2 questions. 1 on Field theory, the other on Ads/CFT.

[sox]

My first question:

QED has U(1) symmetry. The transformation is a function of x (psi -> psi*theta(x)). How does this depend on the spacetime? Do interactions therefore differ between curved space and flat space? Is this what leads on to QFT on curved spacetime research?

My second question:

If we believe that the curvature of spacetime in our universe is positive, why do string theorists like to play around with the Ads/CFT correspondance? Ads is a space with negative curvature is it not? If so, then surely any correspondance between string theory and gauge theory, whilst interesting, is ultimatley incorrect based on the fact that the string theory has been set up in an unphysical space?

Any illumination here would be appreciated.

Sox

 

[Demystifier]

Do interactions therefore differ between curved space and flat space? Is this what leads on to QFT on curved spacetime research?

Yes and yes. The interactions are obtained by replacing ordinary derivatives by covariant derivatives, which are nontrivial in curved spacetime.

 

[sox]

Oh yeah! I had forgotten that the derivative changes in curved spacetime.
Thanks.

Anyone got an answer for the second one?

 

[chrispb]

We don't actually believe that spacetime has positive curvature, at least as I understand it; currently, the measurements are consistent with a flat spacetime (no curvature). What you may be confusing this with is the statement that our universe is expanding. That means that it's inflating faster and faster with respect to time, or the second derivative of the size of the universe with respect to time is positive. This is uncorrelated with the curvature of the universe. The cosmological constant does indeed contribute to the expansion rate of the universe, but so do matter and energy.

One could imagine a 4d gauge group that appeared to be (at least approximately) conformal above, say, 10 TeV, but confined below that. That would be dual to AdS space with boundaries (D3-branes or something of the sort). That's one situation in which one could imagine using AdS/CFT in our universe. Also, you could pretend that our universe was in fact AdS, but with a damn big radius of curvature, meaning a 3d CFT would exist at the boundary. This avenue of research is less popular since it's a little less clear what to do to make progress. However, it's certainly an interesting possibility.