# The spin and weight of a primary field in CFT

+ 3 like - 0 dislike
208 views

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?

This post imported from StackExchange Physics at 2014-11-28 20:52 (UTC), posted by SE-user Anne O'Nyme

+ 4 like - 0 dislike

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 $s$.

This post imported from StackExchange Physics at 2014-11-28 20:52 (UTC), posted by SE-user Frederic Brünner
answered Nov 28, 2014 by (1,120 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.