### 3.1 Reduced phase-space quantization

The reduced phase-space quantization is simply the quantization of the reduced space of the
constrained Hamiltonian system whenever this is possible. Notice that the process of taking quotients is
highly non-trivial and many desirable regularity properties need not be preserved. In the models that
we consider in this paper we will suppose that no obstructions appear so that the reduced
phase space is well defined. Even in this case some difficulties may (and in practice do) arise, in
particular:
- The characterization of the quotient space is usually very difficult even when the quotient
itself is well defined. In practice this reduced phase space is effectively described by using a
gauge fixing procedure that picks a single field configuration from each gauge orbit in a smooth
way (whenever this is possible).
- In general, there is no guarantee that will be the cotangent space of a reduced configuration
manifold . Although there are techniques that may allow us to tackle this situation (i.e.,
geometric quantization [226]) they are not always straightforward to apply.
- It may be difficult to extract physics from the reduced phase-space description. In practice,
even when the reduction can be carried out in an explicit way, it is very difficult to reexpress
the results in terms of the original variables in which the problem is naturally written.

The reduced phase-space description amounts to the identification of the true physical degrees
of freedom of the system. As a rule, for many physical models (and certainly in the case of
gravitational theories) this description is either unavailable or extremely difficult to handle. In these
cases one is forced to learn how to live with the redundant descriptions provided by gauge
theories and how to handle the constraints both at the classical and quantum levels. Finally, it is
important to notice that whenever the reduced phase space can be characterized by means of
a gauge fixing, the quantization ambiguities that may arise do not originate in the different
gauge choices – as long as they are acceptable – but rather in the possibility of having different
quantizations for a given classical model. This is so because from the classical point of view
they are explicit representations of the same abstract object: the reduced phase space [113].
There are several approaches to the quantization of gauge systems that we will briefly discuss
next.