I'm doing an exercise on vertex operators in the CFT book by Di Francesco & al.; exercise 9.2 p.329 :

Using mode expansion show that: $$\langle\tilde{\phi(z)}\tilde{\phi(w)}\rangle= - \text{ln} \;(1-\frac{w}{z})$$

I managed to complete the calculation but I'm bothered by the fact that the right hand side is not apparently translation invariant.

It seems that we have no charges (at infinity or wherever) and this should be a function of $(z-w)$. Am I missing out on something?

The same way one line below we have:

$$\langle V'_{\alpha_1}(z_1)\ldots V'_{\alpha_n}(z_n)\rangle = \Pi_{i<j} (z_i-z_j)^{2\alpha_i \alpha_j}z_i^{-2 \alpha_i \alpha_j}$$

which again doesn't seem translation invariant because of the last term.

This post imported from StackExchange Physics at 2014-08-24 09:13 (UCT), posted by SE-user Learning is a mess