= Canonical stress-energy tensor
{title2=$T^{\mu\nu}_{\mathrm{can}}$}
= Canonical energy-momentum tensor
{synonym}
The <canonical stress-energy tensor> is the <Noether current> associated with <translation symmetry>. For a <Lagrangian density> with no explicit coordinate dependence,
$$
T^\mu{}_{\nu}=\sum_a
\frac{\partial\mathcal L}{\partial(\partial_\mu\Phi_a)}
\partial_\nu\Phi_a-\delta^\mu_\nu\mathcal L.
$$
The <Euler-Lagrange field equations> imply $\partial_\mu T^\mu{}_\nu=0$. Thus $T^0{}_\nu$ are conserved densities and $T^i{}_\nu$ their fluxes; integrated charges are conserved when the boundary flux vanishes. For a <real scalar field> with $\mathcal L=\tfrac12(\partial\Phi)^2-\tfrac12m^2\Phi^2$, the <energy density> is $T^{00}=\tfrac12\dot\Phi^2+\tfrac12|\nabla\Phi|^2+\tfrac12m^2\Phi^2$ in <natural units>. The general <canonical stress-energy tensor> need not be symmetric. <Quantum Field Theory by David Tong>[Section 1.3.2, equations 1.43–1.49]{ref}
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