Editing Noether's theorem (section)

=== Comments ===
Noether's theorem is an [[on shell]] theorem: it relies on use of the equations of motion—the classical path. It reflects the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that

: <math>\int_{\partial N} J^\mu ds_{\mu} \approx 0.</math>

The quantum analogs of Noether's theorem involving expectation values (e.g., <math display="inline">\left\langle\int d^{4}x~\partial \cdot \textbf{J} \right\rangle = 0</math>) probing [[off shell]] quantities as well are the [[Ward–Takahashi identity|Ward–Takahashi identities]].