In order to test the canonical quantization programme for general relativity
we introduce a reduced model for a real sector of complexified Ashtekar gravity
which captures important properties of the full theory. While it does not
correspond to a subset of Einstein's gravity it has the advantage that the
programme of canonical quantization can be carried out completely and
explicitly, both, via the reduced phase space approach or along the lines of
the algebraic quantization programme. This model stands in close correspondence
to the frequently treated cylindrically symmetric waves. In contrast to other
models that have been looked at up to now in terms of the new variables the
reduced phase space is infinite dimensional while the scalar constraint is
genuinely bilinear in the momenta. The infinite number of Dirac observables can
be expressed in compact and explicit form in terms of the original phase space
variables. They turn out, as expected, to be non-local and form naturally a set
of countable cardinality.Comment: 32p, LATE
