Comment by Dittrich: these observables are not diffeo-invariant, however. Indeed, an interesting issue, says the speaker.

The interesting and important question is to identify the quantities in terms of which one defines the coarse graining. One choice is to look at intrinsic and extrinsic curvature [as in the definition of coherent states], another is to look at volume operator [it has been considered in the loop quantum cosmology context].

Move on to the definition of the decoherence functional in this context. Fine-grained histories of spin networks are the individual spin foams, weighted by the “fundamental” spin foam amplitudes that the theory associated to them. Coarse graining is imposed on “bulk configurations”, that is on the spin foams being summed over. More precisely, one coarse grains all histories/spin foams that correspond to the same diff-invariant properties [identified in terms of suitable observables].

notion of coherent states as semiclassical states peaking on the classical phase space variables associated to a given graph [no superposition of graphs considered, here]

Summary of the dynamics: analog of path integral [sum-over-histories] is sum over all 2-complexes and sum over colorings, weighted by quantum amplitude.

Move on to the covariant formulation of loop quantum gravity, that is to the spin foam formalism. Brief summary of the kinematical states of the theory [cylindrical functions and spin networks, as in lattice gauge theory]. Summary of the covariant definition of the dynamics: spin foams as histories of spin networks, defined by 2-complexes with the same type of coloring as spin network states. 

More on the properties of the decoherence functional. Generalization of these concepts and definitions to the field theory [formulated in terms of path integrals] context and to GR [in particular in its Hamiltonian formulation].

More detailed definition of partition and coarse-graining of histories in this example. definition of class operator and decoherence functional for the histories. identification of the [decoherence] conditions that are necessary to be able to assign probabilities to the histories [basically amounting exactly to the suppression of interference between histories).

Summary of idea of sum over histories as a way to describe in a covariant way the dynamics of a quantum system. Example of the [fine-grained] histories of a single particle and basic notion of coarse graining of such histories.

One motivation to use decoherence is that this Copenhagen story is questionable in many ways. One more motivation is that in the general relativistic context, no time variable can be used to describe when the collapse happens, and a covariant version of the suppression of superposition and interference should be looked for.