First class constraint

Summary

In Dirac's theory of constrained Hamiltonian systems, a constraint is first class when its Poisson bracket with every constraint vanishes on the constraint surface. Otherwise it is second class. First class constraints generate gauge transformations: motions along them change the description of the state but not its physical content. Second class constraints instead remove pairs of variables, and they are handled with the Dirac bracket. This distinction is central in the Hamiltonian treatment of gauge theories and general relativity. See the definition.