There is a problem 3.3 in Schwartz’s QFT:
Ambiguities in the energy-momentum tensor:
(a) If you add a total derivative to Lagrangian ${\cal{L}} \rightarrow {\cal{L}} + \partial_\mu X^\mu$, how does the energy-momentum tensor change?
(b) Show that the total energy $Q = \int d^3x \; {\cal{T^{00}}}$ is invariant under such changes
Given the definition of energy-momentum tensor:
$$ {\cal{T}}^\mu{}_\nu = \sum_n \frac{\partial{\cal{L}}}{\partial \left( \partial_\mu \phi_n \right)} \partial_\nu \phi_n - g_{\mu\nu} {\cal{L}} $$
It changes as:
$$ \delta {\cal{T}}^\mu{}_\nu = \sum_n \left( \partial_\rho \frac{\partial{X^\rho}}{\partial \left( \partial_\mu \phi_n \right)} \right) \partial_\nu \phi_n - g_{\mu\nu} {\partial_\rho X^\rho} $$
$T^{00}$ changes as:
$$ \delta {\cal{T}}^{00}= \sum_n \left( \partial_\rho \frac{\partial{X^\rho}}{\partial \dot{\phi_n}} \right) \dot{\phi_n} - \partial_\rho X^\rho $$
From this point I’m stuck. The first term doesn’t seem to disappear at all after integration over space, and the second disappears only partially:
$$ \int d^3x \; \dot{X^0} $$
So even if I’d have assumed that $X$ depends on position, and not on the fields - the total energy doesn’t seem to be invariant under it.
What is the best way to proceed forward with this?
