  The problem of observables in classical and quantum gravity is a
long-standing one. It is sometimes argued that observable quantities should be
diffeomorphsm invariant, following the philosophy of Dirac. We argue that
diffeomorphism invariance in classical and quantum gravity is not primarily a
case of gauge invariance but rather an example of spontaneous symmetry breaking
of the diffeomorphism group. As a consequence, observables fall into two
classes, Dirac observables which are invariant under the full diffeomorphism
group and more general observables which take on different (expectation) values
in the different phases of the broken (diffeomorphims) symmetry group. To this
latter class belong for example scalar functions of the metric tensor.
