Dietrich: Not sure she agrees that in LQG spin networks and spin foams are BI in the relevant sense. There are questions about how we get emergent manifolds just as in Quantum graphity, so can we really divide the two approaches.
Oriti: geometrogenesis doesn’t itself mark out particular theories – he mentioned it in the context of LQG. One of the important distinctions is that in some models (quantum graphity?) we have introduction of a time variable and hence perhaps less background independence.
Mark: understood Q. graphity as not having a single notion of time not sum over histories.
Oriti: we can understand it as a sum over histories but there’s still externally introduced time.
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.
In what sense are dualities like AdS/CFT implementations of BI? Bulk diffeomorphisms that leave the boundary unchanged are gauge symmetries of the bulk theory, so there’s background independence here in our first sense.
Can we go further than our 3 approaches to classical BI?
Smolin: dimension, topology, differential structure and signature are fixed in GR and do count as background.
(Wuthrich – some of these (like dimension) aren’t usually thought of as varying between models of GR! … so we need to bear in mind that we’re thinking about a bigger class of GR-like models than we might think.)
So Mark wants to ask if we can create an approach to QG that’s more background independent than GR.
A question for the end of the talk: Does BI lead to The End of Time (Barbour) or Time Reborn (Smolin)?
He’ll set things up by telling us about classical varieties of BI, so…
3 views of Background Independence:
a. Diffeomorphism Invariance: Following Guilini and Rickles <M,D,B> where D are dynamical fields and B are background fields. We have general covariance if diffeomorphisms applied to both B and D take solutions to solutions. We have invariance if we can transform only the D fields and take solutions to solutions. So GR, with no B fields is both covariant and invariant under diffeomorphisms…
b. Absolute v. Dynamical objects: Anderson and Friedman. Absolute objects are those that are the same in every solution of the theory… standard Anderson line.
c. A matter of degree?: Following Gordon Belot – Backgrounf is a fixed stage on which the physics of the theory is set. We have clear cases of BI when we have fixed fields determining geometry. Need to define an abstract geometry – fully background dependent theories are then those that only realize one abstract geometry (so again, every solution has the same abstract geometry). In full BI, no two solutions have the same abstract geometry. Near background dependence – family of abstract geometries realized in the theory is finite dimensional. Near background independence – family of abstract geometries is infinite dimensional. (Carlip QG in 2+1 dimensions is a case of near background dependent).
Oriti: isn’t the actual geometry here fixed as minkowski? Only topological degrees of freedom.
Mark’s going to try to get a handle on the mystery of background independence (henceforth BI) – try to understand how we can think about it in classical field theories, and identify QG analogues to these. He’s not going to try to tell us why and whether we should care about BI.
Question about connection with observables and diffeomorphisms. Replies never used diff talk. Rejects invariance in favor of covariance. See Trautmann: need actual points, and never move them alone, only individuated by fields.
Is main idea that we introduce points? No. Doesn’t introduce them primitively. Thinks active diffs don’t make much sense: like building a building, seeing you don’t need it, and knocking it down.
Accused of adding points and having more information. Responds that you only get them after the fact. But what if have two manifolds? Why? Question continues: moving points around gives same physical situation. Pitts: we all agree that points exist in GR, just not individuated until have the metric.
This was a confusing QA session. Not sure what the actual worry of the questions was. Butterfield tries to redirect from point type worries to the actual issue of whether we need to coordinate the constraints in order to generate gauge transformations.
I could not really follow Dittrich’s point. If anyone is reading this, perhaps that can be filled in here.
Argument at length about the connection between phase space and observables generally. Chair cut it off.
Pitts tells us that perhaps there is no coherent notion of Bergmann observables. On the other hand we can tell what Dirac observables are supposed to be – so his conception can be fixed.
So Pitts gets Earman’s fondness for Hamiltonian formulation and Maudlin’s desire for change to live happily together. Should have been obvious, he says, because Lagrangian and Hamiltonian formulations should be equivalent.
Result: Change is having no time-like Killing field, and we can get that in homogeneous GR.