# Traceless energy momentum tensor and energy spectrum

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We have a $D$ dimensional flat minkowskian spacetime, and a field theory with $T_{\mu\nu}$ symmetric, traceless ($T^{\mu}_{\mu} = 0$) and conserved ($\partial^{\mu} T_{\mu \nu} = 0$). We also assume that the operator $E = \int d^{D-1} x \, T_{00}$ is well defined and semi-positive definite. Given a state $|{\Phi}\rangle$, consider $\mathcal E(t, x) \equiv \langle{\Phi} |T_{00} (t, x) |{\Phi}\rangle \ .$Show that for all positive energy state $|\Phi\rangle$ the average square radius of the region in which $\mathcal E$ is not zero grows with time at a speed which rapidly approaches the speed of light.

Now, this theory is scale-invariant (due to tracelessness of the energy-momentum tensor), thus intuitively I expect that its excitations should be massless and therefore travel at the speed of light. However, I cannot find a way to prove it formally.

I should stress that this problem was not given to me as a homework, but it has been part of the admission PhD test in SISSA, in 2014. It can be found in their web page.

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