First of all, I read many questions but they don't seem to answer my specific question. So, here it goes

According to Francesco's Conformal Field Theory and many other books, a scale transformation

\begin{equation} x^\mu\rightarrow{x'}^\mu=\lambda x^\mu \end{equation}

affects primary fields in the following way \begin{equation} \phi(x)\rightarrow\phi'(x')=\lambda^{-\Delta }\phi(x) \end{equation}

Here the prime sign ' is very important: the formula is talking about the transformed field in the new coordinate system. This transformation means that if we only transform the field we will get

\begin{equation} \phi(x)\rightarrow\phi'(x)=\lambda^{-\Delta }\phi(\lambda^{-1}x) \end{equation}

My question is this: In every reference I check, they seem to define a scale invariant field as

\begin{equation} \phi(x)=\lambda^{-\Delta}\phi (\lambda x) \end{equation}

But that implies that scale invariant fields transform like

\begin{equation} \phi'(x)=\phi(\lambda^{-2}x) \end{equation}

which makes no sense at all, since I would think that a scale invariant field satisfies

\begin{equation} \phi'(x)=\phi(x) \end{equation}

which would imply quite an opposite definition of scale invariance, more like

\begin{equation} \phi(x)=\lambda^{-\Delta}\phi (\lambda^{-1} x) \end{equation}

Can anyone explain where am I misunderstanding scale invariance?


The transformation for primary fields is equation 4.32 of Francesco and it's repeated on Qualls' lectures on equation 2.41 and many others. The definition of scale invariance is on the wikipedia page of scale invariance: https://en.wikipedia.org/wiki/Scale_invariance#Scale_invariance_of_field_configurations I also found it on other sites.

  • $\begingroup$ Can you give us a reference which says this? $\endgroup$
    – Prahar
    Apr 4, 2019 at 4:03
  • $\begingroup$ sure! I'll edit the post $\endgroup$ Apr 4, 2019 at 4:06
  • $\begingroup$ There were recent discussions that might be possibly related. Such as this and this (and links therein). $\endgroup$
    – MannyC
    Apr 4, 2019 at 4:14


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