# Derivation of some basic properties of conformal transformations [duplicate]

I was trying to understand the properties of conformal transformations in $d$ dimensions, and I am confused about something very basic. The idea is discussed in many books but not very clearly. For example page 22-23 of "Lectures on CFT". I have a few questions about the discussion there:

1) From considering infinitesimal coordinate transformation, I can obtain Eqn-2.5 $\partial_\mu \epsilon_\nu + \partial_\nu \epsilon_\mu = \frac{2}{d}(\partial.\epsilon) \eta_{\mu\nu}$

Few lines below, the author applies $\partial_\nu$ to both sides to get Eqn-2.6.

$\partial_\mu (\partial.\epsilon) + \partial^2 \epsilon_\mu = \frac{2}{d} \partial_\mu (\partial . \epsilon)$

Here somehow the derivative $\partial_{\nu}$ does not seem to act on $\eta_{\mu\nu}$. Why is this so ? Why do I not have an additional term that looks like

$(\partial . \epsilon) \partial^\nu \eta_{\mu\nu}~?$

I suppose the argument is that $\eta_{\mu\nu}(x)$ before the transformation is the constant flat metric. Is this correct?

2) This is a subjective/general question. What is the general motivation / structure for the sequence of arguments here?

3) A more minor question about [page 22 of "Lectures on CFT"]: Shouldn't the transformation above Eqn 2.1 be

$g'_{\rho\sigma}(x') = \frac{\partial x^\mu}{\partial x'^\rho} \frac{\partial x^\nu}{\partial x'^\sigma} g_{\mu\nu}(x)$

instead of what is given in the notes ? This will be consistent with this post.

• About point 1), it seems that your derivation is taking place in flat space, so your metric is constant. If it weren't, you would have covariant derivatives in your Killing equation.
– M.Jo
Commented Jun 15, 2018 at 12:00
• You're also correct about point 3) Commented Jun 15, 2018 at 13:24
• Concerning point 2, the general motivation is to investigate the conformal group in $d$ spacetime dimensions, cf. e.g. this & this Phys.SE posts. Commented Jun 15, 2018 at 18:30