction $\mu$ which are mentioned here are those in which the integer coordinate $n_{\mu}$ associated to that direction is, respectively, increasing and decreasing in magnitude. The notations $\Delta_{\ell}\varphi$ and $\Delta_{\mu}\varphi(s)$ are equivalent because, given a site $s$ and a positive direction of $\mu$ starting from it, a certain link $\ell$ is uniquely determined,

We see that the finite derivative is a variable naturally associated to an oriented link, not to sites. In this way, we may now define a new action functional as

$$S[\varphi]=\sum_{\ell}(\Delta_{\ell}\varphi)^{2}+\sum_{s}\varphi^{2}(s),$$

which also satisfies the two fundamental conditions. The first sum, as symbolized by the subscript $\ell$, extends over all the links of the lattice, that is, it is an abbreviation for a double sum, over all sites and all the directions of the lattice. We may write explicitly that

$$\sum_{\ell}\equiv\sum_{\mu}\sum_{s}\equiv \sum_{\mu=1}^{d}\sum_{n_{1}=1}^{N}\ldots\sum_{n_{d}=1}^{N}.$$

Clearly, we may generalize this expression for the action, without any violation of the two conditions, if we multiply each one of the two terms by any positive coefficients. We will be specially interested in the particular form of this action given by

  $$S_{0}[\varphi]= \frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2} +\frac{\alpha_{0}}{2}\sum_{s}\varphi^{2}(s),$$ (1.3.1)

where $\alpha_{0}\geq 0$. The model defined by this action is called the Gaussian model or the free scalar field. As we shall see, this is the sole type of model over which we have complete analytical control for any value of the lattice size $N$ in any of the dimensions $d$ in which we are interested. For this reason, it will be extensively studied here as a way to probe into the concepts, issues and problems within the theory.

In terms of the physical interpretation, we may say that the non-Euclidean version of this model in $d=4$ represents the dynamics of non-interacting plane waves, or of free particles, which do not interact with each other. Despite this limitation in scope, we will see that many important and useful things can be learned from this model, which in fact holds a few surprises for us, and will help to clarify the very foundations of the theory. A careful and complete understanding of this model is also essential as a preparation for the future study of interacting fields.

Problems

1. Show that, if $\alpha_{0}<0$, then the action $S_{0}$ of the theory of the free field has no lower bound.

2. Determine the range of values of the parameters $\alpha$ and $\lambda$ for which the functional

   
   

   $$S[\varphi]=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2} ... ...sum_{s}\varphi^{2}(s) +\frac{\lambda}{4}\sum_{s}\varphi^{4}(s)$$
   
   

   satisfies the conditions for an action functional. Determine the value of the lower bound of this action as a function of $\alpha$ and $\lambda$.