On the vacuum expectation value of the σ-term

Performing the equal-time limit under the integral of the spectral representation for the vacuum expectation value of the current commutator it is shown (as already suggested by models) that the vacuum expectation value of the equal-time commutator of a hermitean charge and the divergence of the corresponding (hermitean) vector or axial vector current (“σ-term”) vanishes only if the current is conserved. This then implies that a nonconserved hermitean charge cannot annihilate the vacuum, in agreement with Coleman's theorem.