Induction of Symmetries for a Set of Mesons of Both Parities

A bootstrap model for hypothetical sets of odd-parity mesons ($M$) and even-parity mesons ($M^{\prime }$), developed recently by the author, is extended. Nontrivial solutions to the model must involve mesons of both parities. New consistency conditions are obtained by considering two-particle scattering amplitudes for which the product of the intrinsic parities of the four external particles is odd. It is shown that the symmetric ($\mathrm{MM}{M}^{\prime }$ and $M^{\prime }{M}^{\prime }{M}^{\prime }$) interactions, as well as the antisymmetric interactions, must transform as simple representations of a Lie group. Many Lie groups do not lead to solutions. A solution based on $\mathrm{SU}\left(n\right)\mathrm{}$ does exist; a crucial property of these groups (for $n>\mathrm{}2\mathrm{}$) that is necessary for the solution is the existence of a symmetric interaction involving the regular representation states only. The consistency conditions are all expressible in terms of commutator conditions similar to those applied to total charges in current algebra.