Finite Fourier Transforms

We are now in a position to define the finite Fourier transform of our field, as well as its inverse. As we shall see, the orthogonality of the modes we defined establishes that this transform will take us to the normal modes of oscillation of the field within the box. On our cubic $N$-lattice with periodical boundary conditions we define the finite Fourier transform of the field $\varphi(\vec{n})$ as

$$\widetilde\varphi (\vec{k})=\frac{1}{N^{d}}\sum_{\vec{n}} e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}}\varphi(\vec{n}).$$

This is a linear transformation of coordinates in the space of field configurations, taking us from the $N^{d}$ coordinates $\varphi(\vec{n})$ to the also $N^{d}$ coordinates $\widetilde\varphi (\vec{k})$. The inverse transform, taking us back to the coordinates in terms of site positions, is given by

$$\varphi(\vec{n})=\sum_{\vec{k}} e^{-\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}}\widetilde\varphi (\vec{k}),$$

where the sum is over all the momenta, that is, an abbreviation such as

$$\sum_{\vec{k}}\equiv \sum_{k_{1}=k_{m}}^{k_{M}}\ldots\sum_{k_{d}=k_{m}}^{k_{M}}.$$

Note that, as defined here, $\varphi$ and $\widetilde\varphi$ are both dimensionless. The fact that the two operations defined above are the inverses of one another is a consequence of the orthogonality and completeness relations, as you will have a chance to show in one of the problems (problem 2.6.1). Observe that all these relations are exact on finite lattices, that they do not involve any kind of approximation. What we have here is simply a linear transformation in a finite-dimensional vector space. As we shall see further along, this linear transformation allows us to solve exactly the Gaussian model, which is the Euclidean version of the theory of the free scalar field.

Let us examine now how these transformations behave in the continuum limit. As we did before in section 2.1, in order to take this limit we will introduce an external dimensional scale into our system. Once more we assume that the system is contained in a cubic box of size $L$ in this external system of units. When we make $N\rightarrow \infty$ with constant $L$, the lattice spacing $a=L/N$ of the lattice goes to zero and the sum that defines the Fourier transform approaches a Riemann integral over the box of volume $V=L^{d}$. In this case we may write this relation, in the limit, as

$$\widetilde\varphi (\vec{p}\,)=\frac{1}{V}\int_{V}{\rm d}^{d}x \:e^{\imath\vec{p}\cdot\vec{x}}\varphi(\vec{x}),$$

where $\vec{x}=a\vec{n}$ are the coordinates that describe positions within the continuous box and $\vec{p}=2\pi\vec{k}/L$ are the discrete momenta associated to the vectors with integer components $\vec{k}$. Since we are now within a continuous but still finite box, these modes associated to $\vec{p}$ are still discrete, of course, so that the inverse is given by an infinite series rather than a finite sum, but not by an integral,

$$\varphi(\vec{x})=\sum_{\vec{p}}^{\infty} e^{-\imath\vec{p}\cdot\vec{x}}\widetilde\varphi (\vec{p}\,).$$

Note that the factor of $1/L^{d}$ in the transform guarantees that $\varphi$ and $\widetilde\varphi$ still have the same dimensions. These relations are valid for both the dimensionless field $\varphi$ and the dimensionfull field $\phi$, because they are homogeneous on the fields, and therefore we may write

$$\begin{eqnarray*} \widetilde\phi (\vec{p}\,) & = & \frac{1}{V}\int_{V}{\rm d}^{d... ...infty} e^{-\imath\vec{p}\cdot\vec{x}}\widetilde\phi (\vec{p}\,). \end{eqnarray*}$$

The orthogonality and completeness relations may now be written in the form

$$\begin{eqnarray*} \frac{1}{V}\int_{V}{\rm d}^{d}x\:e^{\imath\vec{x}\cdot(\vec{p}... ...ec{x}-\vec{x}')} & = & V\delta^{d}\left(\vec{x}-\vec{x}'\right), \end{eqnarray*}$$

where, in the limit, the product $N^{d}\delta^{d}(\vec{n},\vec{n}')$ transforms into the product of the volume by the Dirac delta function $\delta^{d}\left(\vec{x}-\vec{x}'\right)$ (problem 2.6.2). We have here the usual relations for the case of the Fourier transform within a continuous finite box, that is, for the Fourier series.

Next we may think about taking the limit $L\rightarrow\infty$, increasing the box until it takes all space, which will lead us to the usual Fourier transforms in infinite space. Since the moment of the lowest non-zero mode in the box has magnitude $\left(2\pi/L\right)$, and since they are equally spaced, the volume occupied by each mode in momentum space is given by $(2\pi/L)^{d}$, which goes to zero as $L\rightarrow\infty$, so that in this case the form of the inverse transformation will also approach an integral. Taking a large but still finite box, the transform and its inverse can be written approximately as

$$\begin{eqnarray*} \widetilde\phi (\vec{p}\,) & = & \frac{1}{V}\int_{V}{\rm d}^{d... ...{d}} \:e^{-\imath\vec{p}\cdot\vec{x}}\widetilde\phi (\vec{p}\,), \end{eqnarray*}$$

where, strictly speaking, we cannot yet take the limits because of the divergent factors of $V$. The orthogonality and completeness relations may now be written as

$$\begin{eqnarray*} \frac{1}{V}\int_{V}{\rm d}^{d}x\:e^{\imath\vec{x}\cdot(\vec{p}... ...ec{x}-\vec{x}')} & = & V\delta^{d}\left(\vec{x}-\vec{x}'\right), \end{eqnarray*}$$

where we see that, in this case, the divergent factors of $V$ cancel out. It is clear that the normalization factors involving $V$ are not at all convenient in the case of the infinite box and, therefore, we will change the normalizations, so that we may in fact take the limits and obtain the Fourier transforms in their usual form, in which $\widetilde\phi _{\infty}$ and $\phi$ have different dimensions,

$$\begin{eqnarray*} \widetilde\phi _{\infty}(\vec{p}\,) & = & \int{\rm d}^{d}x \:e... ...vec{x}-\vec{x}')} & = & \delta^{d}\left(\vec{x}-\vec{x}'\right). \end{eqnarray*}$$

All the relations examined in this section also have their equivalent counterparts in non-Euclidean space. In particular, the complex exponentials $\exp(\imath\vec{p}\cdot\vec{x})$ are, in their non-Euclidean version, plane waves that propagate in infinite space. These plane waves will have an important role to play also in the quantum theory, where they will be associated to free particles. We see here that the Fourier transformation may also be understood as a decomposition of functions of position in terms of plane waves.

We wrote here the equations relating to infinite space only for reference, since in these notes we will seldom be working in an infinite box. With few exceptions it will suffice to consider the definition of quantum field-theoretical models inside finite boxes. The normalization we chose to adopt for the case of finite lattices is appropriate for this case. In fact, note that with this normalization the zero-momentum ( $\vec{k}=\vec{0}\,$) transform of the field is its average value inside the box,

$$\widetilde\varphi (\vec{0}\,)=\frac{1}{N^{d}}\sum_{\vec{n}}\varphi(\vec{n})= \overline{\varphi}.$$

This average is also called the zero mode of the field and it will play a special role in the subsequent development of the theory.

Problems

1. Show, on finite lattices, that the inverse Fourier transform really recovers the original function from its Fourier components.

2. Assuming that $\vec{x}=a\vec{n}$, show that the product $a^{-d}\delta^{d}(\vec{n},\vec{n}')$, where $\delta^{d}(\vec{n},\vec{n}')$ is the $d$-dimensional Kronecker delta, transforms into the $d$-dimensional Dirac delta function $\delta^{d}(\vec{x}-\vec{x}')$ in the continuum limit. In order to do this, build on the lattice expressions that, in the continuum limit, converge to integrals of $\delta^{d}(\vec{x}-\vec{x}')$ over $\vec{x}$ in domains that may or may not include the point $\vec{x}'$, showing that they have the values one would expect of a Dirac delta function.

3. Calculate the finite Fourier transform of the field defined, on a one-dimensional periodical lattice with an even number $N=2M$ of sites, by $\varphi(n)=(-1)^{n}$.

4. Show that, for a real field $\varphi$, $\widetilde\varphi (-\vec{k})=\widetilde\varphi ^{*}(\vec{k})$. In particular, show that $\widetilde\varphi (\vec{0}\,)$ is real. In addition to this, in case we have even $N$ in dimension $d=1$, show that $\widetilde\varphi (N/2)$ is real.