Definition of the Classical Theory

We will now use the action $S_{0}$ to illustrate the relation that exists between our mathematical structure on the lattice and the classical (non-quantum) theory of fields. It should be noted here that our approach to the subject, unlike the traditional one, in both the classical and quantum cases, will not be based on equations of motion but rather on the action functional as the object defining the physical models. This classical topic may not be our fundamental objective but it is useful to illustrate the role of each element of the structure and also to help to orient the reader, comparing the elements that appear here with with the corresponding elements as seen in the traditional approach to the subject, usually in graduate courses.

In order to establish this relation it will be necessary to take the limit $N\rightarrow \infty$ in our theory. Before this, however, there are two other things we must do. First, we must define what we will call a finite classical field theory, on each finite lattice. In addition to this, it will be necessary to introduce one more basic element into the structure, to wit, an external dimensional scale.

In order to define the classical theory on finite lattices we consider the action $S_{N}[\varphi]$ of a given model, on a given lattice of size $N$. We say that a classical solution of the model is a configuration $\varphi_{0}$ which minimizes $S_{N}[\varphi]$ locally, that is, at which the action has a local minimum. The stability condition that we imposed on $S_{N}[\varphi]$ implies that there is at least one local minimum of the action, the one located at the position of the global minimum guaranteed by that condition. For the action $S_{0}$ of the free theory we see immediately that the classical solution is simply the identically null configuration, $\varphi\equiv 0$, for all $N$. Note that, in principle, $S_{N}$ may have more than one local minimum and that, when this happens, we will have more than one classical solution. This is not the case for $S_{0}$, but it is possible to construct actions with this property, as we may see in a future opportunity.

Let us also recall that we are assuming the use of periodical boundary conditions. In general the nature of the classical solutions depends on the boundary conditions. With the introduction of other elements into the structure, such as other types of boundary conditions or other terms in the action, the classical solutions may be much less trivial than the simple example we gave here. In particular, later on we will discuss the concept of external sources, which is very important for the physical interpretation of the theory and in the presence of which the classical solutions will change in significant and important ways.

Having defined what we mean by the classical solution of the theory on each finite lattice, we turn to the limit $N\rightarrow \infty$. We must now introduce into the theory a dimensional scale, that is, a notion of distance in our structure. We assume that, in a certain given system of physical units, external to our model and to be added to our lattice structure, the length of a side of the lattice, which is formed by $N$ consecutive sites and links in a given direction, has the value $L$, a quantity with dimensions of length in that external system of units. In what follows we will make some choices as to the type of limit we will consider here. Both here and in the quantum theory it is possible to take the limit $N\rightarrow \infty$ of the models in several different ways, depending on what is done with the parameters of the model during the limiting process.

Figure 2.1.1: The geometrical elements of a periodical two-dimensional lattice.

Let us imagine that $L$ remains fixed and finite during the limit, which means that we are taking the limit in such as way that our lattice remains perfectly fitted within a cubic box with periodical boundary conditions and volume $V=L^{d}$. We will also make the parameter $\alpha_{0}$ go to zero in the limit, in a certain well-defined way. In addition to this we will assume that the lattice remains symmetrical and homogeneous from the point of view of the lengths induced by the introduction of the external scale $L$ into the theory. With all these assumptions made, we may now define dimensionfull versions of each one of the original dimensionless elements of our structure. For example, all the links of the lattice now have the same length, which we will denote by $a$, which is related to $L$ by $L=Na$, and which goes to zero in the limit $N\rightarrow \infty$. The volume of the box which contains the lattice may now be divided into $N^{d}$ disjoint cubes of volume $a^{d}$, whose union reconstitutes the total volume, as shown in figure 2.1.1.

We may now write the action $S_{0}$ in the form

$$S_{0}[\varphi]=\frac{1}{2}\sum_{s}a^{d}\sum_{\mu} \left[\fra... ...}{2a^{2}}\sum_{s}a^{d} \left[a^{(2-d)/2}\varphi(s)\right]^{2},$$

where all the factors of $a$ that we introduced cancel out. Note that the sums over the sites combined with the factors $a^{d}$ approach Riemannian integrals over the volume of the box. In order to make $\alpha_{0}$ go to zero in the limit, as mentioned before that we must, we choose the relation $\alpha_{0}=(m_{0}a)^{2}$ for some finite $m_{0}$. Besides this, we define the dimensionfull version $\phi$ of the field as $\phi=a^{(2-d)/2}\varphi$, with which we may write for $S_{0}$, still on a finite lattice,

$$S_{0}[\phi]=\frac{1}{2}\sum_{s}a^{d}\sum_{\mu} \left[\frac{\... ...)}{a}\right]^{2} +\frac{m_{0}^{2}}{2}\sum_{s}a^{d}\phi^{2}(s).$$

At this point it becomes clear that, since in the $N\rightarrow \infty$ limit with constant $L$ we have $a\rightarrow 0$, the sums indeed approach integrals over the volume of the lattice, with integration element $dv={\rm d}^{d}x=a^{d}$, while the ratios between the finite differences of the field and $a$ approach partial derivatives $\partial_{\mu}\equiv\partial/\partial x_{\mu}$. In short, we may write for $S_{0}$, in this limit, the expression

$$S_{0}[\phi]=\frac{1}{2}\int_{V}{\rm d}^{d}x\sum_{\mu} \left[... ...} +\frac{m_{0}^{2}}{2}\int_{V}{\rm d}^{d}x\;\phi^{2}(\vec{x}).$$

We will refer to this limit as the continuum limit of the classical theory, because in it the lattice spacing $a$ goes to zero while the lattice acquires an increasing number of points and tends to occupy densely the volume $V$ of the interior of the box. The object that results from this process is the usual classical theory of the free scalar field within a box with periodical boundary conditions. In the limit the dimensionfull coordinates $0\leq x_{\mu}\leq L$ describe the continuous interior of the box and, on finite lattices, they relate to the dimensionless coordinates $n_{\mu}$ by $x_{\mu}=n_{\mu}a$.

The functional $S_{0}[\phi]$ above is the usual action that defines the classical dynamics of the free scalar field within a box. Note that, for the dimensionfull mass parameter $m_{0}$ that appears in the second term to be finite in the limit, it is necessary that the dimensionless parameter $\alpha_{0}$ go to zero as $1/N^{2}$, for any value of the mass in the limit. This type of behavior for the dimensionless parameters of the theory is very general. Usually there is a particular set of values of the parameters of the theory that they must approach in any continuum limit which is to be of physical interest. We refer to these special values as critical, for reasons that will become clearer later. In our case here the value $0$ is a critical point of the parameter $\alpha_{0}$.

In this continuum limit the classical solution of the model is given by the Euler-Lagrange equation, which in this case is no more than a generalization of the $d$-dimensional Laplace equation, including the mass term. We can derive this equation by means of the direct application of the principle of minimum action. In order to do this we make a generic variation $\delta\phi(\vec{x})$ of the fields, which is infinitesimal but may be different in each point, and then determine the condition that the field must satisfy so that the action does not change to first order, as a consequence of this variation. Calculating the variation $\delta S_{0}$ to first order in $\delta\phi$ we obtain

$$\delta S_{0}[\phi]=\int_{V}{\rm d}^{d}x\left\{\sum_{\mu} [\p... ...\vec{x})] +m_{0}^{2}\phi(\vec{x})\delta\phi(\vec{x}) \right\}.$$

Using now the easily verifiable fact that $\delta[\partial_{\mu}\phi(\vec{x})]=\partial_{\mu}[\delta\phi(\vec{x})]$ we obtain

$$\delta S_{0}[\phi]=\int_{V}{\rm d}^{d}x\left\{\sum_{\mu} [\p... ...\vec{x})] +m_{0}^{2}\phi(\vec{x})\delta\phi(\vec{x}) \right\}.$$

We may now integrate the first term by parts. There is no surface term, due to the periodical boundary conditions, and we therefore have

$$\begin{eqnarray*} \delta S_{0}[\phi] & = & \int_{V}{\rm d}^{d}x \left\{-\sum_{\m... ... \partial^{2}_{\mu}\phi(\vec{x})+m_{0}^{2}\phi(\vec{x})\right\}. \end{eqnarray*}$$

If we now impose the condition of minimum for $S_{0}$, that is, that $\delta S_{0}=0$ to first order for any variation $\delta\phi(\vec{x})$, we obtain the relation

  $$-\partial^{2}\phi+m_{0}^{2}\phi=0,$$ (2.1.1)

where $\partial^{2}=\sum_{\mu}\partial_{\mu}\partial_{\mu}$ is the Laplacian operator in $d$ dimensions. We refer to this equation, using the usual terminology of physics, as the equation of motion, although it may have nothing to do with movement, for example in the three-dimensional case, in which there is no temporal coordinate. The non-Euclidean version of this equation, in the case $d=4$, is known as the Klein-Gordon equation and is related to the relativistic dynamics of free particles with mass $m_{0}$ and spin zero.

Observe that it is also possible to derive an equation corresponding to this one on finite lattices, because the integration by parts which is used for the derivation of this equation in the classical continuum theory has an exact counterpart on finite lattices. In order to see this we write explicitly the term containing the derivatives, for simplicity in only one dimension,

$$\begin{eqnarray*} \lefteqn{\sum_{l}\left(\Delta_{\ell}\varphi\right)^{2}} && \\ ... ... -\sum_{n}\varphi_{n}(\varphi_{n-1}-2\varphi_{n}+\varphi_{n+1}), \end{eqnarray*}$$

where we denoted the dependency on the position by means of indices, for simplicity of notation. With a detailed examination of the algebraic passages illustrated above it becomes clear that the regrouping of the terms can be done all around the circle, resulting in the final form, which relates a sum over links with a sum over sites,

$$\sum_{l}\left(\Delta_{\ell}\varphi\right)^{2} =-\sum_{s}\varphi(s)\Delta^{2}\varphi(s),$$

where the Laplacian operator on finite one-dimensional lattices is defined as

$$\Delta^{2}\varphi(n)=\varphi(n-1)-2\varphi(n)+\varphi(n+1).$$

Note that the Laplacian has values naturally defined on sites, like the field, not on links. The generalization of this definition to lattices of higher dimensions is immediate, the algebraic operation described can be repeated on all the directions and therefore it suffices to add a sum over the directions,

  $$\Delta^{2}\varphi(\vec{n})=\sum_{\mu} [\varphi(n_{\mu}-1)-2\varphi(n_{\mu})+\varphi(n_{\mu}+1)].$$ (2.1.2)

With this we have on finite lattices the equation of movement which determines the classical solution, whose derivation will be left to the reader (problem 2.1.1),

  $$-\Delta^{2}\varphi+\alpha_{0}\varphi=0.$$ (2.1.3)

In three dimensions and in the continuum limit the zero-mass version of our equation of movement reduces to the Laplacian on the torus, since we are using here periodical boundary conditions. This is a rather familiar situation, since it is just the electrostatics of a torus without internal charges. Something analogous to this happens in the case of four dimensions, in which we obtain an Euclidean version of the wave equation for the scalar potential, which is also a part of classical electrodynamics. If we write equation (2.1.1) explicitly in four dimensions we obtain

$$-\partial_{x}^{2}\phi-\partial_{y}^{2}\phi-\partial_{z}^{2}\phi -\partial_{t}^{2}\phi+m_{0}^{2}\phi=0,$$

where $x$, $y$ and $z$ are the three spacial Cartesian coordinates and $t$ corresponds to the time. The process of passing from Euclidean space to Minkowski space can be effected by the exchange of $t$ for $\imath t$ in this expression, which takes us to

$$-\vec{\nabla}^{2}\phi+\partial_{t}^{2}\phi+m_{0}^{2}\phi=0.$$

where $\vec{\nabla}^{2}=\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2}$ is the three-dimensional Laplacian. In the case $m_{0}=0$ this is the usual wave equation. In general, the passage to Minkowski space is effected identifying within the answers obtained in Euclidean space the metrical tensor $g_{\mu\nu}=\delta_{\mu\nu}$ of this space and changing the sign of its diagonal term which corresponds to the temporal coordinate. In the example above we may write the equation in Euclidean space as

$$-\sum_{\mu,\nu}g_{\mu\nu}\partial_{\mu}\partial_{\nu}\phi+m_{0}^{2}\phi=0$$

and making the transformation to Minkowski space by transforming the metric

$$g_{\mu\nu}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & ... ...& 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{array}\right).$$

This process of de-Euclideanization may always be realized in this fashion, either in position space or in momentum space.

Problems

1. Derive the equation of movement (2.1.3) on finite lattices, applying the principle of minimum action to the action of the free theory given in equation (1.3.1).