  In fundamentally discrete approaches to quantum gravity such as loop quantum
gravity, spin-foam models, group field theories or Regge calculus observables
are functions on discrete geometries. We present a bra-ket formalism of
function spaces and discrete calculus on abstract simplicial complexes equipped
with geometry and apply it to the mentioned theories of quantum gravity. In
particular we focus on the quantum geometric Laplacian and discuss as an
example the expectation value of the heat kernel trace from which the spectral
dimension follows.
