Higher-Order Noncompact Groups as the Relativistic Quark-Model Representations of Current Algebra at Infinite Momentum

Abstract

It is suggested that systematic use be made of higher-order noncompact groups in the problem of the relativistic quark-model representations of the current algebra at infinite momentum. The method includes the idea of relying upon one single unitary irreducible representation of $G$, namely, $SL(2, c)\ensuremath{\subset}G$, rather than considering infinitely reducible representations of $SL(2, c)$. As an illustration, an idealized quark model is discussed, where the particle states at rest belong to the Hilbert space $d\ensuremath{\bigotimes}g$. $d$ denotes the ordinary four-dimensional space of Dirac spinors, while $g$ symbolizes one particular unitary infinite-dimensional irreducible representation of the group $SO(4, 2)$. For simplicity, we consider only a totally degenerate mass spectrum, but the trivial mass operator is not typical, and the method can be extended to more realistic quark models with more interesting mass spectra.