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A primary field in Conformal Field Theory transforms as $$\phi (z,\bar{z}) =\left(\frac{dz}{dz'} \right)^h \left(\frac{d\bar{z}}{d\bar{z}'} \right)^\bar{h}\phi (z',\bar{z}') $$ under a conformal transformation.
I read in chapter 2 page 41 in Strings, Conformal Fields and M-theory by M.Kaku that $h+\bar{h}$ is called a conformal weight and $h-\bar{h}$ a conformal spin.
What is the motivation, especially for the spin-one, for these names?
Both $h$ and $\tilde{h}$ are usually called weights. Their sum, $\Delta=h+\tilde{h}$ is the (scaling) dimension of the operator, while the difference, $s=h-\tilde{h}$ is called the spin. This is due to their association with scale transformations (dilatations) and rotations, respectively. To see this, note that the dilatation operator is given by $D=z\partial+\bar{z}\bar{\partial}$ and the rotation operator by $L=z\partial-\bar{z}\bar{\partial}$. The eigenvalues of a primary under these transformations are given by its scaling dimension $\Delta$ and its spin $s$.
