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  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.

asked May 27, 2020 in Theoretical Physics by Bubi (15 points) [ no revision ]

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