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This Letter presents the fine structure of energy levels for the edge states of a Haldane chain. In order to investigate the edge states, we have performed high field and multifrequency electron spin resonance (ESR) measurements of finite length S=1 antiferromagnetic chains in Y2BaNi0.96Mg0.04O5. Owing to the high spectral resolution by high fields and high frequencies, observed ESR signals can be separated into the contributions of the finite chains with various chain lengths. Our results clearly show that the edge spins actually interact with each other through the quantum spin chain and the interaction depends on the chain length N. This N dependence has been obtained experimentally for the first time, and shows that the correlation length xi in the real system is somewhat larger than that calculated by a simple Heisenberg model.

A phenomenological model for decays of $\ensuremath{\psi}$ and ${\ensuremath{\psi}}^{\ensuremath{'}}$ into normal hadronic modes is proposed. This model can explain the decay rate $\ensuremath{\Gamma}(\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\rho})$ as well as the puzzlingly large rate $\ensuremath{\Gamma}(\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$ in comparison to $\ensuremath{\Gamma}(\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$. It also predicts many other relations which can be easily tested experimentally.

Continuity arguments for transitions between various subgroups of the chiral-symmetry group are presented. Some interesting sum rules are obtained in this way.

We discuss the convergence and the superconvergence properties of the invariant amplitudes which occur in suitable combinations of the propagator functions of the vector and the axial-vector currents, based on the use of symmetry arguments for the asymptotic behavior. The sum rules so obtained for the spectral functions are in good agreement with experiment.

An analysis of the ${A}_{1}$ and the $\ensuremath{\rho}$ decays is made using current algebra, partial conservation of axial-vector currents, and dispersion relations; the pions are treated as "soft" in the sense ${K}^{2}\ensuremath{\rightarrow}0$ instead of $K\ensuremath{\rightarrow}0$ which is usually assumed.

We propose that the recently discovered resonance in ${e}^{+}{e}^{\ensuremath{-}}$ annihilation is a member of the 15 \ensuremath{\bigoplus} 1 dimensional representation of the SU(4) group. This hypothesis is consistent with the various experimental features reported for the resonance. In addition, we make a prediction for the masses of the charmed vector mesons belonging to the same representation.

A low-energy theorem is derived for the structure-dependent axial-vector form factor in the radiative decay $\ensuremath{\pi}\ensuremath{\rightarrow}l\ensuremath{\nu}\ensuremath{\gamma}$ using current-algebra techniques. By saturation of the sum rule with a few low-lying resonances, an estimate is made which is compared with experiments.

It is pointed out that one may test the role of divergenceless currents in strangeness nonconserving ( mu nu ) interactions by looking for certain striking effects in the angular correlations in K/sub mu 3/ decay. lt is assumed that only positive chiral states are coupled. A unique formula for the angular distribution is obtained which reverses the former expectations. Predictions are made from this. (M.H.R.)