Dynamical Groups and the Bethe-Salpeter Equation

Abstract

We have considered a class of equations whose solutions are the $n$-dimensional spherical harmonics. We have proven that the set of solutions can be accomodated into a single irreducible representation of the Lie algebra ${B}_{\frac{(n+1)}{2}}$ for $n$ odd, or ${D}_{\frac{(n+2)}{2}}$ for $n$ even, and also into a single irreducible representation of the group $\mathrm{SO}(n, 2)$. Examples of such equations are the Schr\"odinger equation for the H atom as well as its $(n\ensuremath{-}1)$-dimensional analog, the Schr\"odinger equation with $O(4)$-invariant potentials, as well as their $(n\ensuremath{-}1)$-dimensional analogs; and the Schr\"odinger equation of an $n$-dimensional rigid rotator. In the previous $(n\ensuremath{-}1)$-dimensional cases and in the case of the $n$-dimensional rotator, the solutions can be accomodated into a single irreducible representation of the algebra ${B}_{\frac{(n+1)}{2}}$ for $n$ odd, or ${D}_{\frac{(n+2)}{2}}$ form $n$ even, and the group $\mathrm{SO}(n, 2)$ is the dynamical group of the equation. In the cases of the H atom and $O(4)$-symmetric potentials, we have the algebra ${D}_{3}$ and the group SO(4,2). The class also includes the Bethe-Salpeter (BS) equation for two scalar quarks interacting via the exchange of a scalar boson of zero mass to form a bound state of zero mass, in which case we have---after the Wick rotation---the algebra ${B}_{3}$ and the dynamical group SO(5,2). When the mass of the bound state is different from zero, the SO(5,2) representation splits into $\ensuremath{\Sigma}\ensuremath{\bigoplus}\mathrm{SO}(4, 2)$. The BS equation is transformed to an infinite-component wave equation in the case that the mass of the bound state is zero and in the case that the mass is different from zero. In the first case the known eigenvalue spectrum is obtained. In the second case perturbation theory is applied, and the eigenfunctions and eigenvalues to first and second order in perturbation theory, respectively, are given. Finally, in an Appendix, the H atom and the BS equation representations are written in the canonical basis.