I'm trying to show that [itex]\int d^3x \,x^\mu \left(\partial_\mu \partial_0-g_{\mu 0} \partial^2 \right)\phi^2(x)=0 [/itex]. This term represents an addition to a component of the energy-momentum tensor [itex]\theta_{\mu 0} [/itex] of a scalar field and I want to show that this does not change the dilation operator [itex]D=\int d^3 x \, x^\mu \theta_{\mu 0} [/itex](adsbygoogle = window.adsbygoogle || []).push({});

What I have is that:

$$

\int d^3x \,x^\mu \left(\partial_\mu \partial_0-g_{\mu 0} \partial^2 \right)\phi^2(x)=\int d^3x \left[ \,x^j \left(\partial_j \partial_0 \right)\phi^2(x) +x^0\nabla^2\phi^2(x) \right]

$$

and I can argue that the 2nd term is zero, since x_{0}can be pulled out of the integral, and you are integrating a divergence and because boundary conditions are periodic, this term is zero. But I can't argue that the first term is zero. Is there a simple reason that the 1st term is zero? We are not allowed to use anything like equations of motions for the fields.

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# Improved energy-momentum tensor changing dilation operator

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