Extended observables in Hamiltonian theories with constraints

In a classical Hamiltonian theory with second-class constraints the phase space functions on the constraint surface are observables. We give general formulas for extended observables, which are expressions representing the observables in the enveloping unconstrained phase space. These expressions satisfy in the unconstrained phase space a Poisson algebra of the same form as the Dirac bracket algebra of the observables on the constraint surface. The general formulas involve new differential operators that differentiate the Dirac bracket. Similar extended observables are also constructed for theories with first-class constraints which, however, are gauge-dependent. For such theories one may also construct gauge-invariant extensions with similar properties. Whenever extended observables exist the theory is expected to allow for a covariant quantization. A mapping procedure is proposed for covariant quantization of theories with second-class constraints.