Nonlinear realizations and effective Lagrangian densities for nonlinear o-models

Field: Physics and Astronomy (Physical Sciences)

<p>Nonlinear realizations of the groups <i>SU(N), SO(m)</i> and <i>SO(t,s)</i> are analysed, described by the coset spaces <img alt=" 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"/>. The analysis consists of determining the transformation properties of the Goldstone Bosons, constructing the most general possible Lagrangian for the realizations, and as a result identifying the coset space metric. We view the &lambda; matrices of <i>SU(N)</i> as being the basis of an (<i>N</i><sup>2</sup> - 1) dimensional real vector space, and from this we learn how to construct the basis of a Cartan Subspace associated with a vector. This results in a mathematical structure which allows us to find expressions for coset representative elements used in the analysis. This structure is not only relevant to <i>SU(N)</i> breaking models, but may also be used to find results in <i>SO(m)</i> and <i>SO</i>(1, <i>m</i> - 1) breaking models.</p>