  We investigate using plane fronted gravitational wave space-times as model
systems to study loop quantization techniques and dispersion relations. In this
classical analysis, we start with planar symmetric space-times in the real
connection formulation. We reduce via Dirac constraint analysis to a final form
with one canonical pair and one constraint, equivalent to the metric and
Einstein equations of plane-fronted with parallel rays waves. Due to the
symmetries and use of special coordinates general covariance is broken.
However, this allows us to simply express the constraints of the consistent
system. A recursive construction of Dirac brackets results in non-local
brackets, analogous to those of self-dual fields, for the triad variables
chosen in this approach.
