Making use of generally accepted assumptions about the equal-time commutator algebra of axial charges and axial divergences, the spin and helicity dependence of Weinberg's algebraic relations is entirely determined for pions in arbitrary partial waves. It is shown that the algebraic structure of axial-vector coupling matrices may be given by the Lie algebra of the group $\mathrm{SO}(4,3)\mathrm{}$, and so hadron states must be assigned to unitary representations of this group. Furthermore it is proved that the mass-spectrum operator is given as a sum of a scalar and a component of a 35-dimensional totally antisymmetric irreducible tensor of the group $\mathrm{SO}(4,3)\mathrm{}$. The general form of the mass spectrum is exhibited as a linear combination of the Clebsch-Gordan coefficients of the group $\mathrm{SO}(4,3)\mathrm{}$. Application to hadron states of fixed intrinsic quantum numbers leads to the conclusion that mass-squared values of hadrons must be a linear function of spin. This result is a unique and exact consequence of the structure of certain algebraic relations.

• Received 18 September 1970

DOI:https://doi.org/10.1103/PhysRevD.3.3047