Consider a field theory, and a rescaling transformation of the coordinates \begin{equation} T_\epsilon[\phi(x)]=\phi((1+\epsilon)x). \end{equation} From what I understand, one usually requires that, under such a transformation, the field transform as \begin{equation} T_\epsilon[\phi(x)]=\phi((1+\epsilon)x)=(1+\epsilon)^b \phi(x) \end{equation} where $b$ is chosen so that the Lagrangian is scale invariant. Given this (and provided it is correct), I'm searching for a compact conditions that tells me wether a field $\phi$ indeed has this transformation property. The easiest is the original condition itself \begin{equation} \phi((1+\epsilon)x)=(1+\epsilon)^b \phi(x), \end{equation} but I was wondering if there exists some condition which contains derivatives of $\phi$. I tried to consider $\epsilon$ small and expand \begin{equation} \phi(x)+\epsilon \ (x \cdot \nabla \phi) =(1+b\epsilon) \phi(x)+\mathcal{O}(\epsilon), \end{equation} which leads to \begin{equation} (x \cdot \nabla \phi)=b\phi, \end{equation} but this looks wrong for some reason. Is this reasoning correct? Can I look at the space of all possible fields and choose the fields having the right transformations property by only looking at this condition?
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