I think I've seen reduced density matrix used to refer to the density matrix with off diagonal terms set to 0.
My guess is that "projector" refers to a function that projects a quantum state onto a basis vector of the Hilbert space.
The purpose of quantum gravity is to create a unified and consistent theory that reconciles quantum mechanics with general relativity, i.e., explains all the phenomena of both theories, and perhaps to predict new phenomena.
The most obvious obstacle is that quantum mechanics has been...