- #1

snypehype46

- 12

- 1

- Homework Statement
- Determine the transformation of ##X## under conformal transformations

- Relevant Equations
- None

I'm trying to do the following question from David Tong's problem sheets on string theory:

> A theory of a free scalar field has OPE $$\partial X(z)\partial X(w) = \frac{\alpha'}{2}\frac{1}{(z-w)^2}+...$$. Consider the putative candidate for the stress energy tensor $$T(z) = \frac{1}{\alpha '}: \partial X (z) \partial X (z) : -Q \partial^2 X(z)$$. Use ##TX## OPE to determine the transformation of ##X## under conformal transformations ##\delta z = \epsilon(z)##

Now to determine ##T(z)X(w)##, I thought I would contract the normally ordered field with ##X(w)##. So to get:

$$2\langle \partial X (z) \partial X (w) \rangle -Q^2 \partial^3 X(z)$$

Is that the correct way of proceeding? I'm not sure if I need to contract the not-normally ordered fields ##\partial^2 X(z)## with ##X(w)## as well?

Also, I don't quite understand how would I continue with this question after I worked out the OPE of ##TX##.

> A theory of a free scalar field has OPE $$\partial X(z)\partial X(w) = \frac{\alpha'}{2}\frac{1}{(z-w)^2}+...$$. Consider the putative candidate for the stress energy tensor $$T(z) = \frac{1}{\alpha '}: \partial X (z) \partial X (z) : -Q \partial^2 X(z)$$. Use ##TX## OPE to determine the transformation of ##X## under conformal transformations ##\delta z = \epsilon(z)##

Now to determine ##T(z)X(w)##, I thought I would contract the normally ordered field with ##X(w)##. So to get:

$$2\langle \partial X (z) \partial X (w) \rangle -Q^2 \partial^3 X(z)$$

Is that the correct way of proceeding? I'm not sure if I need to contract the not-normally ordered fields ##\partial^2 X(z)## with ##X(w)## as well?

Also, I don't quite understand how would I continue with this question after I worked out the OPE of ##TX##.