BI in Loop Quantum Gravity:
In what sense are spin networks and spin foams implementations of BI? We set things up such that these fundamental objects are diffeo invariant so type 1 BI is implemented.
But can we have BI in another sense? How about approach 3?
Will tackle this notion by looking at Quantum Graphity – a theory that features emergent matter degrees of freedom that create geometry and gravity. He starts off by giving us some graph theory preliminaries (see the slides in the handout section). In this theory, geometrogenesis works via string-net condensation – what emerges is a view of spacetime and matter that Mark claims is a Machian view because spacetime depends (entirely?) on matter fields. (I’m puzzled here because then there’s talk about spacetime degrees of freedom, but we’re going through technical material quite fast here, so I’m sure I’ve missed something!)
Glossing over the details here, what we end up with is an emergent manifold coming out of the graph theoretic description – as long as we’re looking at large enough scales that our really discrete structure looks continuous, we get approximate manifold structure.
So how much BI in this model? Ultimate fulfilment of the Anderson project! (But now we’re going through slides a little fast because we’re running out of time).
What about time? Type 1 background independence (e.g. in LQG) leads to the elimination of time because dynamical evolution is described as a sum over histories. Type 2 might have a notion of fundamental time.