  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.
