Eigenvalues and Eigenvectors of the Laplacian

Having discussed before the relevant realizations of the finite-difference operators $\Delta_{\mu}$ and $\Delta^{2}$, we are now in a position to study their eigenvalues and eigenvectors. For simplicity, it is more convenient to start the discussion by the eigenvalues of the symmetrical realization $\Delta_{\mu}^{\rm (c)}$, but it is also interesting to compare the results for the various realizations, as we shall do.

The eigenvectors of the finite-difference operators with periodical boundary conditions are the exponential functions $\exp(\imath 2\pi\vec{k}\cdot\vec{n}/N)$ that appear in the Fourier transformations. We will also refer to these eigenvectors as eigenfunctions. We have, for example, by the direct application of the definition of eigenvector followed by a simple calculation (problem 2.7.1), that

$$\Delta_{\mu}^{\rm (c)}e^{\imath\frac{2\pi}{N}\vec{k}\cdot\ve... ...t{n}_{\mu}\right) e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}},$$

where $\hat{n}_{\mu}=\hat{x}_{\mu}$ is the versor in the direction $\mu$. Observe that the eigenvalue is complex and that, therefore, the operator is not Hermitian. We may define the quantity

$$\rho_{\mu}^{\rm (c)}(k_{\mu}) =\sin\left(\frac{2\pi}{N}\vec{... ...\hat{n}_{\mu}\right) =\sin\left(\frac{2\pi k_{\mu}}{N}\right),$$

that plays a role similar to that of the linear momentum in the case of the continuum formalism. For the realization $\Delta_{\mu}^{(+)}$, with values on links, we have

$$\Delta_{\mu}^{(+)}e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}... ...}_{\mu}}-1\right) e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}}.$$

For $\Delta_{\mu}^{(-)}$ we have a similar relation. Note that, as in the case of $\Delta_{\mu}^{(c)}$, the eigenvalue of $\Delta_{\mu}^{(+)}$ is also complex. We may write this eigenvalue as

$$2\imath e^{\imath\frac{\pi}{N}\vec{k}\cdot\hat{n}_{\mu}} \sin\left(\frac{\pi}{N}\vec{k}\cdot\hat{n}_{\mu}\right),$$

or, defining a new version of the quantity $\rho_{\mu}(k_{\mu})$, that plays the role of the linear momentum, by

  $$\rho_{\mu}(k_{\mu}) =2\sin\left(\frac{\pi}{N}\vec{k}\cdot\hat{n}_{\mu}\right) =2\sin\left(\frac{\pi k_{\mu}}{N}\right),$$ (2.7.1)

the remaining factors are simply the imaginary unit $\imath$ that comes from the finite differentiation and a phase

$$e^{\imath\frac{\pi}{N}\vec{k}\cdot\hat{n}_{\mu}} =e^{\imath \vec{p}\cdot\hat{x}_{\mu}a/2},$$

indicating that the natural location of the result is the middle of the link that points in the positive $\mu$ direction starting from the site in question! The dimensionfull quantity $p^{(N)}_{\mu}=\rho_{\mu}(k_{\mu})/a$ is the quantity on the lattice that really corresponds to the physical linear momentum of the states and particles of the theory. We will also refer to the quantities $\rho_{\mu}(k_{\mu})$ as the dimensionless momenta on the lattice. The lattice momenta $p^{(N)}_{\mu}$ reduce in the continuum limit to the continuum momenta $p_{\mu}$, so long as these are finite in the limit, that is, for modes with integer coordinates $k_{\mu}$ that are much smaller than $N$ on large lattices,

$$p_{\mu}^{(N)}=\frac{2}{a}\sin\left(\frac{\pi k_{\mu}}{N}\rig... ...Na}=\frac{2\pi k_{\mu}}{L}=p_{\mu},\mbox{~~for~~}k_{\mu}\ll N.$$

In the case of the operator $\Delta^{2}$ (problem 2.7.2) we have for each finite second-difference $\Delta_{\mu}^{2}$ (with no sum over $\mu$),

$$\Delta_{\mu}^{2}e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}}=... ...right)\right]^{2} e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}}.$$

Hence we see that the eigenvalues of $\Delta^{2}_{\mu}$ are $-\rho^{2}_{\mu}$ and we have, therefore,

  $$\Delta^{2}e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}} =-\rho^{2}e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}},$$ (2.7.2)

where $\rho^{2}$ is given by

$$\rho^{2}=\sum_{\mu}\rho_{\mu}^{2} =4\left[\sin^{2}\left(\fra... ...ight) +\ldots+\sin^{2}\left(\frac{\pi k_{d}}{N}\right)\right].$$

In this case the eigenvalues are real and we have a Hermitian operator. Again, the eigenvalues are related to the dimensionless versions of the momenta on the lattice, $\rho_{\mu}(k)$. All the quantities that we will calculate in the theory will end up written in terms of these quantities, more often in terms of $\rho^{2}$.

Note that the fact that the complex exponentials are eigenvectors of the Laplacian implies that some of them are also solutions of the classical theory in its non-Euclidean version. In order to see this it suffices to directly apply the equation to the functions, resulting in

$$(-\Delta^{2}+\alpha_{0})e^{\imath\frac{2\pi}{N}\vec{k}\cdot\... ...ho^{2}+\alpha_{0})e^{\imath\frac{2\pi}{N}\vec{k}\cdot\vec{n}},$$

thus showing that the equation is solved by modes for which $\rho^{2}=-\alpha_{0}$. Since the parameter $\alpha_{0}$ is positive, this can only be satisfied in the non-Euclidean version of the theory, in which the sum $\rho^{2}=\sum_{\mu}\rho_{\mu}^{2}$, which is here manifestly positive, changes so as to have a negative element,

$$\rho^{2}=-\rho_{0}^{2}+\sum_{i=1}^{d-1}\rho_{i}^{2}.$$

It is clear that, on finite lattices and depending on the value of $\alpha_{0}$, there might be no mode such that $\rho^{2}=-\alpha_{0}$, which just shows the discrete nature of the solutions within a finite box, even in the non-Euclidean case. However, if we take the so-called infra-red limit, making the size $L$ of the box tend to infinity, the separations between consecutive square momenta $\rho^{2}$ of the modes become infinitesimal and in this case it is always possible to find a mode with $\rho^{2}$ arbitrarily close to any given positive value of $\alpha_{0}$. The relation $\rho^{2}+\alpha_{0}=0$ is referred to as the on-shell condition and is a characteristic of the plane waves that constitute the relativistically invariant classical solutions of the theory in non-Euclidean space.

Problems

1. Show, using directly the definition of the finite-difference operators, that the functions $f_{\vec{k}}(\vec{n})=\exp(\imath 2\pi\vec{k}\cdot\vec{n}/N)$ are eigenfunctions of these operators. In order to do this apply to these functions the definition of the operators at an arbitrary internal site and remember that the boundary conditions are periodical.

2. Show, using the definition of the Laplacian that follows from the definitions of the finite-difference operators $\Delta_{\mu}^{\pm}$, that the functions $f_{\vec{k}}(\vec{n})=\exp(\imath 2\pi\vec{k}\cdot\vec{n}/N)$ are eigenfunctions of the finite-difference Laplacian. In order to do this apply to these functions the definition of the Laplacian at an arbitrary internal site and remember that the boundary conditions are periodical.