The diffeomorphism invariance of scalars is often written as: $$ \phi'(x') = \phi(x).\tag{1}$$ However, while scaling transformation is a type of diffeomorphism, in many places (say Di Francesco, Matthieu and Senechal page 38), you see the following for scalar fields: $$\phi'(\lambda x)=\lambda^{-\Delta} \phi(x).\tag{2.121}$$ This is taken to define the scaling dimension $\Delta$. Aren't these two definitions incongruent unless the scaling dimension is 0? I'm guessing no, I just have trouble seeing what is supposed to be happening here.

The diffeomorphism invariance of scalars is often written as: $$ \phi'(x') = \phi(x).\tag{1}$$ However, while scaling transformation is a type of diffeomorphism, in many places (say Di Francesco, Matthieu and Senechal page 38), you see the following for scalar fields: $$\phi'(\lambda x)=\lambda^{-\Delta} \phi(x).\tag{2.121}$$ This is taken to define the scaling dimension $\Delta$. Aren't these two definitions incongruent unless the scaling dimension is 0? I'm guessing no, I just have trouble seeing what is supposed to be happening here.

EDIT: To clarify a little bit, I don't fully understand why the prime appears in the second equation.

The diffeomorphism invariance of scalars is often written as: $$ \phi'(x') = \phi(x) $$ However, while scaling transformation is a type of diffeomorphism, in many places (say Di Francesco, Matthieu and Senechal page 38), you see the following for scalar fields: $$\phi'(\lambda x)=\lambda^{-\Delta} \phi(x)$$ This is taken to define the scaling dimension $\Delta$. Aren't these two definitions incongruent unless the scaling dimension is 0? I'm guessing no, I just have trouble seeing what is supposed to be happening here.

The diffeomorphism invariance of scalars is often written as: $$ \phi'(x') = \phi(x).\tag{1}$$ However, while scaling transformation is a type of diffeomorphism, in many places (say Di Francesco, Matthieu and Senechal page 38), you see the following for scalar fields: $$\phi'(\lambda x)=\lambda^{-\Delta} \phi(x).\tag{2.121}$$ This is taken to define the scaling dimension $\Delta$. Aren't these two definitions incongruent unless the scaling dimension is 0? I'm guessing no, I just have trouble seeing what is supposed to be happening here.