= Noether current for a spacetime symmetry
{c}
{title2=$j^\mu=\sum_a\Pi_a^\mu Q_a+\mathcal L\xi^\mu-K^\mu$}
For $x'^\mu=x^\mu+\varepsilon\xi^\mu$ and $\phi'_a(x')=\phi_a(x)+\varepsilon\Delta_a$, the fixed-coordinate field variation is $Q_a=\Delta_a-\xi^\nu\partial_\nu\phi_a$. Suppose action invariance is expressed as $\delta_0\mathcal L/\varepsilon+\partial_\mu(\mathcal L\xi^\mu)=\partial_\mu K^\mu$. Writing $\Pi_a^\mu=\partial\mathcal L/\partial(\partial_\mu\phi_a)$ and $E_a=\mathcal L_{\phi_a}-\partial_\mu\Pi_a^\mu$, the product rule gives $\partial_\mu j^\mu=-\sum_a E_aQ_a$. Thus $j^\mu$ is a <conserved current> <on shell>. A vanishing spatial boundary flux is additionally required for its <Noether charge> to be conserved. This formula includes internal transformations, translations and <Lorentz transformations>.
Back to article page
