WSpin for Any Spin

The collinear group ${\mathrm{SU}\left(6\right)\mathrm{}}_W$ enables us to discuss vertices, form factors, and other collinear processes in an $\mathrm{SU}\left(6\right)\mathrm{}$ theory which is consistent with relativistic invariance. It leads, however, to a classification of the particles which is different from that of the "static" ${\mathrm{SU}\left(6\right)\mathrm{}}_S$. The general "$W$-spin" properties of an arbitrary spin state constructed from any number of basic spin-½ objects are discussed in detail. Explicit formulas for expressing the eigenstates of ${\mathbf{W}}^2$ as linear combinations of ordinary spin states are given and some properties of the transformation matrices are discussed. The relation between $W$ spin and ordinary $S$ spin in the framework of the $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(2\right)\mathrm{}$ algebra is generalized to an arbitrary Lie algebra of the form $G\otimes G$. Some examples of such generalized $W$-type algebras are considered and the special case of ${\mathrm{SU}\left(6\right)\mathrm{}}_W$ and ${\mathrm{SU}\left(6\right)\mathrm{}}_S$ is discussed in detail. An explicit formula for calculating the ${\mathrm{SU}\left(6\right)\mathrm{}}_W$ properties of an arbitrary component of a representation of the nonchiral $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$ is given. Some explicit transformation matrices between the eigenstates of $W$ spin and $S$ spin are shown.