Applications of the Chiral<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>6</mml:mn><mml:mo>)</mml:mo><mml:mn/><mml:mo>⊗</mml:mo><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>6</mml:mn><mml:mo>)</mml:mo><mml:mn/></mml:math>Algebra of Current Densities

Consequences of the local commutation relations of vector and axial currents proposed by Gell-Mann are explored: (1) A recipe for detecting and isolating Schwinger terms in the commutators, proportional to derivatives of the $\ensuremath{\delta}$ function, is discussed. (2) Under assumptions of smooth asymptotic behavior of form factors for forward scattering of the isovector current from a proton, we show that the $U(3)\ensuremath{\bigotimes}U(3)$ algebra for the time components of the currents implies the $U(6)\ensuremath{\bigotimes}U(6)$ algebra for space components, at least for spin-averaged diagonal single-particle states. (3) The derivation of the Adler-Weisberger formula for $\frac{{G}_{A}}{{G}_{V}}$ is sharpened by giving arguments that, at fixed energy, the forward $\ensuremath{\pi}\ensuremath{-}p$ Green's function satisfies an unsubtracted dispersion relation in the pion mass. (4) A lower bound for inelastic electron-nucleon scattering at high momentum transfer is derived on the basis of $U(6)\ensuremath{\bigotimes}U(6)$. (5) The contribution of very virtual photons to the hyperfine anomaly in hydrogen is shown to be related to an equal-time commutator of currents; this contribution is crudely estimated to be 4 parts per million (ppm). (6) The logarithmically divergent part of electromagnetic mass differences of hadrons is shown to be proportional to matrix elements of the equal-time commutator of the electromagnetic current with its time derivative. It is suggested that this "divergent" part be identified with the Coleman-Glashow "tadpoles"; this suggestion is discussed in the framework of a simple quark model. (7) The logarithmically divergent part of the electromagnetic correction to the process ${\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}+{e}^{\ensuremath{-}}+\overline{\ensuremath{\nu}}$ is, on the basis of the $U(6)\ensuremath{\bigotimes}U(6)$ current algebra, shown to be nonvanishing, and is computed. (8) A speculative argument is presented that the rate ${e}^{+}+{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{hadrons}$ is comparable to the rate ${e}^{+}+{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}+{\ensuremath{\mu}}^{\ensuremath{-}}$ in the limit of large energies.