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Adding ${\partial }_{\rho }{B}^{\rho \mu \nu }$ with $B^{\rho \mu \nu }=-B^{\mu \rho \nu }$ to a stress-energy tensor preserves its divergence identically, since commuting partial derivatives annihilates the antisymmetric pair. The total four-momentum changes only by a spatial boundary term. Improvements can make local densities symmetric or gauge-invariant; equality of integrated charges requires the chosen boundary conditions. Equation-of-motion terms can also relate representatives on shell.

The two photon orders in a tree-level charged-fermion amplitude must be summed. Contracting one photon vertex with its momentum writes the contracted slash as a difference of inverse Dirac propagators. The external Dirac equations remove the on shell inverse factors; the two remaining terms have equal size and opposite sign. Thus the sum vanishes under the momentum replacement although individual diagrams generally do not. This implements the gauge equivalence of photon polarization vectors.