  When quantum fields are studied on manifolds with boundary, the corresponding
one-loop quantum theory for bosonic gauge fields with linear covariant gauges
needs the assignment of suitable boundary conditions for elliptic differential
operators of Laplace type. There are however deep reasons to modify such a
scheme and allow for pseudo-differential boundary-value problems. When the
boundary operator is allowed to be pseudo-differential while remaining a
projector, the conditions on its kernel leading to strong ellipticity of the
boundary-value problem are studied in detail. This makes it possible to develop
a theory of one-loop quantum gravity from first principles only, i.e. the
physical principle of invariance under infinitesimal diffeomorphisms and the
mathematical requirement of a strongly elliptic theory.
