  Pure gravitational plane waves are considered as a special case of spacetimes
with two commuting spacelike Killing vector fields. Starting with a
midisuperspace that describes this kind of spacetimes, we introduce
gauge-fixing and symmetry conditions that remove all non-physical degrees of
freedom and ensure that the classical solutions are plane waves. In this way,
we arrive at a reduced model with no constraints and whose only degrees of
freedom are given by two fields. In a suitable coordinate system, the reduced
Hamiltonian that generates the time evolution of this model turns out to
vanish, so that all relevant information is contained in the symplectic
structure. We calculate this symplectic structure and particularize our
discussion to the case of linearly polarized plane waves. The reduced phase
space can then be described by an infinite set of annihilation and creation
like variables. We finally quantize the linearly polarized model by introducing
a Fock representation for these variables.
