Second-class constraint ($\det \{{\chi }_a,{\chi }_b\}\ne 0$)

A collection of Hamiltonian constraints is second class when its Poisson bracket matrix $C_{ab}=\{{\chi }_a,{\chi }_b\}$ is invertible on the constraint surface. These conditions eliminate canonical variables rather than generate a gauge symmetry. Their consistent reduced bracket is the Dirac bracket, $\{F,G{\}}_D=\{F,G\}-\{F,{\chi }_a\}(C^{-1}{)}^{ab}\{{\chi }_b,G\}$. The zero temporal canonical momentum of a massive Proca field and its massive Gauss law form such a pair; imposing an independent temporal canonical commutation relation would contradict the constraint.