External Sources and Green Functions

In this section we will introduce a new fundamental element, still in the context of the classical theory. This is the concept of an external source for the field. This new object will have great importance also in the quantum theory. In our paradigmatic model, the theory of the free scalar field, the introduction of a dimensionless external source $j$ is implemented by the addition of a new term to the action, which becomes

  $$S_{0}[\varphi]=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2} +\frac{\alpha_{0}}{2}\sum_{s}\varphi^{2}(s)-\sum_{s}j(s)\varphi(s).$$ (2.10.1)

The external source is a given function of the sites, over which we make no restrictions except that it have finite values on finite lattices. Our objective here is to determine how the introduction of this new term affects the classical solutions of the theory.

First of all, it is necessary to note that the introduction of this term, which is not necessarily positive, changes the local minima of the action without, however, causing it to become unbounded from below. It is easy to see that the action still has a global lower bound in the case in which $j$ is a constant over the whole lattice and we adopt periodical boundary conditions, because in this case we have discrete translation invariance in the model. Under these conditions one can show (problem 2.10.1) that the minimum of the action must be achieved for a constant field. In this case the derivatives of the field, which can only contribute positively for $S_{0}$, are zero and the corresponding term of $S_{0}$ assumes its minimum. So long as $\alpha_{0}$ is positive, the second-degree polynomial that remains in the actions certainly has a lower bound, which is located at $\varphi=j/\alpha_{0}$. Note that, due to the adoption of periodical boundary conditions, it is essential that $\alpha_{0}$ be strictly positive, for in the case $\alpha_{0}=0$ there is no lower bound for $S_{0}$. In this case we may make $S_{0}\rightarrow-\infty$ taking a constant $\varphi$ over the whole lattice and making its value go to $\pm\infty$, depending on the sign of $\sum_{s}j(s)$. This is just another consequence of the fact that the $\alpha_{0}=0$ model has a zero mode on the torus.

Even for an arbitrary function $j(s)$, so long as it is finite, it is still true that there is a global minimum of the action, although in this case it is not so immediate to find it. This is due to the fact that, in order to make infinitely negative the only term of the action that can be negative, it is necessary to make $\varphi$ tend to $\pm\infty$ at one or more points. However, in this case the quadratic terms at each site will always tend to $+\infty$ faster than the corresponding linear terms can tend to $-\infty$. Due to this it is not possible to make $S_{0}$ tend to $-\infty$ by any changes of the fields, but only to $+\infty$, which implies that $S_{0}$ has a lower bound. It is, however, possible to show rigorously that the action has a lower bound so long as $\alpha_{0}>0$, rewriting it in momentum space and completing a square (problem 2.10.2).

We can find the classical solution in the presence of the external source using the principle of minimum action, as we already did before for the free theory without external source. If we make an infinitesimal variation $\delta\varphi(s)$ of the fields, possibly different at each site, the corresponding variation of the action will be given by

$$\begin{eqnarray*} \delta S_{0} & = & \delta\left[\frac{1}{2}\sum_{\ell}(\Delta_{... ...s)\left[-\Delta^{2}\varphi(s)+\alpha_{0}\varphi(s) -j(s)\right], \end{eqnarray*}$$

were we made the “integration by parts” for periodical boundary conditions as was already explained before in section 2.1. If we now impose that $\delta S_{0}=0$ for any $\delta\varphi(s)$, we obtain the equation of motion

$$\begin{eqnarray*} -\Delta^{2}\varphi(s)+\alpha_{0}\varphi(s)-j(s) & = & 0 \Rightarrow \\ \left[-\Delta^{2}+\alpha_{0}\right]\varphi(s) & = & j(s). \end{eqnarray*}$$

This is the non-homogeneous version of the equation of motion obtained before for this same model, where the non-homogeneous term is the external source. We can find the general solution of this equation using the usual techniques of the theory of linear differential equations. The general solution of a non-homogeneous linear equation is always obtained as the sum of the general solution of the homogeneous equation with a particular solution of the non-homogeneous equation. However, we are not interested here in writing explicitly the general solution, but rather in finding the solution for the particular type of boundary condition that we adopted. This solution can be obtained by the use of the finite Fourier transforms. In order to do this, we write the field and the external source in terms of their Fourier transforms as

$$\begin{eqnarray*} \varphi(\vec{n}) & = & \sum_{\vec{k}}e^{-\imath\frac{2\pi}{N}\... ...ac{2\pi}{N}\vec{k}\cdot\vec{n}}\,\widetilde\jmath (\vec{k}). \\ \end{eqnarray*}$$

The substitution of these expressions in the equation of motion (2.1.3) result in

$$\sum_{\vec{k}}\widetilde\varphi (\vec{k})(-\Delta^{2}+\alpha... ...frac{2\pi}{N}\vec{k}\cdot\vec{n}}\,\widetilde\jmath (\vec{k}).$$

Since the exponentials are eigenfunctions of the Laplacian with eigenvalues given by $-\rho^{2}(\vec{k})$, as we saw in equation (2.7.2), we obtain

$$\sum_{\vec{k}}e^{-\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}}\l... ...c{k})+\alpha_{0}\right]-\widetilde\jmath (\vec{k}) \right\}=0.$$

Since the exponentials form a complete set of functions, in order for the linear superposition to be zero it is necessary that all the coefficients be zero, and from this we conclude that, for all $\vec{k}$,

$$\widetilde\varphi (\vec{k})\left[\rho^{2}(\vec{k})+\alpha_{0}\right]=\widetilde\jmath (\vec{k}).$$

In this way the differential equation reduces to an algebraic equation for the Fourier components of the field. The solution may now be written explicitly both in momentum space and in position space, in this second case by means of a simple inverse transformation,

$$\begin{eqnarray*} \widetilde\varphi (\vec{k}) & = & \frac{\widetilde\jmath (\vec... ...\frac{\widetilde\jmath (\vec{k})}{\rho^{2}(\vec{k})+\alpha_{0}}. \end{eqnarray*}$$

We see in this way that it is possible to find the exact form of the classical solutions of the free theory in the presence of arbitrary external sources. The form of the solutions in momentum space is very simple, but it is not so easy to visualize the solutions in position space, because in this case the solutions are written as superpositions of all the Fourier modes. In order to be able to visualize the solutions in position space, we will examine a particularly simple case which is, however, of extreme importance. This is the case of an external source which is zero at all sites except one. We refer to this external source as a point source or as a point charge, a reference to its analogy with the familiar case of electrostatics. We write the point source of magnitude $j_{0}$ located at the site $\vec{n}'$ in the form

$$j(\vec{n})=j_{0}\delta^{d}(\vec{n},\vec{n}'),$$

where a Kronecker delta function appears. The finite Fourier transform of this point source is given by

$$\begin{eqnarray*} \widetilde\jmath (\vec{k}) & = & \frac{1}{N^{d}} \sum_{\vec{n}... ...\frac{j_{0}}{N^{d}}e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}'}. \end{eqnarray*}$$

The solution of the classical theory in momentum space may now be written as

$$\widetilde\varphi (\vec{k})= j_{0}\frac{1}{N^{d}\left[\rho^{... ...lpha_{0}\right]} e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}'},$$

and the solution in position space reduces to

$$\varphi(\vec{n})=j_{0} \sum_{\vec{k}}\frac{1}{N^{d}\left[\rh... ...ght]} e^{-\imath\frac{2\pi}{N}\vec{k}\cdot(\vec{n}-\vec{n}')}.$$

We may simplify a little these expressions by choosing the position $\vec{n}'=\vec{0}$ for the point source, which can always be done by means of an adequate choice of the intervals of variation of the integer coordinates of the sites. These solutions represent the response of the system to the presence of a point source. The function of the momenta that appears in these solutions,

$$\widetilde g(\vec{k})\equiv\frac{1}{N^{d}\left[\rho^{2}(\vec{k})+\alpha_{0}\right]},$$

is called the propagator or the Green function of the system, written in momentum space. Its finite inverse Fourier transform, which can be written as

$$g(\vec{n}-\vec{n}')= \sum_{\vec{k}}\frac{1}{N^{d}\left[\rho^... ...ght]} e^{-\imath\frac{2\pi}{N}\vec{k}\cdot(\vec{n}-\vec{n}')},$$

is the Green function in position space, the response of the theory to the presence of a unit point source, with $j_{0}=1$. Observe that this is a kind of double inverse transform, where the two factors $\exp(-\imath 2\pi\vec{k}\cdot\vec{n}/N)$ and $\exp(\imath 2\pi\vec{k}\cdot\vec{n}'/N)$ appear, since in general the function $g$ is a function of two points, $g(\vec{n},\vec{n}')$. The fact that $g$ is a function only of the difference $\vec{n}-\vec{n}'$ is a specific property of the periodical boundary conditions, for which we have discrete translation invariance on the lattice. These lattice versions of the Green functions are both dimensionless. Our solutions for the point charge may now be written as

Figure 2.10.1: Periodical Green function in the case $d=1$.

Figure 2.10.2: Periodical Green function in the case $d=2$.

$$\begin{eqnarray*} \widetilde\varphi (\vec{k}) & = & j_{0}\widetilde g(\vec{k})e^... ...t\vec{n}'}, \\ \varphi(\vec{n}) & = & j_{0}g(\vec{n}-\vec{n}'), \end{eqnarray*}$$

which, for $\vec{n}'=\vec{0}$, simplify to

$$\begin{eqnarray*} \widetilde\varphi (\vec{k}) & = & j_{0}\widetilde g(\vec{k}), \\ \varphi(\vec{n}) & = & j_{0}g(\vec{n}). \end{eqnarray*}$$

Figure 2.10.3: Periodical Green function in the case $d=3$.

Figure 2.10.4: Periodical Green function in the case $d=4$.

In addition to this, the solution for an arbitrary external source in momentum space may now be written as

  $$\widetilde\varphi (\vec{k})=N^{d}\widetilde\jmath (\vec{k})\widetilde g(\vec{k}),$$ (2.10.2)

and the corresponding solution in position space, with a little more work (problem 2.10.3) and the use of the orthogonality and completeness relations, as

  $$\varphi(\vec{n})=\sum_{\vec{n}'}j(\vec{n}')g(\vec{n}-\vec{n}').$$ (2.10.3)

Let us consider quickly the continuum limit of this expression. It is easy to verify (problem 2.10.4) that, in order for the complete action in the continuum limit to be written as

  $$S_{0}[\phi]=\int{\rm d}^{d}x\left\{ \frac{1}{2}\sum_{\mu}\left[\p... ...ght]^{2} +\frac{m_{0}^{2}}{2}\phi^{2}(\vec{x})-J(\vec{x})\phi(\vec{x})\right\}$$ (2.10.4)

it is necessary that the dimensionfull version of $j$ be defined as $J=a^{-(d+2)/2}j$. On the other hand, we may define the dimensionfull version of $g$ as $G=a^{2-d}g$ and, since with our normalization the functions and their Fourier transforms have the same dimensions, $\widetilde G=a^{2-d}\widetilde g$. With these definitions we have that

$$\widetilde G(\vec{k})=\frac{1} {L^{d}\left[\rho^{2}(\vec{k})/a^{2}+\alpha_{0}/a^{2}\right]},$$

so that, in the continuum limit, the expression for the Green function is

$$\widetilde G(\vec{p}\,)=\frac{1}{V\left(p^{2}+m_{0}^{2}\right)},$$

where, as we already saw in section 2.7, $\rho^{2}(\vec{k})/a^{2}\rightarrow p^{2}$, with $\vec{p}=2\pi\vec{k}/L$, and $\alpha_{0}/a^{2}\rightarrow m_{0}^{2}$. In this way one can verify (problem 2.10.5) that our expression for the solution in the presence of an arbitrary external source in terms of the Green function becomes, in the continuum limit,

  $$\phi(\vec{x})=\int{\rm d}^{d}x\;J(\vec{x}')G(\vec{x}-\vec{x}').$$ (2.10.5)

We see therefore that $G$ is in fact the Green function of the non-homogeneous equation of motion in the usual sense in which the term is used in the theory of linear differential equations. Note that, due to the existence of a zero mode on the torus, for which $\rho^{2}=0$, this Green function is well defined only if $\alpha_{0}>0$.

We may acquire an intuitive idea of how the response of the system to the presence of a point source looks, in position space, drawing a few graphs of this function around the point where the external point source is located. Figure 2.10.1 shows the function $g$ for dimension $d=1$, mass parameter $\alpha_{0}=3$ and external source $j_{0}=1$ on a lattice with $N=25$. We see here that the field assumes non-zero values along all the lattice, with a maximum at the position of the source. This is the solution which we denominate qualitatively as the “circus tent”, given its form. The same happens in larger dimensions. A similar example for $d=2$ can be found in figure 2.10.2, for the same values of $\alpha_{0}$, $N$ and $j_{0}$. In this case the functions falls off in a somewhat more pronounced way when we go away form the position of the point source. This is due to the larger number of neighbor sites that are connected by links to the site there the point source is located. This effect becomes more even pronounced in larger dimensions. In the graphs contained in figures 2.10.3 and 2.10.4 one can see similar examples for $d=3$ and $d=4$, still with the same values of $\alpha_{0}$, $N$ and $j_{0}$.

Note that the response of the system to an external point source is progressively more localized in the immediacy of the position of the source, as the dimension of space-time increases. For $d=3$ in the continuum limit in infinite space, with appropriate boundary conditions and $m_{0}=0$, which is possible in this case due to the different boundary conditions, the solution becomes the Coulomb solution, the electrostatic potential of a point charge. In all cases the maximum value of the solution is proportional to $j_{0}$ and, in the limit in which $j_{0}$ goes to zero, the solution becomes identically zero, which is the solution that we discussed before for the theory without external sources. We see that the response of the system consists of a deformation of the field centered at the position of the charge, with an intensity proportional to its magnitude and, in general, a finite range. This range depends on $\alpha_{0}$, as one can verify by drawing other graphs of the functions (problem 2.10.6), a fact that will be of great importance in the quantum theory.

Problems

1. Show that, on a finite lattice with periodical boundary conditions and a constant external source $j_{0}$, the minimum value of the action $S_{0}[\varphi]$ is achieved for a field $\varphi(s)=\varphi_{0}$, which is constant over the whole lattice.

2. ($\star$) Show that the action $S_{0}[\varphi]$ given in equation (2.10.1) has a lower bound, for any finite external source $j$, so long as $\alpha_{0}>0$. In order to do this, write the action in terms of the Fourier transforms $\widetilde\varphi (\vec{k})$ of the field and $\widetilde\jmath (\vec{k})$ of the external source, then complete a square in order to show that the dependence on the field is contained solely within a manifestly positive term of the resulting expression for the action. Determine also the value of the lower bound as a function of the external source.

3. Show, starting from the general solution in momentum space given in equation (2.10.2), using the expressions for the direct and inverse Fourier transforms, and using the orthogonality and completeness relations, that the general solution in position space is the one given in equation (2.10.3).

4. Show, using the definition of the dimensionfull field in terms of the dimensionless one, as well as the other scaling relations, that the dimensionfull external source must be given by $J=a^{-(d+2)/2}j$ so that the action in the continuum limit may be written as shown in equation (2.10.4).

5. Starting from equation (2.10.3), and using the necessary scaling relations, show that the general solution in position space in the continuum limit is indeed the one given by equation (2.10.5).

6. ($\star$) Write a program to calculate the Green function $g(n)$ in position space in one dimension. Use it to plot a series of graphs of $g$ like the one in the text, with $N=25$, but using various values of $\alpha_{0}$ between $1$ and $10$, so as to verify in a quantitative way how the range of the deformation of the field due to the presence of the external source varies as a function of this parameter.