2013-09-29 12:26:50: 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.
2013-09-29 12:47:26: 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!
2013-09-29 13:04:33: We get the Ricci flatness condition (my latex skills are up to that!)
Tiziana offers to take some extra time for us and explain everything so we understand it properly – apparently we can skip lunch and our flights aren’t till the evening anyway! The audience laughs nervously. Nick Huggett points out that there’s no lunch to skip…
Why does string theory have to be conformally invariant? There are consistency reasons connected to the fact that the only recipe for calculating scattering amplitudes involves taking the partition function on the moduli Riemann space…
Why is this emergence? We’re discussing an inter-theoretic relation between GR and string theory. We need a Batterman/Butterfield approach to emergence. Notes that although Butterfield and Batterman agree about emergence involving limits, they disagree as to whether this happens before or after the limit and hence on whether emergence is compatible with reduction. Notes that the notion of novelty applicable here will be epistemic – we do get novel robust behaviour characterised by length scale that can’t be (explanatorily?) captured by the reducing theory.
What have we been doing here? Looks like a case of reduction and emergence but depends how we think about conformal invariance – this comes in as a kind of guiding principle – we’re not deriving GR from within string theory itself. There is a no man’s land between the theory in which we have to use the guiding principle.
2013-09-29 13:16:18: Muntean (IM): This is a case in which novelty in emergence is very obvious – the eisntein field equations are completely unexpected!
TV: Yes, it’s magic!
IM: But if we’re talking about Batterman and Butterfield’s approaches where’s the limit that we’re taking in the Polyakov action?
TV: Take a full expansion and then takes a low energy limit - but the emergence isn’t at this stage – this is just a matter of focussing attention on the lowest level of perturbation. The deduction of GR comes after we take the limit a la Butterfield. But she’s worried that she’s not really doing the right kind of deduction because it involves assumptions about conformal invariance which isn’t an intrinsic formal feature of the theory. If we think the conformal invariance condition is external, then we have a stronger emergence a la Batterman.
Pitts: If we’re integrating over the worldsheet in the action, how can we contract the metric – we have worldsheet indices running only from 1-2, so how do we get out vanishing of the whole Ricci tensor?
TZ: If we think about conformal invariance, then we do need vanishing of the Ricci tensor.