### 3.3 Quantization with partial gauge fixing

As mentioned above the reduced phase space is the space of gauge orbits endowed with a
symplectic structure inherited from the original one in the full phase space. A strategy
that is useful in the context of midisuperspaces is to partially fix the gauge. In practice this
means that the dimensionality of the constraint hypersurface (and, as a consequence, of the
gauge orbits) is reduced. This may be useful if one is interested in leaving some residual gauge
symmetry in the model on purpose (such as radial diffeomorphisms in spherically-symmetric
models [54]) to check if one can deal with it in some quantization scheme. In other situations
the natural gauge fixing conditions simply fail to fix the gauge completely; this happens, for
example, in the compact Gowdy models [165]. In such cases the residual gauge invariance is
usually treated by employing Dirac’s procedure, though other approaches are, of course, possible.
A very attractive feature of the resulting formulation is that the quantum dynamics of the
model is given by a “time” dependent Hamiltonian that can be studied in great detail due to its
relatively simple structure. This is possible because its meaning can be understood by using results
developed in the study of the time-dependent harmonic oscillator (see [100] and references
therein).