  The quantisation of gauge theories usually procedes through the introduction
of ghost fields and BRST symmetry. In the case of quantum gravity in the
presence of boundaries, the BRST-invariant boundary value problem for the gauge
field operators is non-elliptic, and consequently the definition of the
effective action using heat-kernel techniques becomes problematic. This paper
examines general classes of BRST-invariant boundary conditions and presents new
boundary conditions for quantum gravity which fix the extrinsic curvature on
the boundary and lead to a well-defined effective action. This prompts a
discussion of the wider issue of non-ellipticity in BRST-invariant boundary
value problems and when the use of gauge-fixing terms on the boundary can
resolve the issue.
