# Vertex operator - state mapping in Polchinski's book

+ 5 like - 0 dislike
779 views

In Polchinski's textbook String Theory, section 2.8, the author argues that the unit operator $1$ corresponds to the vacuum state, and $\partial X^\mu$ is holomorphic inside couture $Q$ in figure 2.6(b), so operators $\alpha_m^\mu$ with $m>=0$ vanishes.

I am a bit confused about why $\partial X^\mu$ has no pole inside the contour. Before this section $\partial X^\mu$ always has the singularity part ($1/z^m$). Therefore would it be possible for you to give a more mathematical argument what condition requires $\partial X^\mu$ having no poles in this case?

Thanks a lot for your time!

This post imported from StackExchange Physics at 2014-04-14 16:20 (UCT), posted by SE-user Han Yan
The main point is that the operator-state correspondence maps all the annihilation operators to zero, so that an operator-valued Laurent series in $z$ and $\bar{z}$ maps to a ket-state-valued power series in $z$ and $\bar{z}$.
 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$\hbar\varnothing$sicsOverflowThen 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). Please complete the anti-spam verification