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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$\begingroup$ Your second equation is in general false. That being said, for those functions $\phi$ for which it is true, the rest of equations are true as well, cf. Wikipedia. $\endgroup$– AccidentalFourierTransformMar 8, 2020 at 22:26
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$\begingroup$ Are you referring to the fact that a scaling transformation acts as $T_\epsilon[\phi(x)]=(1+\epsilon)^b \phi((1+\epsilon) x)$ instead of what I wrote? $\endgroup$– TanatofobicoMar 8, 2020 at 22:57
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$\begingroup$ your first equation is fine; the second one is not. $\endgroup$– AccidentalFourierTransformMar 8, 2020 at 23:02
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$\begingroup$ Could you please be more specific? Why is the second equation not fine? I'm guessing you are referring to the scaling properties of the field, but I'm not sure. From what I understand of what you wrote, there is no requirement that fields are homogeneous functions; the scaling transformation of the field is wrong as I wrote it, and the right one is that which I wrote in the previous comment. $\endgroup$– TanatofobicoMar 9, 2020 at 22:08
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1$\begingroup$ the general definition of a dilatation is $(T_\lambda\phi)(x):=\lambda^\Delta\phi(\lambda x)$ (cf. this PSE post). In general, the equation $\phi(\lambda x)=\lambda^b\phi(x)$ is false. So the first equality of your second equation is fine; the second equality is not. $\endgroup$– AccidentalFourierTransformMar 9, 2020 at 22:26
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