Relativistic SU(4) and Quaternions

Abstract

A classification of hadrons and their interactions at low energies according to SU(4) allows to identify combinations of the fifteen mesons $\pi$, $\omega$ and $\rho$ within the spin-isospin decomposition of the regular representation \rhdmulti{15}. Chirally symmetric SU(2)$\times$SU(2) hadron interactions are then associated with transformations of a subgroup of SU(4). Nucleon and Delta resonance states are represented by a symmetric third rank tensor \rhdmulti{20} whose spin-isospin decomposition leads to $4\oplus 16$ `tower states' also known from the large-N$_c$ limit of QCD. Towards a relativistic hadron theory, we consider possible generalizations of the stereographic projection {\bf S}$^{2}$ $\to$ {\bf C} and the related complex spinorial calculus {\it on the basis of the division algebras with unit element}. Such a geometrical framework leads directly to transformations in a quaternionic projective `plane' and the related symmetry group SL(2,{\bf H}). In exploiting the Lie algebra isomorphism sl(2,{\bf H}) $\cong$ su$*$(4) $\cong$ so(5,1), we focus on the Lie algebra su$*$(4) to construct quaternionic Dirac-like spinors, the associated Clifford algebra and the relation to SU(4) by Weyl's unitary trick. The algebra so(5,1) contains the de Sitter-algebra so(4,1) which can be contracted to the algebra of the Poincar\'e group.