It is pointed out that the coupling characterizing theψ-γ vertex must change substantially between the limits,ψ on mass-shell which occurs inψ→e⁺e⁻ and photon on mass-shell which is relevant in radiative decays likeψ→ππγ, ψ→ηγ and photoproduction ofψ. This has the consequence that the value ofψN total cross section must be larger than what is inferred from the use of naive vector dominance in photoproduction.

Mandelstam’s argument that PCAC follows from assigning Lorentz quantum numberM=1 to the massless pion is examined in the context of multiparticle dual resonance model. We construct a factorisable dual model for pions which is formulated operatorially on the harmonic oscillator Fock space along the lines of Neveu-Schwarz model. The model has bothm_π andm_ϱ as arbitrary parameters unconstrained by the duality requirement. Adler self-consistency condition is satisfied if and only if the conditionmϱ²−mπ²=1/2 is imposed, in which case the model reduces to the chiral dual pion model of Neveu and Thorn, and Schwarz. The Lorentz quantum number of the pion in the dual model is shown to beM=0.

A general derivation of the coupling constant relations which result on embedding a non-simple group like SU_L (2) ⇔ U(1) in a larger simple group (or graded Lie group) is given. It is shown that such relations depend only on the requirement (i) that the multiplet of vector fields form an irreducible representation of the unifying algebra and (ii) the transformation properties of the fermions under SU_L (2). This point is illustrated in two ways, one by constructing two different unification groups containing the same fermions and therefore have same Weinberg angle; the other by putting different SU_L (2) structures on the same fermions and consequently have different Weinberg angles. In particular the value sin²ϑ=3/8 is characteristic of the sequential doublet models or models which invoke a large number of additional leptons likeE₆, while addition of extra charged fermion singlets can reduce the value of sin² ϑ to 1/4. We point out that at the present time the models of grand unification are far from unique.