# Thought on T-duality?

1. Jul 3, 2015

### "pi"mp

I was recently familiarizing myself with T-duality and had a thought that I'm not sure is correct:

In QFT when we probe too short a distance, the theory explodes and gives infinite results. In string theory, from what I understand, we don't really have any business asking about length scales below $\sqrt{\alpha'}$. Yet it seems that instead of exploding like QFT, the theory politely "cushions" us when we try to do so, and places us at some larger radius thanks to T-duality.

Is this a sensible way to interpret the result? Or is there a better way? Seems like something along these lines should hold also for strong/weak dualities in M-theory

2. Jul 5, 2015

### julian

In string theory, even ignoring quantum mechanics, point particles are replaced by strings so space-time become fuzzy' at scales of about <tex>\sqrt{\alpha}</tex>. The full theory employing string tension and quantum mechanics is only beginning to form shape, it is said.

Something that shocked me the other week was watching a video of Witten accepting some Newton prize
(not shocked by the accolation - but something he said) . He said that something that had motivated strings over particles is that in GR space-time points have no operational meaning. Here was me thinking this point was the reserve of LQG/General relativists! He said that GR cannot have a gauge potential that is a function of a space-time point because space-time points have no operational meaning in GR (or so it seems) so we should replace point particles by strings!!!!

I think Ashtekar may have mentioned something about how there was some theorem that said what they were trying wasn't possible, but they found a loop-hole (pun intended).

For example, as Rovelli has pointed out Wightman functions or n-point functions encode all the physical information of a QFT, we calculate scattering amplitudes from them. Rovelli et al have been able to give a background-independent meaning to scattering amplitudes by assigning an operational meaning to them as actual experiments that are performed...

At the same time string-theory/M-theory is still yet to have a background-independent formulation, despite progress (e.g AdS/CFT). Brian Greene, for example and Smolin think a way forward to unifying string theories and its dualities is background-independence?

Something T-duality reminds me of vaguely is weaves...so before spin networks were introduced (in LQG) as eigen-vectors of area/volume operators we had Wilson-loop' approximations of a given spatial metric. People were surprised that when they refined the loop beyond the Plank scale, instead of getting a better approximation of the spatial geometry it started approximating a different geometry, in fact the more loops you added the bigger the space got?

In case you didn't know background-independence, in classical GR this means you can replace one metric with another so that small and large distances' between abstractly defined space-time points are gauge-equivalent, this is a profound shift from SR/Newton that has not been fully understood in any quantum theory of gravity yet let alone string theory? But might be relevant to your question.

Last edited: Jul 5, 2015
3. Jul 5, 2015

### julian

You did ask for a thought'...that all it is.

4. Jul 5, 2015

### julian

In a theory that is background-independent, as GR is and QM versions should be, it is the gauge equivalence between small and large distances that cushions the blow. Thiemann has show in his book for example that is precisely the removal of background-independence that introduces infinities into the QFT.

Last edited: Jul 5, 2015
5. Jul 7, 2015

### "pi"mp

I'm not sure I understand what you mean by background-independence in GR.

In the same talk I believe you're referring to above, Witten also said that we need a background-independent string theory where the fundamental geometric objects are not points or lines, but rather the string worldsheets themselves. This makes sense to me, as we wouldn't need to embed the worldsheets into any ambient background. But how can GR be background-independent, when the metric describes the structure of the universe as a whole?

6. Jul 7, 2015

### julian

A solution to Einstein's equations isn't particular space-time geometry, a particular background geometry, it is really an equivalence of distinct space-time geometries related to each other through (what mathematicians would call) diffeomorphisms. This follows from Einstein's hole argument. I posted a an explanation here http://physics.stackexchange.com/qu...endence-and-how-important-is-it/192989#192989 (ivan44) if you want to have read of it . (Do excuse mistakes in my post above and lack of proper explanation, was a bit tired then). To get more complete explanation of these conceptual matters of classical GR you should have a look at Rovelli's book "Quantum Gravity" - there is a free draft version at http://www.cpt.univ-mrs.fr/~rovelli/book.pdf.

I may not have got through the whole of Witten's Newton 2010 lecture: (here's a link to part 1) someone gave a link to the whole lecture. I'll have to listen to it again.

Rovelli gives a quote of Witten on background-independence in string theory, and Rovelli makes comments about the background-independence of string theory in: "Strings, loops and others: a critical survey of the present approaches to quantum gravity" http://arxiv.org/pdf/gr-qc/9803024.pdf - that was a while ago.

Last edited: Jul 7, 2015