Finite-Difference Operators

We will now elaborate a little the notion of operators that act on the lattice, related to finite differences of the fields. In the usual numerical methods for the solution of differential equations these operators are looked at as approximations on finite lattices for the corresponding objects in the continuum limit, which are differential operators. In order to define the action functionals to be used in the quantum theory it is useful to first study the relation between the discrete and continuum objects in the scope of the classical theory. We will then, in this section, imagine that we are starting from the continuum objects and trying to model them, approximating them by discrete objects. It is clear that this is not our main point of view, but there is no harm in adopting it temporarily to illustrate the discussion of the nature of the relation between these mathematical objects.

Let us recall, then, that we are only talking about the classical theory of fields, described on finite lattices, with the intention of taking the classical continuum limit. It is therefore implicitly understood in all this discussion the existence of an external dimensional scale $L$ and of the corresponding lattice spacing $a$. For ease of presentation, we will use in this section the dimensional coordinates $\vec{x}$ and the corresponding versors $\hat{x}_{\mu}$, for each one of the $\mu=1,\ldots,d$ direction of the space. Let us consider now in more detail the definition on the lattice of the finite difference operator $\Delta_{\mu}$. Considered as an approximation of the differential operator “partial derivative” it may be represented in several different ways, for example as the forward difference operator $\Delta_{\mu}^{(+)}$, which when applied to a function $f(\vec{x})$ on the lattice produces

$$\Delta_{\mu}^{(+)}f(\vec{x})=f(\vec{x}+a\hat{x}_{\mu})-f(\vec{x}),$$

or as the backward difference operator $\Delta_{\mu}^{(-)}$, which is defined by

$$\Delta_{\mu}^{(-)}f(\vec{x})=f(\vec{x})-f(\vec{x}-a\hat{x}_{\mu}),$$

or even as the symmetrical difference operator or central difference operator $\Delta_{\mu}^{\rm (c)}$, which is given by

$$\Delta_{\mu}^{\rm (c)}f(\vec{x})= \frac{1}{2}[f(\vec{x}+a\hat{x}_{\mu})-f(\vec{x}-a\hat{x}_{\mu})].$$

The reason for the existence of all these different representations is that the finite difference operator is in fact an operator with its arguments on sites and its values on links, and there is no unique or natural way to represent it only on sites. Note the absence of factors of $a$ in the denominator in these definitions. In the continuum limit the partial differentiation operator $\partial_{\mu}$ may be identified with $\Delta_{\mu}/a$ in any of the forms given above,

$$\partial_{\mu} =\lim_{a\rightarrow 0}\frac{\Delta_{\mu}^{(+)... ...)}}{a} =\lim_{a\rightarrow 0}\frac{\Delta_{\mu}^{\rm (c)}}{a}.$$

It is clear that the operator $\partial_{\mu}$ can only be applied to differentiable functions in the continuum limit, unlike the realizations of $\Delta_{\mu}$ on the lattice, which can be applied to any functions defined on the lattice.

Our action functional $S_{0}$ depends only on the field and its first derivatives (or rather, finite differences), therefore we are ready to write it in a more explicit way. It can be verified that from the use of either the $\Delta_{\mu}^{(+)}$ form or the $\Delta_{\mu}^{(-)}$ form of the finite-difference operator results the same action on the lattice, with only next-neighbor couplings among the values of the field at the various sites,

$$S_{0}[\varphi] =\frac{1}{2}\sum_{\ell}\left[\Delta^{(+)}_{\e... ...varphi\right]^{2} +\frac{\alpha_{0}}{2}\sum_{s}\varphi^{2}(s).$$

In both these cases, expanding the square of the finite-difference operator, one may write $S_{0}$ as (problem 2.3.1)

$$S_{0}[\varphi]= -\sum_{\ell}\left[\varphi(\vec{x})\varphi(\v... ...eft(d+\frac{\alpha_{0}}{2}\right)\sum_{s}\varphi^{2}(\vec{x}).$$

What is understood by “next-neighbor couplings” is that, in the terms of the action which involve products of values of the field at different sites, the values involved are in adjacent sites, connected by a link. Note that the inter-site interaction that appears above is similar to an interaction between two spins at neighboring sites, such as the one that appears in the Ising model of magnetism. This elementary interaction among neighboring sites should not be confused with what is usually referred to in the theory as “interaction terms”, which supposedly couple together the Fourier modes of the fields and thus cause plane waves to interact with each other, giving rise to scattering processes in the non-Euclidean version of the theory. These interaction terms will be discussed in a future volume, but are outside the scope of this book.

The use of the realization $\Delta_{\mu}^{\rm (c)}$, however, produces from the continuum action a lattice action with more than interactions between next-neighbor sites. The association of the finite-difference operator with links is intuitive and very attractive, as well as closely related to the realization of the lattice of gauge theories, as will be seen in a future volume. This fact, plus the simplicity of having to deal only with couplings between next neighbors, would be sufficient to decide the question as to the type of finite-differencing scheme to choose for the definition of the theory on the lattice. But we will see in what follows, in a direct way, that the use of $\Delta_{\mu}^{\rm (c)}$ may also be reduced to exactly the same theory on the lattice, and that the finite-difference operator is inevitably defined on links.

In order to see this we consider the integration by parts of the first term of the action $S_{0}$ which, as commented previously, is only a not too appropriate name for an algebraic operation involving sums of differences along the lattice. In this way we may write the action as

$$S_{0}[\varphi] =-\frac{1}{2}\sum_{s,\mu}\varphi(\vec{x})\Del... ...phi(\vec{x}) +\frac{\alpha_{0}}{2}\varphi^{2}(\vec{x})\right].$$

In spite of the fact that in order to do this with the usual mathematical language of the continuum it may seem necessary to interpret the term $(\Delta_{\mu}\varphi)^{2}$ in a mixed way, with different realizations of the finite-difference operator $\Delta_{\mu}$ for each factor involved, for example as $(\Delta_{\mu}^{\rm (c)}\varphi)(\Delta_{\mu}^{(+)}\varphi)$, integration by parts is just an application of the Stokes theorem and it therefore a simplicial operation which is exact on the lattice. The form above for the action is exactly equal to the previous ones, with the use of any consistent realization of $\Delta_{\mu}$. With the use of $\Delta_{\mu}^{(\pm)}$ the Laplacian operator $\Delta^{2}$ that appears above may be defined through its action on lattice functions as

$$\Delta^{2}f(\vec{x})=\sum_{\mu}\Delta_{\mu}^{2}f(\vec{x}),$$

where the definition of the action on the functions of the “second difference” $\Delta_{\mu}^{2}$ in the direction $\mu$ is given by (no sum over $\mu$ here)

$$\Delta_{\mu}^{2}f(\vec{x})= f(\vec{x}+a\hat{x}_{\mu})-2f(\vec{x})+f(\vec{x}-a\hat{x}_{\mu}),$$

resulting therefore in

$$\Delta^{2}f(\vec{x})=\sum_{\mu} [f(\vec{x}+a\hat{x}_{\mu})-2f(\vec{x})+f(\vec{x}-a\hat{x}_{\mu})].$$

Note that these operators are naturally defined with values on sites, as is necessary for the expression of the action in terms of the Laplacian. The second derivative maps values of the function $f$ at the points $\vec{x}$, $\vec{x}+a\hat{x}_{\mu}$ and $\vec{x}-a\hat{x}_{\mu}$ to a resulting value to be associated to the point $\vec{x}$. One can show (problems 2.3.2 and 2.3.3) that the second-derivative operator is obtained by the iterated application of $\Delta_{\mu}^{(+)}$ (which maps $\vec{x}+a\hat{x}_{\mu}$ and $\vec{x}$ on $\vec{x}$) and $\Delta_{\mu}^{(-)}$ (which maps $\vec{x}-a\hat{x}_{\mu}$ and $\vec{x}$ on $\vec{x}$).

It is necessary to emphasize at this point that the iteration of $\Delta_{\mu}^{\rm (c)}$ does not produce the operator $\Delta^{2}$ defined above but, instead of that, results in a different realization of it, $\Delta^{2}_{\rm (c)}$, related to the differences of second order given by

$$\Delta^{2}_{\rm (c)}f(\vec{x})= \frac{1}{4}\sum_{\mu}[f(\vec{x}+2a\hat{x}_{\mu}) -2f(\vec{x})+f(\vec{x}-2a\hat{x}_{\mu})].$$

In the context of the classical theory this is related to a higher-order approximation $\Delta^{2}_{\rm (c)}$ to the continuum operator $\Delta^{2}$. Note that, with its use, the action $S_{0}$ would involve more than couplings between next neighbors. It is clear that our point here is not to obtain better approximations to the solutions of the classical theory and we will never use these higher-order realizations of the finite-difference operator. For us the important realizations are those that appear in the various forms of the action that we have already seen, which may be obtained by the direct application of the realization on the links, be it in the form $\Delta_{\mu}^{(+)}$ or in the form $\Delta_{\mu}^{(-)}$.

Figure 2.3.1: A one-dimensional double lattice.

Of course we could consider the definition of the theory with the finite-difference operator $\Delta_{\mu}^{\rm (c)}$ defined on sites and the higher-order realization $\Delta^{2}$ given above. The interesting thing is that the theory would still be the same in any case, and that we would be simply rescaling the continuum limit by a factor of two. In fact one can verify that, in $d$ dimensions with even $N$ and periodical boundary conditions, the use of $\Delta_{\mu}^{\rm (c)}$ and $\Delta^{2}_{\rm (c)}$ corresponds exactly to the representation on the lattice of $2^{d}$ simultaneous and non-interacting copies of the same model, with a lattice spacing parameter rescaled by two, equal therefore to $2a$. For odd $N$ the use of the central finite-difference operator will produce a multiple cover of the torus and we end up with a single realization of the model wrapped up $2^{d}$ times around the torus, with both $a$ and $L$ rescaled by a factor of two. Figure 2.3.1 may help in the visualization of these facts in the case $d=1$. In the figure the lines with arrows point to the two subsets of sites which are related by the dynamics of the theory. In one dimension one of the two sets is the set of sites with even integer coordinates, and the other is the set of sites with odd integer coordinates. According to the dynamics of the theory, each one of these two sets interacts internally, but each one of them does not interact at all with the other one (problems 2.3.4 and 2.3.5).

Observe that this realization by means of $\Delta_{\mu}^{\rm (c)}$ does complicate the counting of the degrees of freedom of the models. Where we thought we had a single field value per site we end up with $2^{d}$ independent field values, which are non-interacting or interacting only through the boundary conditions. It is interesting to observe that the realization on the lattice of fermionic fields also involves multiplications of the spectrum of particles by factors of $2^{d}$, a phenomenon that remains as one of the main open problems for that type of field. In that case the problem appears in momentum space rather than position space, but it seems likely that in that case too the problem may reduce to the question of counting degrees of freedom.

We see therefore that in these realizations the finite-difference operator ends up once more associated in a natural way to links, now with length $2a$, a fact that leads us to think that the association of finite differences to links has a certain character of inevitability. The same is true for more complex models such as, for example, the polynomial models and the sigma models, so long as consistent use is made in them of either the $[\Delta_{\mu}^{(\pm)},\Delta^{2}]$ or the $[\Delta_{\mu}^{\rm (c)},\Delta^{2}_{\rm (c)}]$ realizations. Since we do not have any interest in having to deal with several identical copies of the same model sharing the same lattice, in what follows we will restrict the discussion to only the case of next-neighbor couplings.

Problems

1. Show, by expanding the term that contains the square of the finite-difference operator, that the action $S_{0}$ of the free scalar field can be written as

   
   

   $$S_{0}[\varphi]= -\sum_{\ell}\left[\varphi(\vec{x})\varphi(\v... ...eft(d+\frac{\alpha_{0}}{2}\right)\sum_{s}\varphi^{2}(\vec{x}).$$
   
   

2. Show that one can obtain the second-difference operator $\Delta_{\mu}^{2}$ by the iteration of the forward-difference operator $\Delta_{\mu}^{(+)}$ and the backward-difference operator $\Delta_{\mu}^{(-)}$, in any order, that is, show that

   
   

   $$\Delta_{\mu}^{2}=\Delta_{\mu}^{(+)}\Delta_{\mu}^{(-)} =\Delta_{\mu}^{(-)}\Delta_{\mu}^{(+)}.$$
   
   

   Remember that $\Delta_{\mu}^{(+)}$ maps $\vec{x}+a\hat{x}_{\mu}$ and $\vec{x}$ to $\vec{x}$ and that $\Delta_{\mu}^{(-)}$ maps $\vec{x}-a\hat{x}_{\mu}$ and $\vec{x}$ to $\vec{x}$, and consider in detail the action of these operators over lattice functions, calculating for example $\Delta_{\mu}^{(+)}[\Delta_{\mu}^{(-)}f(\vec{x})]$ and $\Delta_{\mu}^{(-)}[\Delta_{\mu}^{(+)}f(\vec{x})]$.

3. Show that the twice-repeated iteration of either $\Delta_{\mu}^{(+)}$ or $\Delta_{\mu}^{(-)}$ does not reproduce the second-difference operator as defined in the text, that it, show that

   
   

   $$\Delta_{\mu}^{(+)}\Delta_{\mu}^{(+)}\neq\Delta_{\mu}^{2}\neq \Delta_{\mu}^{(-)}\Delta_{\mu}^{(-)}.$$
   
   

   Verify in each case that, with respect to the definition given in the text, there is a displacement of the point to which the value of the operator should be associated, and examine the consequences of this displacement in the case of fixed boundary conditions, and in the case of periodical boundary conditions.

4. Consider the action $S_{0}[\varphi]$ of the free scalar field written on a lattice with $N=2N'$ sites in one dimension, with the use of the central finite-difference operator $\Delta_{(c)}$. Let $s_{o}$ run over the $N'$ sites with odd integer coordinates and $s_{e}$ over those with even integer coordinates. Show that it is possible to write the action as

   
   

   $$S_{0}[\varphi]=S_{o}[\varphi]+S_{e}[\varphi],$$
   
   

   where $S_{o}[\varphi]$ depends only on the fields $\varphi(s_{o})$ at the odd sites and $S_{e}[\varphi]$ only on the fields $\varphi(s_{e})$ at the even sites. In this way one sees that the classical dynamics of the system separates in two independent parts.

5. Apply the Euler-Lagrange equation or the principle of minimum action to the actions $S_{o}[\varphi]$ and $S_{e}[\varphi]$ of problem 2.3.4 and show that the equations of motion relative to each one of the two sets of sites do not involve at all the variables at the sites of the other set. In this way one sees that the classical dynamics of the system decouples into independent dynamics for each one of the two sets of sites.