Lie Groups, Lie Algebras, and the Troubles of RelativisticSU(6)

Some of the difficulties of relativistic $\mathrm{SU}\left(6\right)\mathrm{}$ are examined. Those arising from the use of continuous groups can be avoided by the use of algebras of finite sets of operators which are sufficient to give the desired properties of elementary particles. The nonconservation of probability associated with the relativistic separation of space and spin is pointed out. Quantum electrodynamics applied to atomic structure is shown to exhibit the type of peculiar symmetry which leaves the interaction invariant but is broken by free Dirac propagators. The implications of this analogy for $\mathrm{SU}\left(6\right)\mathrm{}$ are discussed. The mixing of physical and non-physical states (positive- and negative-energy quark states) leads to noninvariance of the vacuum under the symmetry group, and to a degenerate vacuum in the exact symmetry limit. The existence of open inelastic channels for low-mass boson production is relevant to unitarity calculations and is implied in all energy regions where the symmetry is not badly broken.