Sources and Fixed Boundary Conditions

We may introduce external sources in systems with fixed boundary conditions, in the same way as we did for periodical boundary conditions. The form of the action of the free theory in this case is the same as before, as well as the form of the equation of motion,

  $$(-\Delta^{2}+\alpha_{0})\varphi(\vec{n})=j(\vec{n}).$$ (2.11.1)

The difference is that, in order to solve the equation in this case, we should use the basis of eigenfunctions $f^{N}_{\vec{k}}(\vec{n})$ which is appropriate to this type of boundary conditions,

$$f^{N}_{\vec{k}}(\vec{n})= 2^{d/2} \sin\left(\frac{\pi k_{1}n... ...N+1}\right)\ldots \sin\left(\frac{\pi k_{d}n_{d}}{N+1}\right),$$

which satisfy the orthogonality and completeness relations

$$\begin{eqnarray*} \sum_{\vec{n}}f^{N}_{\vec{k}}(\vec{n})f^{N}_{\vec{k}'}(\vec{n}... ...{\vec{k}}(\vec{n}') & = & (N+1)^{d}\delta^{d}(\vec{n},\vec{n}'). \end{eqnarray*}$$

The functions $\varphi(\vec{n})$ and $j(\vec{n})$ may be written in terms of their transforms as

$$\begin{eqnarray*} \varphi(\vec{n}) & = & \sum_{\vec{k}}f^{N}_{\vec{k}}(\vec{n})\... ...sum_{\vec{k}}f^{N}_{\vec{k}}(\vec{n})\widetilde\jmath (\vec{k}). \end{eqnarray*}$$

Substituting these expressions in (2.11.1) we have

$$(-\Delta^{2}+\alpha_{0}) \sum_{\vec{k}}f^{N}_{\vec{k}}(\vec{... ...m_{\vec{k}}f^{N}_{\vec{k}}(\vec{n})\widetilde\jmath (\vec{k}).$$

Since the functions $f^{N}_{\vec{k}}(\vec{n})$ are eigenvectors of the Laplacian with eigenvalues $-\rho_{f}^{2}$, we have

$$\sum_{\vec{k}}f^{N}_{\vec{k}}(\vec{n})\left[ \widetilde\varp... ...(\rho_{f}^{2}+\alpha_{0})-\widetilde\jmath (\vec{k})\right]=0.$$

We see therefore that, once more, the differential equation reduces to an algebraic equation for the components of the field in momentum space, in fact, the very same equation we had before,

Figure 2.11.1: Green function with fixed boundary conditions for $d=1$.

Figure 2.11.2: Green function with fixed boundary conditions for $d=2$.

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

the sole difference being that both the transforms and the eigenvalues $\rho_{f}^{2}$ relate now to the eigenfunctions of the Laplacian with fixed boundary condition. We have now solutions that are similar to the ones we had before,

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

For the case of the point source we have for the external source and its transform

$$\begin{eqnarray*} j(\vec{n}) & = & j_{0}\delta^{d}(\vec{n},\vec{n}'), \\ \wideti... ...{n}') \\ & = & \frac{j_{0}}{(N+1)^{d}}f^{N}_{\vec{k}}(\vec{n}'), \end{eqnarray*}$$

which implies that the solutions of the classical theory may be written as

$$\begin{eqnarray*} \widetilde\varphi (\vec{k}) & = & j_{0}\widetilde g_{f}(\vec{k... ...g_{f}(\vec{k})f^{N}_{\vec{k}}(\vec{n}')f^{N}_{\vec{k}}(\vec{n}), \end{eqnarray*}$$

and the quantity that we will call the propagator or the Green function, in momentum space, is given by

$$\widetilde g_{f}(\vec{k})= \frac{1}{(N+1)^{d}\left[\rho_{f}^{2}(\vec{k})+\alpha_{0}\right]},$$

while the Green function in position space is, as in the periodical case, defined as a double inverse transform of this function, involving the points $\vec{n}$ e $\vec{n}'$,

$$g_{f}(\vec{n},\vec{n}')=\sum_{\vec{k}} \frac{f^{N}_{\vec{k}}... ...n})} {(N+1)^{d}\left[\rho_{f}^{2}(\vec{k})+\alpha_{0}\right]}.$$

Just as in the periodical case, this function represents the response of the system to the presence of a unit external source, with $j_{0}=1$. Note that in this case $g_{f}$ is not a function only of the difference $\vec{n}-\vec{n}'$ but rather of $\vec{n}$ and $\vec{n}'$ separately, because with fixed boundary conditions there is no discrete translation invariance on the lattice. We may also, just as in the periodical case, write the general solution for $\varphi(\vec{n})$ in terms of the Green function,

$$\varphi(\vec{n})=\sum_{\vec{n}'}j(\vec{n}')g_{f}(\vec{n},\vec{n}').$$

The expression for the Green function in position space may be somewhat simplified if we choose the position of the source right in the middle of the lattice, so that $n'=(N+1)/2$, which can be done without difficulty for the case of odd $N$. In figures from 2.11.1 to 2.11.4 we show the Green function on a lattice with $N=23$ and this value of $n'$, in dimensions from $1$ to $4$. As one can see, the behavior is similar to that of the case of periodical boundary conditions, with the exception that in our case here the functions are always zero at the boundary.

Figure 2.11.3: Green function with fixed boundary conditions for $d=3$.

Figure 2.11.4: Green function with fixed boundary conditions for $d=4$.

Using the usual rescalings of all quantities in terms of the lattice spacing $a$, we may write dimensionfull versions of all these quantities. As before, we define the dimensionfull version $\widetilde G_{f}$ in momentum space as $\widetilde G_{f}=a^{2-d}\widetilde g_{f}$, resulting in

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

so that in the continuum limit we obtain once more

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

where in this case we have $\vec{p}_{f}=\pi\vec{k}/L$.

Note that in this case, just as in the case of periodical boundary conditions, the Green function in momentum space always has the same form when written in terms of the eigenvalues of the Laplacian, be it on finite lattices, in the continuum inside a finite box or in infinite space. What changes is the number and character of these eigenvalues, of which there is a finite number on finite lattices, a discrete infinity in the continuum inside a finite box, and an uncountable, continuous infinity in infinite space. On the other hand, the Green functions in position space change significantly when one passes from one case to the other. Hence, we see that the transformation by the eigenfunctions of the Laplacian effectively filters out the detailed effect of the boundary conditions and of the finite volume of the box over the physically relevant results of the theory, and that these effects remain manifested in momentum space only by the existence of an infrared cutoff in finite boxes, which is removed when we go to infinite space. This fact will be of particular importance for the correlation functions to be defined in the quantum theory, of which the Green function discussed here is an example.

Clearly, despite the fact that the imposition of the condition $\varphi=0$ at the border changes the detailed form of the results of the theory in position space, both on finite lattices and in the continuum limit inside a finite box, it does not prevent us from seeing the basic structure of the continuum theory in infinite space, so long as we use the representation of the theory in momentum space, when we deal with the theory on finite lattices. In addition to this, the fact that the Green function is a function of the coordinates $k_{\mu}$ only through the combination $\rho_{f}^{2}(\vec{k})$ is clearly related to the underlying rotational invariance of the Euclidean theory in infinite space which, in this form, may be indirectly detected from within a finite continuous box or even from within the confines of a discrete lattice. These two properties allow us to treat models in a very specific and practical way, in any particular set of boxes and lattices that we may be using, without loosing sight of the invariances that the theory should have when one goes to infinite continuous space.