ApproximateSU3and Its Nonrelativistic Limit

Hadron states with given parity and a complete set of Poincaré and ${\mathrm{SU}}_3$ labels are discussed in purely Liealgebraic terms. Since P must vanish for these states, the mass operator becomes $M=c^{-1\mathrm{}}{(-P_{\mu }{P}^{\mu })}^{\frac{1}{2}}=c^{-1\mathrm{}}{P}_0$. Having thus resolved the question of whether the Gell-Mann-Okubo splitting of $M$ should be formulated in terms of the Hamiltonian $H=c{P}_0$ or of $M$ itself (which is $c^{-1\mathrm{}}{P}_0$ for these states), we consider the implied commutators of $P_0$ with the ${\mathrm{SU}}_3$ generators. We take these commutators as given, but do not assume that $P_0={P_0}^{\mathrm{}\left(0\right)\mathrm{}}+{P_0}^{\mathrm{}\left(1\right)\mathrm{}}$, with ${P_0}^{\mathrm{}\left(0\right)\mathrm{}}$ an ${\mathrm{SU}}_3$ scalar and ${P_0}^{\mathrm{}\left(1\right)\mathrm{}}\ll {P_0}^{\mathrm{}\left(0\right)\mathrm{}}$. The nonrelativistic Galilean limit $c\to \infty$ is explored for any particle representation of the Poincaré algebra, and it is concluded that $P_0$ is made up of two parts, with $c^{-1\mathrm{}}{P_0}^{\mathrm{}\left(0\right)\mathrm{}}\to \mathfrak{M}$, the Galilean mass, and $c{P_0}^{\mathrm{}\left(1\right)\mathrm{}}\to -H_0$, the internal energy. If it is assumed that the fundamental commutators of $c{P}_0$, $P_0$, and $c^{-1\mathrm{}}{P}_0$ with the ${\mathrm{SU}}_3$ generators remain finite as $c\to \infty$, one obtains the limit of the Gell-Mann-Okubo formulation. The only such limit describes hadrons with the same spin and parity as if they were eigenstates of a single isolated system of (undefined) nonrelativistic particles. $\mathfrak{M}$ is an ${\mathrm{SU}}_3$ scalar, while $H_0$ either is a scalar or has eigenvalues given by the familiar Gell-Mann-Okubo formula, but now obtained as an exact rather than a first-order perturbation result.