{c}
{title2=$j^\mu$}
{wiki=Noether's_theorem}

A continuous variational symmetry gives a current satisfying $\partial_\mu j^\mu=0$ <on shell>. Integrating $j^0$ over space gives a conserved charge when boundary flux vanishes.

Noether current

The translation changes the <Lagrangian density> by $\delta\mathcal L_0=\epsilon^\nu\partial_\nu\mathcal L_0=\partial_\mu(\epsilon^\mu\mathcal L_0)$. The <Noether current> is therefore
$$
j^\mu=\frac{\partial\mathcal L_0}{\partial(\partial_\mu\psi)}\delta\psi-\epsilon^\mu\mathcal L_0
=\epsilon_\nu T^{\mu\nu},
\qquad
T^{\mu\nu}=i\bar\psi\gamma^\mu\partial^\nu\psi-\eta^{\mu\nu}\mathcal L_0.
$$
This is the <canonical stress-energy tensor>. Directly,
$$
\partial_\mu T^{\mu\nu}
=\left(\frac{\partial\mathcal L_0}{\partial\psi}
-\partial_\mu\frac{\partial\mathcal L_0}{\partial(\partial_\mu\psi)}\right)\partial^\nu\psi
$$
up to the analogous adjoint-field term, so the <Euler-Lagrange field equations> imply $\partial_\mu T^{\mu\nu}=0$ <on shell>. The <Dirac equation> also gives $\mathcal L_0=0$ on shell.

Solved by gpt-5.6-sol high.

Solution

Paper 301 / 1 / a / Solution

After using the <Clifford algebra>, integrations by parts and the <Dirac equation>, every allowed Lorentz-covariant symmetric rank-two expression with one derivative reduces, modulo terms vanishing <on shell> and identically conserved improvements, to an overall multiple of
$$
\bar\psi\gamma^{(\mu}\overleftrightarrow{\partial}^{\nu)}\psi.
$$
Thus the general nontrivial tensor in the stated class is
$$
\widehat T^{\mu\nu}
=C\left[
\bar\psi\gamma^\mu\partial^\nu\psi
+\bar\psi\gamma^\nu\partial^\mu\psi
-(\partial^\nu\bar\psi)\gamma^\mu\psi
-(\partial^\mu\bar\psi)\gamma^\nu\psi
\right].
$$
It is manifestly symmetric. Differentiating it, commuting partial derivatives and using
$$
i\gamma^\rho\partial_\rho\psi=m\psi,
\qquad
i(\partial_\rho\bar\psi)\gamma^\rho=-m\bar\psi,
$$
makes the terms cancel pairwise, proving $\partial_\mu\widehat T^{\mu\nu}=0$ on shell. The normalization $C$ remains free at this stage.

Solved by gpt-5.6-sol high.

Paper 301 / 1 / b / Solution

{wiki}

A field configuration or identity is on shell when the <Euler-Lagrange field equations>[equations of motion] hold.

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 Incoming links: On shell