Scaling Dimension of a Field in CFT

In summary, "Scaling Dimension of a Field in CFT" discusses the concept of scaling dimensions in conformal field theory (CFT), which characterizes how fields transform under scale transformations. It highlights the significance of scaling dimensions in understanding the behavior of physical systems at different length scales and their implications for the correlation functions and operator product expansions in CFT. The paper also explores the relationship between scaling dimensions and other critical properties, such as conformal symmetry and the classification of operators, emphasizing their role in the study of phase transitions and critical phenomena.
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shinobi20
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TL;DR Summary
I'm quite unsure about how the scaling dimension of a field is devised. I need clarifications on small details in order to make sure that my concepts are clear.
I'm studying CFT, and I find the lecture notes and books really confusing and devoid of explanations (more details).

In a scale transformation ##x' = \lambda x##, the field ##\phi(x)## should also be affected by the scale transformation, i.e., ##\phi'(x') = \phi'(\lambda x) = \lambda^{-\Delta} \phi(x)##. I think this should mean that we assume that the field should scale but we do not know by how much, and ##\Delta## quantifies this unknown, which is called the scaling dimension. We put a minus sign because when we plug it in the action, this will avoid a negative dimension (see below)?

If the scale transformation is a symmetry of the theory, then the action must be invariant under this. Particularly, in a free theory,

\begin{align*}
S' & = \int d^d x' \partial'_\mu \phi'(x') \partial'^\mu \phi'(x')\\
& = \int d^d x' g^{\mu \nu} \partial'_\mu \phi'(x') \partial'_\nu \phi'(x')\\
& = \int d^d x \Bigg| \frac{\partial x'}{\partial x} \Bigg| \lambda^{-2} g^{\mu \nu} \partial_\mu \phi'(x') \partial_\nu \phi'(x')\\
& = \int d^d x \lambda^d \lambda^{-2} \partial_\mu \phi'(x') \partial^\mu \phi'(x')\\
\end{align*}

Comparing this with ##S = \int d^d x \partial_\mu \phi(x) \partial^\mu \phi(x)##,

\begin{equation*}
\int d^d x \lambda^d \lambda^{-2} \partial_\mu \phi'(x') \partial^\mu \phi'(x') = \int d^d x \partial_\mu \phi(x) \partial^\mu \phi(x)
\end{equation*}

We can infer that,

\begin{equation*}
\lambda^{\frac{d-2}{2}} \phi'(x') = \phi(x), \quad \text{OR} \quad \phi'(x') = \lambda^{\frac{-(d-2)}{2}} \phi(x)
\end{equation*}

So the scaling dimension is ##\Delta = \frac{(d-2)}{2}##. If we were to not put a minus sign from the start, i.e., ##\phi'(\lambda x) = \lambda^{\Delta} \phi(x)##, then ##\Delta = \frac{-(d-2)}{2}##.

Questions:
1. Can anyone verify if what I'm saying (my statements) above are correct?
2. Can anyone explain more about the minus sign?
3. I'm not sure if what I wrote in the second line of the action is correct, i.e., ##\partial'_\mu \phi'(x') \partial'^\mu \phi'(x') = g^{\mu \nu} \partial'_\mu \phi'(x') \partial'_\nu \phi'(x')##. Is this correct or should it be ##\partial'_\mu \phi'(x') \partial'^\mu \phi'(x') = g'^{\mu \nu} \partial'_\mu \phi'(x') \partial'_\nu \phi'(x')##? If this is the case then ##g'^{\mu \nu}## should also transform, but then I would not get the correct scaling dimension.
 

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