Relationship between Nonlinear and Linear Realizations of ChiralSU(2)×SU(2): Theory and Applications

The problem of converting nonlinear realizations of $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ has been approached in three different ways. Weinberg has introduced a matrix $\Lambda \left(\pi \right)$ with well-defined properties under infinitesimal chiral transformations and has shown that it can be used to linearize nonlinear fields. Coleman, Wess, and Zumino have achieved the same result by treating the pion fields as parameters of finite chiral rotations; and the present author has made use of the fact that linear realizations are eigenstates of the Casimir operators and close upon themselves under the action of chiral operators. Here we show that the three approaches are equivalent to one another. We first calculate the most general expression for $\Lambda \left(\pi \right)$, and show that certain of its matrix elements have the desired properties with respect to the Casimir and chiral operators. We then show that the method of Coleman, Wess, and Zumino is a special case of $\Lambda \left(\pi \right)$. In the course of the analysis, we find that $\Lambda \left(\pi \right)$ is manifestly covariant under redefinitions of the pion field. To illustrate the usefulness of converting nonlinear fields to linear forms, we calculate $\pi -\pi$ scattering lengths and construct weak currents for meson decay.