  The present article is primarily a review of the projection-operator approach
to quantize systems with constraints. We study the quantization of systems with
general first- and second-class constraints from the point of view of
coherent-state, phase-space path integration, and show that all such cases may
be treated, within the original classical phase space, by using suitable
path-integral measures for the Lagrange multipliers which ensure that the
quantum system satisfies the appropriate quantum constraint conditions. Unlike
conventional methods, our procedures involve no $\delta$-functionals of the
classical constraints, no need for dynamical gauge fixing of first-class
constraints nor any average thereover, no need to eliminate second-class
constraints, no potentially ambiguous determinants, as well as no need to add
auxiliary dynamical variables expanding the phase space beyond its original
classical formulation, including no ghosts. Besides several pedagogical
examples, we also study: (i) the quantization procedure for reparameterization
invariant models, (ii) systems for which the original set of Lagrange
mutipliers are elevated to the status of dynamical variables and used to define
an extended dynamical system which is completed with the addition of suitable
conjugates and new sets of constraints and their associated Lagrange
multipliers, (iii) special examples of alternative but equivalent formulations
of given first-class constraints, as well as (iv) a comparison of both regular
and irregular constraints.
