Momentum Space

We will now define a concept which will be of great importance both for the solution of the mathematical problems of the theory and in relation to its physical interpretation, even when it is not possible to find exact solutions. This is the concept of momentum space. The fields, as defined so far, are written as functions of the sites $s$ and hence of their integer coordinates $\vec{n}$. As we saw before, we refer to the space of all possible fields as the space of field configurations. We will call this representation of the fields, as functions of the sites, by the name position space. Another representation of the fields exists in momentum space, which is obtained by means of a linear transformation that effects a change of basis in the space of configurations. The integer coordinates $\vec{n}$ of the sites will be mapped on a new set of integer coordinates $\vec{k}$ that index what we will call the modes of the lattice, in a way similar to the indexing of the sites by $\vec{n}$. The name originates from the classical concept of normal modes of oscillation.

We will begin by showing a simple property of exponentials with discrete arguments involving integers. Let $1\leq n\leq N$ be an integer and consider the exponential function

$$e^{\imath 2\pi n/N}$$

for the set of $N$ possible values of the argument. We have here complex phases, which means that this function assumes values along the unit circle of the complex plane. As $n$ varies from $1$ to $N$ the function goes around the circle, defining along it $N$ equally spaced points. In figure 2.5.1 we have an example with $N=8$, with the $N$ values of $n$ marked for each phase.

Figure 2.5.1: Example of a complex unit circle for $N=8$.

Figure 2.5.2: Example of a complex unit circle for $N=7$.

Note that, since the sum of the complex numbers is equivalent to the sum of the two-dimensional vectors shown in the figure, its symmetry implies that

$$\sum_{n=1}^{N}e^{\imath 2\pi n/N}=0.$$

Note that this argument does not depend on the parity of $N$, but only on the fact that the vectors are equally spaced along the circle. In figure 2.5.2 we show an example with $N=7$.

We will now consider the slightly more complicated case in which we multiply the argument of the exponential by another integer $0\leq k\leq N-1$, obtaining the function

$$e^{\imath 2\pi k n/N}.$$

In this case, as $n$ varies from $1$ to $N$ the function goes around the circle exactly $k$ times. In the case $k\neq 0$ we will still be defining in this way sets of points equally spaced along it. For example, for $k=2$ we have, for the two values of $N$ used before, the phases shown in figures 2.5.3 and 2.5.4.

As one can see, in the case $N=7$ all the possible phases end up occupied a single time, as before, but in a different order. In the case $N=8$ only one half of the possible phases ends up occupied, each one of them twice. Hence, in this latter case, in which $N$ is divisible by $k$, the set of phases ends up reduced to the set of the case $N'=N/k=4$, repeated $k$ times. From the symmetry of the resulting sets of phases in either case, we see that it is still true that the sum of all these phases is zero,

Figure 2.5.3: A double lap on the complex unit circle with $N=8$.

Figure 2.5.4: A double lap on the complex unit circle with $N=7$.

$$\sum_{n=1}^{N}e^{\imath 2\pi k n/N}=0.$$

The same is valid for any other values of $k$ in the interval under consideration, except for $k=0$. In this case we always have, very simply, $N$ times the positive real phase $\exp(0)=1$, so that we may write for our sum of phases

$$\sum_{n=1}^{N}e^{\imath 2\pi k n/N}=N\delta(k,0),$$

where $\delta(k,0)$ is the Kronecker delta, equal to $1$ if $k=0$, equal to $0$ if $k\neq 0$. In order to convince oneself of the truth of this fundamental relation, which will give rise to the relations of orthogonality and completeness that we will frequently use, one may try out a certain number of particular cases, until one acquires a practical understanding of how the sum of phases works. We may also verify it in a simple and elegant way using the formula for the sum of a geometrical progression^2.1, although the numbers involved are complex rather than real. The extension of this formula to the complex domain is a simple process of analytical extension and its validity can be verified algebraically (problem 2.5.1). If we have as the initial element of the progression $a_{1}=\exp(\imath 2\pi k/N)$ and as the ratio $q=\exp(\imath 2\pi k/N)\neq 1$, with $k\neq 0$, we will also have $a_{N}=\exp(\imath 2\pi k)=1$ for any $k$, and the sum is given by

$$\frac{a_{N}q-a_{1}}{q-1}=\frac{e^{\imath 2\pi k}e^{\imath 2\... ... 2\pi k/N}\frac{e^{\imath 2\pi k}-1}{e^{\imath 2\pi k/N}-1}=0.$$

For the case $k=0$ and $q=1$ the formula cannot be used due to the zero in denominator, but in this case the result is obvious because all the elements of the sum are equal. In the problems a different approach to this question is proposed, equally rigorous and more complicated and detailed (problems 2.5.5, 2.5.6 and 2.5.7).

Observe that we may use integer coordinates for the sites with values in the interval $[0,N-1]$, or any other interval containing $N$ consecutive integers, as well as in the interval $[1,N]$, as we have been doing. In addition to this, the exponential that appears in the sum above is symmetrical by the exchange of $n$ e $k$, so that it is equally true that

$$\sum_{k=0}^{N-1}e^{\imath 2\pi k n/N}=N\delta(n,N).$$

One can also see that the modes $k=0$ e $k=N$ are in reality the same mode, in fact $k$ and $k+N$ always represent the same mode (problem 2.5.2). Thus, in the case of the $k$ coordinates we may also choose the extremes of the interval of variation arbitrarily, so long as we always take $N$ consecutive values. For reasons associated to the physical interpretation of these modes, it will be convenient that we take the intervals of variation of $k$ in a way as symmetrical as possible around $0$. For this reason we will adopt the following standard intervals, one for odd $N$,

$$k=-\frac{N-1}{2},\ldots,0,\ldots,\frac{N-1}{2},$$

and another for even $N$,

$$k=-\left(\frac{N}{2}-1\right),\ldots,0,\ldots, \left(\frac{N}{2}-1\right),\frac{N}{2}.$$

We may now write the relations above in a slightly different form, that we will name relation of orthogonality,

  $$\sum_{n=1}^{N}e^{\imath\frac{2\pi}{N}n(k-k')}=N\delta_{k,k'},$$ (2.5.1)

and relation of completeness,

  $$\sum_{k=k_{m}}^{k_{M}}e^{\imath\frac{2\pi}{N}k(n-n')}=N\delta_{n,n'},$$ (2.5.2)

where $k_{m}$ and $k_{M}$ are the minimum and maximum limits of the interval of values of $k$ in each case. Once these relations are established for the one-dimensional case as we have done here, their extensions to higher dimensions is immediate, achieved by means of the use of the properties of the exponential function (problem 2.5.3). Hence in $d$ dimensions we have the relations

$$\begin{eqnarray*} \sum_{n_{1}=1}^{N}\ldots\sum_{n_{d}=1}^{N} e^{\imath\frac{2\pi... ...n}')} &=& N^{d}\delta_{n_{1},n'_{1}}\ldots\delta_{n_{d},n'_{d}}. \end{eqnarray*}$$

The first relation establishes a definition of scalar product between modes, according to which they are all orthogonal to one another. The exponential functions $\exp(\imath 2\pi\vec{k}\cdot\vec{n}/N)$ are called the mode functions of these Fourier modes. This scalar product is defined as a sum over position space of products of two mode functions, which are characterized by $\vec{k}$ and $\vec{k}'$. The second relations involves a sum over momentum space and establishes that any function of the sites on the lattice can be written as a linear superposition of these mode functions, which therefore constitute a complete set of functions on the lattice (problem 2.5.4).

Problems

1. Derive algebraically the expression for the sum of a geometrical progression of $N$ terms with $q\neq 1$ and show that the result is valid independently of $q$ and the first term $a_{1}$ being real or complex.

2. Show that the modes corresponding to $k_{1}$ and $k_{1}+N$ are in fact the same mode, that is, that the corresponding mode functions, $\exp(\imath 2\pi k n/N)$, have the same values at all the lattice sites.

3. Using the properties of the exponential function, derive the orthogonality and completeness relations in $d$ dimensions from the one-dimensional relations given in equations (2.5.1) and (2.5.2).

4. Show that any function $f(\vec{n})$ on the torus can be written as a linear combination of the collection of $N^{d}$ functions $\exp(\imath 2\pi\vec{k}\cdot\vec{n}/N)$ and give explicitly the coefficients of the expansion.

5. In this problem, for simplicity, we will number the sites from $0$ to $N-1$. Given a certain $N$ in $d=1$ and a certain value of $k$, show that the set of phases defined by

   
   

   $$\left\{e^{\imath\frac{2\pi}{N}nk}\mbox{ \Large / }n=0,\ldots,N-1\right\}$$
   
   

   is contained within the set of phases given by

   
   

   $$\left\{e^{\imath\frac{2\pi}{N}m}\mbox{ \Large / }m=0,\ldots,N-1\right\}$$
   
   

   where the integer $m$ is given in terms of $n$ by

   
   

   $$m=nk-pN$$
   
   

   where $p$ is some integer. We say that $n$ is equal to $m$ module $N$. In this way we succeed to map the phases generated by any given $k$ back to the interval described by an integer from $0$ to $N-1$.

6. Show that if $N$ is not divisible by $k$ in problem 2.5.5 then the two sets of phases are identical, that is, the relation between them is a bijection or one-to-one. Use this fact to establish that the sum of the phases for $k\neq 0$ is zero in this case. Hint: show that, if $n_{1}$ is mapped to $m_{1}$ and $n_{2}$ to $m_{2}$, then $m_{1}=m_{2}\Rightarrow n_{1}=n_{2}$.

7. If $N$ is divisible by $k$ in problem 2.5.5, that is, if $N=N'k$ for some $N'$, then show that only the phases corresponding to the lattice of size $N'$ appear in the first set. Show that each one of them appears exactly $k$ times. Use these facts to establish that the sum of the phases for $k\neq 0$ is zero in this case, effectively reducing the problem of the $N$-lattice to the problem of the $N'$-lattice.

8. Write a program to verify the orthogonality relation of the momentum-space mode-functions in $d=1$, for a given fixed $N$ and any given $k$.