The definition of the transformed field in CFT

Question

I am a little puzzled by what people call "the transformed field" in CFT. The usual definition of the scale-invariant function is \begin{equation} \phi(\lambda z) = \lambda^\Delta \phi(z) \end{equation} Clearly, we have same $\phi$ in both LHS and RHS. The natural generalization of this definition is \begin{equation} \phi(f(z)) = \left(\dfrac{\partial f}{\partial z}\right)^\Delta \phi(z) \end{equation} Again, we have same $\phi$ on both sides.

Now, after making the change of variables $z\to f(z)$, (to me) it would make sense to introduce a new function defined as \begin{equation} \phi'(z) \equiv \phi(f(z)) \end{equation}

However, in the standard CFT books I find something completely opposite. Namely, for some reason people put prime in the LHS of the first two equations I wrote.

"Introduction to Conformal Field Theory With Applications to String Theory" by Blumenhagen and Plauschinn:

"Conformal Field Theory" by Di Francesco, Mathieu and Senechal:

I feel like I have a gap in conceptual understanding... Why do they put primes?? Clearly, to define a function $\phi$ having some special property, we need to have it on both sides - just like in the usual definition of the scale-invariant function.

Answer 1

For a function $F(z)$, let us define $F'(z) = \lambda^\Delta F(\lambda z)$. Then $F(z)$ is covariant (with weight $\Delta$) under scale transformations if $F'(z) = F(z)$.

For a field $\phi(z)$ let us define $\phi'(z) = \lambda^\Delta \phi(\lambda z)$. Fields are not supposed to be covariant under scale transformations. Rather, correlation functions are covariant. The covariance of the $N$-point function $\left<\prod_{i=1}^N \phi_i(z_i) \right>$ can be written as $$ \left<\prod_{i=1}^N \phi_i(z_i) \right> = \left<\prod_{i=1}^N \phi'_i(z_i) \right> $$