Applications of the Chiral $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$ 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 $\delta$ 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 $U\left(3\right)\mathrm{}\otimes U\left(3\right)\mathrm{}$ algebra for the time components of the currents implies the $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$ 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 $\pi -p$ 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 $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$. (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 ${\pi }^{-}\to {\pi }^0+e^{-}+\overline{\nu }$ is, on the basis of the $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$ current algebra, shown to be nonvanishing, and is computed. (8) A speculative argument is presented that the rate $e^{+}+e^{-}\to \mathrm{hadrons}$ is comparable to the rate $e^{+}+e^{-}\to {\mu }^{+}+{\mu }^{-}$ in the limit of large energies.