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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: enter image description here

"Conformal Field Theory" by Di Francesco, Mathieu and Senechal: enter image description here

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.

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  • $\begingroup$ I don't think that you have any gaps in your understanding. As usual, there are two ways of looking at symmetry transformation - the active viewpoint (functions change, get dragged along the static coordinates) which is used in the textbook and the passive viewpoint (we look at the same static thing using different coordinates) which you seem to like. These two ways are equivalent. $\endgroup$ – Prof. Legolasov Nov 29 '16 at 10:13
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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 $$ \left<\prod_{i=1}^N \phi_i(z_i) \right> = \left<\prod_{i=1}^N \phi'_i(z_i) \right> $$

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