= Solution
A symmetry preserves the action, possibly up to a boundary term. In a <classical field theory>, this connects transformations of fields to conserved quantities. In a <quantum field theory>, it also constrains states, observables and scattering amplitudes. Continuous symmetries have <Noether currents>; discrete symmetries can be equally important without supplying a current through the same theorem.
For a local density depending on fields $\Phi_a$ and their first derivatives, define $\Pi_a^\mu=\partial\mathcal L/\partial(\partial_\mu\Phi_a)$ and $\mathcal E_a=\partial\mathcal L/\partial\Phi_a-\partial_\mu\Pi_a^\mu$. Suppose a one-parameter transformation has fixed-coordinate variation $\delta\Phi_a=\varepsilon R_a$ and satisfies $\delta\mathcal L=\varepsilon\partial_\mu K^\mu$. The <Noether's first theorem> asserts that
$$
\boxed{j^\mu=\sum_a\Pi_a^\mu R_a-K^\mu,\qquad\partial_\mu j^\mu=0\quad\text{on the field equations}.}
$$
For Grassmann-valued fields the derivatives and ordering are chosen consistently; the same integration-by-parts argument applies. To prove the theorem, expand the density variation and integrate its derivative term once algebraically:
$$
\delta\mathcal L=\sum_a\left(\frac{\partial\mathcal L}{\partial\Phi_a}\delta\Phi_a+\Pi_a^\mu\partial_\mu\delta\Phi_a\right)=\sum_a\mathcal E_a\delta\Phi_a+\partial_\mu\left(\sum_a\Pi_a^\mu\delta\Phi_a\right).
$$
Comparison with the assumed divergence gives $\partial_\mu j^\mu=-\sum_a\mathcal E_aR_a$. On shell every Euler-Lagrange expression vanishes. Integrating this continuity equation over space gives the <Noether charge> $Q=\int j^0d^3x$ and $dQ/dt=-\int_{\partial\mathbb R^3}\mathbf j\cdot d\mathbf S$. Thus $Q$ is conserved whenever the boundary flux vanishes. Boundary conditions are part of the conservation statement; the local identity alone does not discard nonzero flux.
Translations illustrate the theorem in every relativistic field theory. For $R_a=a^\nu\partial_\nu\Phi_a$ and $K^\mu=a^\mu\mathcal L$, the current is $a^\nu T^\mu{}_{\nu}$, where
$$
T^\mu{}_{\nu}=\sum_a\Pi_a^\mu\partial_\nu\Phi_a-\delta^\mu{}_{\nu}\mathcal L.
$$
The four charges are energy and momentum. For the <Klein-Gordon field> they reduce to the expressions derived in Question 1. <Lorentz invariance> gives angular momentum and boost currents. With a symmetric improved <stress-energy tensor>, these are $M^{\lambda\mu\nu}=x^\mu T^{\lambda\nu}-x^\nu T^{\lambda\mu}$. The canonical Dirac expression also carries an intrinsic spin current; improvement incorporates it into the symmetric tensor. Together translations and Lorentz transformations form the <Poincare group>.
Internal symmetries act on field components without moving spacetime points. The <global phase symmetry of a complex scalar field>, $\delta\phi=-i\varepsilon\phi$, has
$$
j^\mu=i(\phi^*\partial^\mu\phi-\phi\partial^\mu\phi^*).
$$
Its charge distinguishes particles and antiparticles. A Dirac phase transformation $\delta\psi=-i\varepsilon\psi$ gives $j^\mu=\bar\psi\gamma^\mu\psi$. This is preserved by the real-scalar <Yukawa interaction>, because the phases of $\bar\psi$ and $\psi$ cancel; consequently fermion number is conserved even while scalar and fermion particles interact. A multiplet of real scalars with a potential depending only on $\sum_a\phi_a^2$ has orthogonal internal rotations. Their antisymmetric currents can be written $j_{ab}^\mu=\phi_a\partial^\mu\phi_b-\phi_b\partial^\mu\phi_a$.
Discrete examples include $\phi\mapsto-\phi$ for an even real-scalar potential, <parity symmetry in quantum field theory>, and <charge conjugation>, which interchanges a complex scalar with its conjugate and reverses the electromagnetic potential. Discrete transformations are not generated by an infinitesimal continuous parameter, so <Noether theorem> does not attach a local conserved current to each of them. Nevertheless they forbid interaction terms and relate physical processes.
A <global symmetry in field theory> uses parameters constant across spacetime and can act nontrivially on physical states. A <gauge redundancy> allows arbitrary spacetime-dependent parameters and relates descriptions of the same physical configuration. In <scalar electrodynamics>, replacing the scalar's global phase by a local phase requires the compensating <gauge transformation> of $A_\mu$ and the <gauge covariant derivative>; the invariant contraction uses the opposite representation for $\phi^*$. In <Yang-Mills theory>, matter transforms in a representation of a non-Abelian gauge group and the gauge potential transforms so that covariant derivatives and field strengths transform covariantly. Gauge fixing, such as <Coulomb gauge>, chooses representatives of these descriptions rather than changing physical predictions. The current-contracted propagator identity of Question 3 illustrates why a noncovariant gauge choice leaves scattering Lorentz invariant.
The arbitrary local parameter produces a stronger statement than one independent conserved charge for each function. For the abelian scalar transformation, vary the action with an arbitrary compactly supported $\alpha(x)$ and integrate its derivative term:
$$
0=\delta S=\int d^4x\,\alpha\left[-\partial_\mu\mathcal E_A^\mu-ie\phi\mathcal E_\phi+ie\phi^*\mathcal E_{\phi^*}\right].
$$
Therefore
$$
\boxed{-\partial_\mu\mathcal E_A^\mu-ie\phi\mathcal E_\phi+ie\phi^*\mathcal E_{\phi^*}\equiv0.}
$$
This consequence of <Noether second theorem>, the <Noether identity for abelian scalar gauge symmetry>, holds off shell: gauge invariance makes the field equations dependent. In Hamiltonian language the same redundancy is reflected by the <Gauss law constraint in gauge theory> and the elimination of unphysical gauge degrees of freedom. Gauge transformations that vanish at the boundary are redundancies; transformations with nontrivial boundary behaviour can instead carry physical surface charges. This is why identifying every gauge transformation with a zero-charge operation would be too strong.
Finally, symmetry of the action need not imply symmetry of a chosen vacuum. For a complex scalar with a symmetry-preserving potential whose minima occur at nonzero $|\phi|$, different constant phases label degenerate vacua. Choosing one breaks the global phase symmetry of the state; the <Goldstone theorem> supplies a massless mode under its usual relativistic assumptions. When the same phase is gauged, the <Higgs mechanism> describes the phase degree of freedom becoming the longitudinal polarization of a massive gauge boson; the local redundancy is still present in the underlying description. At the quantum level a classical symmetry can also fail through a <quantum anomaly>, so a classical Noether derivation alone does not guarantee an exact quantum Ward identity. \b[Conserved charges, physical global transformations and local gauge redundancy are related but distinct consequences of symmetry.]
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