Space of Field Configurations

Linear operators and the spaces on which they act can in general be represented by matrices and vectors, respectively. Having constructed the Laplacian operator both on finite lattices and in the continuum limit, we will now show how to represent it in matrix form on finite lattices. We will have in this way a very concrete representation both of the operators and of the vectors that constitute the space on which they act. This representation is very useful as a concrete mental image of the objects under study, as well as a computational tool, which can be used to good advantage in the context of the stochastic methods which constitute the main calculational approach to the quantum theory.

In order to do this we will introduce and develop to some extent the concept of the space of field configurations, which will be of great importance for the quantum theory. The space of configurations, as previously defined, is the space of all possible field-functions on the lattice. This is in fact a different space for each lattice size, so we are actually talking here of a set or sequence of spaces. For a real field with a single component on a lattice of $N^{d}$ sites in $d$ dimensions this space is $\mathbb{R}^{(N^{d})}$.

For simplicity we will do the construction for the case of periodical boundary conditions. We shall start with the case $d=1$, which is very simple. In this case we can represent each configuration of the field by the collection of its values at the sites along the lattice, in the order in which they appear on the linear chain defined by it,

$$\left[ \begin{array}{c} \varphi_{1} \\ \cdot \\ \cdot \... ...\\ \cdot \\ \cdot \\ \varphi_{N} \\ \end{array}\right].$$

Any function defined on the lattice can be represented by a vector like this one, with absolutely no additional restriction imposed on the nature of the functions. The $n=N^{1}$ components of the vector are the $n$ values assumed by the function on the $n$ sites existent on the lattice. In this simple case the index $i$ of the vector that represents the field is simply the integer coordinate $n_{1}$ of the one-dimensional lattice. Further along the development of the theory one may have scalar fields $\varphi_{a}$ with more than a single component, that is, vectors $\vec{\varphi}$ in some internal symmetry space of the fields, but one may still use this kind of representation for the fields in the space of configurations. One will simply have several vectors like the one illustrated above, one for each field component in the internal space. The ideas involved in the representation of the field as a function over the lattice do not change, the field simply acquires an additional index that does not mix with those referring to space-time.

Is is not difficult to verify that the Laplacian operator is represented by the following matrix in this space formed by the vectors containing the values of the fields at the sites,

  $$\Delta^{2}= \left[ \begin{array}{ccccccccc} -2 & 1 & 0 & 0 &\cdo... ...1 & -2 & 1 \\ 1 & 0 & 0 & 0 &\cdots & 0 & 0 & 1 & -2 \\ \end{array}\right].$$ (2.4.1)

Note that almost all elements are zero, thus indicating the absence of long-range couplings on the lattice. The only non-zero elements are the three central diagonals indicated and the two elements at the corners along the anti-diagonal. These two elements are the ones that establish the periodical boundary conditions. It suffices to apply this matrix to the vector that represents the field to verify that it reproduces the definition of the Laplacian on finite lattices given in (2.1.2).

The same procedure can be realized on lattices with dimensions larger than $1$, however in this case there is a slight complication because if the lattice is not one-dimensional then it does not establish a natural order for the sites. We will establish here a standard order for writing the components of vectors and matrices in the space of field configurations, which will also be very useful for computer simulations. Starting with a two-dimensional lattice with integer coordinates $\vec{n}=(n_{1},n_{2})$, where $n_{1},n_{2}=1,\ldots,N$, consider the integer $\iota$ defined by

$$\iota=1+(n_{1}-1)+(n_{2}-1)N.$$

Observe that $\iota$ varies from $1$ to $N^{2}$ when $n_{1}$ and $n_{2}$ assume all the possible values, from $1$ to $N$. The interesting thing is that this operation is invertible, that is, given a certain value of $\iota$ between $1$ and $N^{2}$ it is possible to determine uniquely the values of $n_{1}$ and $n_{2}$ that correspond to it. Hence, $\iota$ enumerates all the sites in a definite way. This operation, which we call the indexing of the lattice by the index $\iota$, is similar to the operation of writing integers in the base $N$. The inversion of the operation is realized by means of integer division. One can show (problem 2.4.1) that

$$n_{2}=1+\frac{\iota-1}{N},$$

where the division is integer division, that is, there is truncation of the result as is usual in the integer arithmetic of digital computers. Note that due to this the order of operations is important here. Once $n_{2}$ is obtained, we have for $n_{1}$

$$n_{1}=\iota-(n_{2}-1)N.$$

In this way we see that both the pair $(n_{1},n_{2})$ and the index $\iota$ determine uniquely a site. We may now use $\iota$ as the index of the vector that will represent the field on a two-dimensional lattice,

$$\left[ \begin{array}{c} \varphi_{1} \\ \cdot \\ \cdot \... ... \cdot \\ \cdot \\ \varphi_{N^{2}} \\ \end{array}\right],$$

that is, we may represent the function $\varphi(n_{\mu})$ by the vector with components $\varphi_{\iota}$.

This method can be immediately generalized to any dimension $d$. For example in three dimensions, with integer coordinates $(n_{1},n_{2},n_{3})$, the index is given by

$$\iota=1+(n_{1}-1)+(n_{2}-1)N+(n_{3}-1)N^{2},$$

and the inversion operations are (problem 2.4.2)

$$\begin{eqnarray*} n_{3}&=&1+\frac{(\iota-1)}{N^{2}}, \\ n_{2}&=&1+\frac{(\iota-... ...-1)N^{2}}{N}, \\ n_{1}&=&1+(\iota-1)-(n_{3}-1)N^{2}-(n_{2}-1)N. \end{eqnarray*}$$

At this point it is already possible to see the pattern and imagine how to generalize this. In general the index, on a lattice in $d$ dimensions with integer coordinates $n_{\mu}$, will be given by

$$\iota=1+(n_{1}-1)+(n_{2}-1)N+\ldots +(n_{d-1}-1)N^{d-2}+(n_{d}-1)N^{d-1} =1+\sum_{\mu=1}^{d}(n_{\mu}-1)N^{\mu-1},$$

where $\iota$ varies from $1$ a $N^{d}$ along all the extent of the lattice and the recursive inversion operations are an immediate generalization of the relations for the three-dimensional case, given above. For $n_{1}=\ldots=n_{d}=1$ we immediately have $\iota=1$ and for $n_{1}=\ldots=n_{d}=N$ we have $\iota=N^{d}$, as we may verify by inspection,

$$\left. \begin{array}{cc} & (n_{d}-1)N^{d-1} \\ + & (n_{d-... ... + & N & - & 1 \\ + & 1 & & \\ \end{array}\right\}=N^{d}.$$

As one can see, the relation between the index $\iota$ and the integer coordinates is in fact a bijection because, besides defining $\iota$ from a set of integer coordinates $n_{\mu}$, given a value for $\iota$ we may also solve for all the coordinates $n_{\mu}$, dividing $\iota-1$ successively by $N^{d-1},N^{d-2},\ldots,N^{1},N^{0}$, so as to recover each one of the $n_{\mu}$ as the complements of the remainders of the successive divisions. This is, therefore, an unequivocal way to pile up all the sites of a $d$-dimensional lattice into a single vector of size $n=N^{d}$. The same ordering procedure can and should be used for the elements of the matrices acting on these vectors.

The linear operators that act on this space are representable, of course, by $n\times n$ matrices, as we saw before for the Laplacian in one dimension, with periodical boundary conditions. One can verify that the determinant of that matrix is zero (problem 2.4.3), which is a consequence of the fact that the operator $\Delta^{2}$ has a null eigenvector, or a zero mode, on the torus. The matrix form of the operator has a global character, including in its structure the boundary conditions which are adopted. For example, the Laplacian in one dimension with fixed boundary conditions, where each integer coordinate varies from $0$ to $N+1$ and we have a total of $n'=(N+2)^{d}$ sites, is represented by the $n'\times n'$ matrix

  $$\Delta^{2}= \left[ \begin{array}{ccccccccc} 1 & 0 & 0 & 0 &\cdot... ... 1 & -2 & 1 \\ 0 & 0 & 0 & 0 &\cdots & 0 & 0 & 0 & 1 \\ \end{array}\right].$$ (2.4.2)

In this case the operator is, in fact, an operator that acts on a space of dimension $n'$ with values in a sub-space of dimension $n=N^{d}$, since it does not make sense to calculate the Laplacian at the sites of the fixed border, at which we defined it to act as the identity, in order to complete the square matrix above. On the other hand, one can verify (problem 2.4.4) that the determinant of this matrix is not zero, but $(n'-1)(-1)^{n'}$ instead, reflecting the fact that there is no zero-mode for fixed boundary conditions.

Back to the case of periodical boundary conditions, the Euclidean Klein-Gordon operator $-\Delta^{2}+m^{2}$ is represented in this case by the matrix

$$-\Delta^{2}+m^{2}= \left[ \begin{array}{ccccccc} 2+m^2 & -1... ... -1 & 0 & 0 &\cdots & 0 & -1 & 2+m^2 \\ \end{array}\right],$$

which has a non-zero determinant so long as the mass is not zero. This corresponds to the fact that only the theory of the free field with zero mass has a zero mode on the torus and, potentially, problems due to the occurrence of divergences in the infra-red limit, in which one makes the size $L$ of the box tend to infinity, thus including into the structure of the models arbitrarily large wavelengths and, therefore, arbitrarily low frequencies.

Still for periodical boundary conditions, the forward difference operator $\Delta_{\mu}^{(+)}$ has the matrix representation

$$\Delta_{\mu}^{\left(+\right)}= \left[ \begin{array}{cccccccc... ...1 & 0 & 0 & 0 &\cdots & 0 & 0 & 0 & -1 \\ \end{array}\right],$$

and the backward difference operator $\Delta_{\mu}^{(-)}$ the representation

$$\Delta_{\mu}^{\left(-\right)}= \left[ \begin{array}{cccccccc... ...0 & 0 & 0 & 0 &\cdots & 0 & 0 & -1 & 1 \\ \end{array}\right].$$

It can be easily verified (problem 2.4.5), in this one-dimensional case, that they are related to $\Delta^{2}$ by

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

In this way the treatment, both of the fields and of the action on them of linear operators such as the Laplacian, can always be reduced in an explicit way to operations with vectors and matrices in a space with a large but finite dimension, the space of field configuration on the lattice. This is specially useful in programs for the execution of stochastic simulations of models in the quantum theory.

Problems

1. Show that, if the index $\iota$ for a two-dimensional lattice is defined as $\iota=1+(n_{1}-1)+(n_{2}-1)N$, then one can recover from it the integer coordinates $n_{1}$ and $n_{2}$ by means of the operations

   
   

   $$n_{2}=1+\frac{\iota-1}{N}\mbox{~~~~and~~~~}n_{1}=\iota-(n_{2}-1)N,$$
   
   

   in the indicated order, where the division is an integer division, that is, there is truncation of the result as is usually the case in the integer arithmetic of digital computers.

2. Repeat the demonstration described in problem 2.4.1 for the case of three dimensions, in which the index is defined as $\iota=1+(n_{1}-1)+(n_{2}-1)N+(n_{3}-1)N^{2}$ and the inversion operations are, in order,
   $$\begin{eqnarray*} n_{3}&=&1+\frac{(\iota-1)}{N^{2}}, \\ n_{2}&=&1+\frac{(\iota-... ...-1)N^{2}}{N}, \\ n_{1}&=&1+(\iota-1)-(n_{3}-1)N^{2}-(n_{2}-1)N. \end{eqnarray*}$$
   
   
   
   

3. Show, in the case $d=1$ on a lattice with periodical boundary conditions, that the determinant of the Laplacian given in equation (2.4.1) is zero, for any value of $N$.

4. Show, in the case $d=1$ on a lattice with fixed boundary conditions, that the determinant of the Laplacian given in equation (2.4.2) has its value given by $(n'-1)(-1)^{n'}$, where $n'=(N+2)^{d}$.

5. Show, in the one-dimensional case, executing explicitly the matrix products, that the iteration of $\Delta^{(+)}$ and $\Delta^{(-)}$, in any order, has $\Delta^{2}$ as the result,

   
   

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

6. Write explicitly the matrix of the Laplacian operator on a lattice with $N=4$, in two dimensions, with periodical boundary conditions.

7. Write the operations for the inversion of the index for a lattice in $d=4$, with periodical boundary conditions.

8. This one is just for fun: write explicitly the matrix of the Laplacian operator on a one-dimensional lattice with $N=2$ and periodical boundary conditions.