Scalar Densities in Current Commutation Relations and the Partially-Conserved-Axial-Vector-Current Hypothesis

Abstract

In the theory of the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ group for baryons and mesons, there exist eighteen scalar and pseudoscalar densities. Confining ourselves to the $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ subgroup, the pseudoscalar densities are equal to the divergences of the axial-vector currents and therefore can be interpreted as being proportional to the pion field on the basis of the partially-conserved-axial-vector-current hypothesis. An attempt is made to give physical interpretation to the scalar density by calculating the expectation value of its space integral with respect to nucleon states. It is found that this expectation value is small (at most, of the order of the pion mass), which is related to Adler's consistency condition on the pion-nucleon scattering amplitude implied by the PCAC hypothesis. Specific models that are consistent with the PCAC hypothesis, such as the gradient-coupling model and the σ-meson model, are examined on the basis of our result.