Action and Noether’s theorem

Work in flat four-dimensional spacetime, with $\hbar=c=1$ and metric $\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$. The field $\phi(x)$ assigns a value to every spacetime event. An action integrates a local Lagrangian density over spacetime. “Local” here means that the density depends on fields and a finite set of derivatives at the same event. It does not mean that the resulting solution has no correlations between distant points.

Compare $\phi$ with $\phi+\delta\phi$, keeping the variation zero on the boundary. Integration by parts moves a derivative off $\delta\phi$ and produces a boundary contribution. The remaining bulk coefficient must vanish because the interior variation is arbitrary. We have not minimized a positive function: Lorentzian actions generally have stationary points, which can be saddles. Keeping that distinction prevents a misleading picture of nature “choosing the least energy history.”

Suppose a constant infinitesimal parameter $\epsilon$ changes the fields by $\delta\phi_a=\epsilon\Delta_a$ without changing the action, apart from a total derivative. Temporarily let $\epsilon$ depend on position. This is a bookkeeping device, not a claim that the theory has acquired a gauge symmetry. The coefficient multiplying $\partial_\mu\epsilon$ identifies a current. On solutions of the equations of motion, that current has zero divergence.

For a real shift symmetry, $\phi\mapsto\phi+\epsilon$, the massless derivative Lagrangian is unchanged. The associated current is $j^\mu=\partial^\mu\phi$. Its continuity equation is exactly the massless wave equation. Add a potential depending on $\phi$ and the shift usually fails. A symmetry must be tested on the action before one names its charge; recognizing a familiar-looking transformation is not sufficient.

Integrate the continuity equation over a spatial region. A decreasing charge inside can be exactly balanced by current flowing out through its boundary. The integral over all space is constant only when the flux at infinity vanishes, or another stated boundary condition supplies the same result. In a periodic box, opposite faces cancel. In an open detector volume, charge inside the apparatus can change even while the theory respects its symmetry.

Time translations similarly lead to energy conservation when the action has no explicit time dependence. Spatial translations lead to momentum. Currents themselves are not unique: adding the divergence of an antisymmetric tensor leaves the continuity equation intact and, with suitable boundaries, leaves the charge intact. This freedom matters when comparing canonical and improved stress tensors. Conservation is the robust statement; one particular local expression may be a convenient representative.