# Conformal transformation

I am reading some lecture notes on Conformal Field Theory by Joshua D. Qualls (https://arxiv.org/abs/1511.04074).

At the end of page 5 of these notes, it is stated that the four momentum transforms as $$p^{\mu} \mapsto\lambda^{-1} p^{\mu}$$ under a conformal transformation $$x^{\mu}\mapsto\lambda x^{\mu}$$. This seems to make sense if you consider that in natural units, momentum has dimensions of inverse length.

However, if we apply the usual tensor transformation rules, we get $$p^{\mu}\mapsto\lambda p^{\mu}$$.

Writing $$p^{\mu}$$ in its quantum mechanical operator form seems to justify what is given, but the author claims to just deal with classical conformal invariance in the section.

What is wrong here?

• Maybe the author just messed up the upper/lower indices convention? After all, $p_\mu = \frac{\partial}{\partial x^\mu}$, which is consistent with them transforming opposite to each other. – Joe Aug 6 at 15:24
• Yeah, that seems to be correct. Each of the three methods(dimensions,tensor transformation, operator notation) lead to the same answer then. Thanks – Mani Jha Aug 6 at 16:12