- #1
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 , the field should also be affected by the scale transformation, i.e., . I think this should mean that we assume that the field should scale but we do not know by how much, and 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,
Comparing this with ,
We can infer that,
So the scaling dimension is . If we were to not put a minus sign from the start, i.e., , then .
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., . Is this correct or should it be ? If this is the case then should also transform, but then I would not get the correct scaling dimension.
In a scale transformation
If the scale transformation is a symmetry of the theory, then the action must be invariant under this. Particularly, in a free theory,
Comparing this with
We can infer that,
So the scaling dimension is
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.,