This is kind of related to this, Defining a CFT using beta-functions
So what would be the right definition of a CFT even classically?
- Is it true that classically one will call a theory scale invariant only if the action is invariant under scale transformations? (and not by the Lagrangian density)
For example under the scale transformations $x' = \lambda x$, in $3+1$ the scalar field goes as $\phi'(x') = \lambda \phi(x)$ and in $1+1$ it goes as $\phi'(x') = \phi (x)$. This means that in $3+1$ the action of the massless scalar field is not scale invariant but in $1+1$ it is but the Lagrangian density goes the otherway. But from the point of view of beta-functions isn't it more consistent to call the $1+1$ theory as a CFT but not the one in $3+1$?
Isn't a massless scalar field theory in $3+1$ guaranteed to produce mass by RG flow whereas the $1+1$ theory will not?