I'm trying to solve this exercise:
Suppose an arbitrary theory (Flat space-time?) with a single field (Is a scalar field?) invariant under dilations, i.e. $x\mapsto b x$ and $\phi \mapsto \phi$. Show that the stress energy tensor is traceless.
Writing the transformations as $x\mapsto e^\theta x$, $\phi\mapsto e^{\omega\theta}\phi$, and $\partial_\mu\phi\mapsto e^{(\omega-1)\theta}\partial_\mu\phi$, for $\omega=0$.
I get the variations as $\delta x_\mu=\theta x_\mu$, $\delta\phi=\omega\theta\phi=0$, and $\partial_\mu\phi=(\omega-1)\theta\partial_\mu \phi=-\theta\partial_\mu\phi$; and I've tried to get some useful expression bassed on the variation of lagrangian: $$ \delta L=\frac{\partial L}{\partial\phi}\delta \phi+\frac{\partial L}{\partial(\partial_\mu\phi)}\delta({\partial_\mu\phi})=-\theta\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\mu\phi. $$ On the other hand, the trace of the tensor takes the form $$ T_\mu^\mu=\eta_{\mu\nu}T^{\mu\nu}=\eta_{\mu\nu}(-\eta^{\mu\nu}L+\frac{\partial L}{\partial(\partial_\mu\phi)}\partial^\nu\phi)=-2L+\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\mu\phi $$ Thus, $$ T_\mu^\mu=-2L-\frac{\delta L}{\theta}. $$ Obviously, if the lagrangian is invariant then the Tensor isn't traceless. So I don't have idea how to proceed.
