ad 2). What is vacuum?
It surely isn’t a simple object. Maybe one of the following?
– lowest-energy state
– invariant under symmetries of the background space-time
– encode the geometry of Minkowski space
– cyclic state which generates the Fock space.
In LQG, we have a kinematical vacuum. (Simple: it’s a box!)
In all seriousness: It’s a peculiar state. In LQG, it’s maximally squeezed state, in the sense that all spatial geometry operators have zero expectation value on it, and zero fluctuations! It’s a state of no spatial geometry. Truly nothing!
As in non-gravitational quantum theory, we can build a highly excited state from the vacuum by applying loads of ‘creation operators’, based on graphs.
The LQG representation of the kinematical observable algebra is unique! I.e., it is
– irreducible, and it
– carries a representation of spatial diffeos.