By Malcolm T Sanford
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If it commutes with the generators Q we have already discussed, we can make two of these generators simultaneously 44 CHAPTER 3. SYMMETRIES OF MESON AND BARYON SYSTEMS diagonal. 73) the diagonal matrices are λ3 ˆ 3 and Q ˆ 8 are chosen as diagonal generators. and λ8 , so that Q The physical meaning of these two operators is expressed by the following new notation T3 = Y = ˆ3 Q 2 ˆ √ Q 8 3 . 84) ˆ 3 with T3 , the third component of the isospin, is motivated The identification of Q by the fact that λ3 contains τ3 as a submatrix.
Since the states within a multiplet are labeled by the eigenvalues of the two operators T3 and Y they have to be represented by points in the two-dimensional (t3 , y) plane. A certain multiplet is determined by specifying which points are occupied and how often. The general shape of a multiplet-boundary in the (t3 , y) plane can be inferred from symmetry considerations. First, we know that SU (2) is a subgroup of SU (3). Therefore, within a given isospin multiplet all states from −t3 to +t3 (for fixed y) must belong to the SU (3) multiplet.
The T = 1 multiplet could be obtained by coupling two nucleons each with T = 1/2. 28). If this representation is realized in nature then we should find a triplet of particles with the same mass and different electric charges. 30)). Symmetry of the Pion Lagrangian Another example for an isospin triplet is given by the three pions. The pions carry the quantum numbers J π = 0− and are thus pseudoscalar particles. 0 MeV); it is natural to associate the somewhat higher mass of the two charged pions with the effect of the repulsive Coulomb interaction.
4H - Beekeeping Basic Manual by Malcolm T Sanford