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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?

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    $\begingroup$ 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. $\endgroup$ – Joe Aug 6 at 15:24
  • $\begingroup$ Yeah, that seems to be correct. Each of the three methods(dimensions,tensor transformation, operator notation) lead to the same answer then. Thanks $\endgroup$ – Mani Jha Aug 6 at 16:12

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