Getting GR out of string theory – we’re doing perturbative expansion of the classical action, but that only works at weak curvature. If it were high curvature, we’d need a different approach.

Show how QFT imposes constraints on how the string theory has to interact with the background metric – mathematically beautiful!

Puts up a slide with the weyl transformations on the screen – notes that conformal transformations act on coordinates and weyl transformations act on the metric.

QFT will break weyl invariance…

Now puts up the Polyakov action on the screen and explains the terms – reminds us that we have a worldsheet metric and a target metric (conformal symmetry wrt string metric, not target metric!)

Let’s quantize this action… Perturb the classical solution and move to a quantum description. Transform coordinates until we find Riemannian normal reference system in which everything looks tidy. Perturb the metric and get a term that depends on Riemann tensor. Plug this back into the action and get a divergence. (Offers to demonstrate this, but the audience agree to take her at her word). We want to introduce a length scale to fix this, but we had a conformally invariant theory! Apparently if we understand how this can be the case, we’ll understand how the Ricci flatness condition comes out!

We do a bit of work on the action integral, and the Ricci tensor starts to appear. (This blogger apologises for the fact that her latex equation skills aren’t quite up to the task of typing all this out for the reader.) If we look closely, we see that things aren’t conformally invariant after all!

Tiziana’s going to talk about emergenet spacetime in string theory…

When we have a theory that’s so far ahead of our ability to test it, emergence of spacetime is going to be an important test.

But of course we need to define a notion of emergence! She’s going to focus on emergence of ordinary 4D GR spacetime – not discussing emergence of, say, one exotic spacetime from another in the case of dualities.

2 ingredients: First, string theory predicts the existence of spin-2 particles (gravitons). What does this mean? The particle appears in the spectrum of states of the quantum state – has the right mathematical properties to describe the postulated physical particle. Let’s keep that first ingredient in the back of our mind, but focus on the second…

Second ingredient: Derivation of GR from string theory (as discussed by Huggett yesterday).

So on the one hand we have gravitons, and on the other the idea that we get a background metric that obeys the Einstein Field Equations. Together these give us emergence of spacetime.

Questions:

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?

Geometrogenesis:

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.

AdS/CFT 

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.