= Solution
A symmetry of a <classical field theory> maps solutions to solutions; for the <Noether theorem> we require a continuous variational symmetry of the action, allowing its density to change by a total divergence. This is stronger than simply observing an accidental symmetry of one solution. Consider first-derivative fields $\phi_a$ and define
$$
\Pi_a^\mu=\frac{\partial\mathcal L}{\partial(\partial_\mu\phi_a)},\qquad
E_a=\frac{\partial\mathcal L}{\partial\phi_a}-\partial_\mu\Pi_a^\mu.
$$
The <Euler-Lagrange field equations> are $E_a=0$. For a fixed-coordinate variation $\delta_0\phi_a=\varepsilon Q_a$, the product rule gives the off-shell identity
$$
\frac{\delta_0\mathcal L}{\varepsilon}
=\sum_a E_aQ_a+\partial_\mu\!\left(\sum_a\Pi_a^\mu Q_a\right).
$$
This identity is the proof's essential integration-by-parts step.
For a general spacetime symmetry write $x'^\mu=x^\mu+\varepsilon\xi^\mu$ and $\phi'_a(x')=\phi_a(x)+\varepsilon\Delta_a$. At a fixed point $Q_a=\Delta_a-\xi^\nu\partial_\nu\phi_a$, while the volume-element variation contributes $\partial_\mu(\mathcal L\xi^\mu)$. Suppose action invariance takes the form
$$
\frac{\delta_0\mathcal L}{\varepsilon}+\partial_\mu(\mathcal L\xi^\mu)=\partial_\mu K^\mu.
$$
Combining with the preceding identity proves
$$
\boxed{j^\mu=\sum_a\Pi_a^\mu Q_a+\mathcal L\xi^\mu-K^\mu,
\qquad\partial_\mu j^\mu=-\sum_aE_aQ_a=0\ \text{on shell}.}
$$
This is the <Noether current for a spacetime symmetry>. It also covers internal symmetries, for which $\xi=0$. A constant parameter corresponds to one <Noether current>; several independent parameters give several currents. Integrating the continuity equation gives
$$
\frac{dQ}{dt}=-\int_{\partial\Sigma}\mathbf j\cdot d\mathbf S,
\qquad Q=\int_\Sigma j^0\,d^3x.
$$
Thus the <Noether charge> is conserved when the fields have suitable falloff or no flux through the spatial boundary. These are classical identities; quantum symmetries may additionally require absence of an <quantum anomaly>.
\b[Translations conserve energy and momentum.] If the density has no explicit coordinate dependence, take the active fixed-coordinate variation $Q_a=\partial_\nu\phi_a$ for translation in direction $\nu$. Then $\delta_0\mathcal L/\varepsilon=\partial_\nu\mathcal L$, and the internal-form proof with $K^\mu=\delta^\mu{}_{\nu}\mathcal L$ gives the <canonical energy-momentum tensor>
$$
T^\mu{}_{\nu}=\sum_a\Pi_a^\mu\partial_\nu\phi_a-\delta^\mu{}_{\nu}\mathcal L,
\qquad \partial_\mu T^\mu{}_{\nu}=0.
$$
Its four spatial integrals are the conserved energy and momentum. For a real <scalar field> with density $\tfrac12(\partial\phi)^2-V(\phi)$ this becomes $T^{\mu\nu}=\partial^\mu\phi\partial^\nu\phi-\eta^{\mu\nu}\mathcal L$. Direct differentiation gives $\partial_\mu T^{\mu\nu}=(\Box\phi+V'(\phi))\partial^\nu\phi$, which vanishes by the field equation.
\b[Lorentz symmetry conserves angular momentum and boost charges.] For that scalar example, $T^{\mu\nu}$ is symmetric, so the <Lorentz transformation> currents can be written
$$
J^{\mu\rho\sigma}=x^\rho T^{\mu\sigma}-x^\sigma T^{\mu\rho}.
$$
Their divergence is $T^{\rho\sigma}-T^{\sigma\rho}=0$. Spatial pairs give <angular momentum>, and time-space pairs give the conserved boost charges. For a <spinor field> or a <vector field>, the intrinsic field variation supplies an additional spin current; equivalently the <Belinfante-Rosenfeld stress-energy tensor> combines spin and orbital pieces into a symmetric stress tensor. The nonunitarity of the finite component <Spinor representation of the Lorentz group> does not obstruct conservation of these physical spacetime charges.
\b[Global phase symmetry conserves particle charge.] For a complex <scalar field> with density $\partial_\mu\Phi^*\partial^\mu\Phi-V(\Phi^*\Phi)$, the constant transformation $\Phi\mapsto e^{-i\alpha}\Phi$ leaves the density invariant. Using $Q_\Phi=-i\Phi$ and $Q_{\Phi^*}=i\Phi^*$ gives the <Noether current>
$$
j^\mu=i(\Phi^*\partial^\mu\Phi-\Phi\partial^\mu\Phi^*).
$$
For a <Dirac field>, the corresponding variations $Q_\psi=-i\psi$, $Q_{\bar\psi}=i\bar\psi$ yield instead $j^\mu=\bar\psi\gamma^\mu\psi$, the <Dirac current>. Its divergence vanishes by the <Dirac equation> and <adjoint Dirac equation>. The <Yukawa interaction> of part 2 also preserves this phase symmetry, because each bilinear has one $\psi$ and one $\bar\psi$. Thus it preserves fermion number and forbids annihilation of two fermions into a single scalar. These currents express charge or particle-minus-antiparticle-number conservation, rather than conservation of total particle count in arbitrary interacting processes.
A <global symmetry in field theory> uses a spacetime-independent parameter and ordinarily relates physically distinct configurations. A <gauge symmetry> instead has arbitrary local parameters and represents redundancy in the description. For example, the <Maxwell field> is unchanged physically by $A_\mu\mapsto A_\mu+\partial_\mu\alpha(x)$: its <electromagnetic field tensor> is unchanged. For charged matter use $D_\mu=\partial_\mu+iqA_\mu$ together with $\psi\mapsto e^{-iq\alpha(x)}\psi$. Differentiating shows $D'_\mu\psi'=e^{-iq\alpha}D_\mu\psi$, so the coupled <Dirac action> is locally gauge invariant. A local phase applied to free matter alone produces extra derivatives of $\alpha$ and is not an invariance; the compensating gauge field is necessary.
Local arbitrariness gives identities among field equations, rather than an independent physical charge for every function $\alpha$. In the pure <Maxwell field>, action invariance under a compactly supported $\alpha$ implies
$$
0=\int d^4x\,E^\nu\partial_\nu\alpha
=-\int d^4x\,\alpha\partial_\nu E^\nu,
\qquad E^\nu=\partial_\mu F^{\mu\nu}.
$$
Thus $\partial_\nu E^\nu=0$ holds off shell, directly because $F^{\mu\nu}$ is antisymmetric. This exemplifies the <Noether second theorem>. Gauge fixing and the <Gupta-Bleuler null-state quotient> remove redundant photon components; they do not remove a physical global charge. Gauge transformations that are nontrivial at a boundary can carry boundary charges, so the redundancy statement concerns transformations satisfying the chosen boundary conditions. Finally, discrete symmetries such as the sign reversal of a real scalar or <parity symmetry in quantum field theory> can be important without furnishing a continuous-parameter <Noether current>.
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