On to part 2, Duality!

Strings can wrap around a compact dimension, which leads to a number w, the ‘winding number’ of the string.  Classically one would expect that to be fixed, but when we get interactions w will be able to change, i.e. be dynamic, and will have a corresponding quantum number.

Huggett goes on to explain T-duality.  Winding and momentum can change roles (n <–> w) in the Hamiltonian (and R <–> 1/R) .  The dynamics of the spatial wavefunction becomes the dynamics of the winding wavefunction in the dual, and vice-versa…  the pattern of observed quantities is preserved.  T-duality is like a translation manual between two theories.

This raises Huggett’s 3rd question: How should we understand T-duality?  Do duals describe physically distinct situations?

Huggett raises a new question: should the compact dimensions be considered “spatial” in the same sense of as the large dimensions?  Or are they better thought of as representing internal degrees of freedom?  Are there really 26 dimensions to spacetime in bosonic string theory?  

Somebody keeps editing my posts on Huggett.  The good news is that the edits are truth-preserving.

Huggett now is plowing through his simple classical example, the solutions for the string’s behaviour and stress-energy. One thing that pops out are the Virasoro constraints, which correspond to the conformal symmetry of the string on the worldsheet, which Huggett mentioned earlier.  Using these constraints one can show that the mass is a function of (or “comes from”) the vibration modes of the string.  

Huggett moves on to introducing the technical ideas and apparatus, beginning with a classical example.  Simple string in an n-dim relativistic spacetime.  Introduces the notion of a string worldsheet and its internal coordinates.  The action (Nambu-Goto action) giving the law for this string will be one that minimizes the relativistic area.  A special metric h is introduced on the 2-d worldsheet of the string, which is not connected to the full spacetime metric g, and should not have physical import in the end.

Huggett starts with some words on how to take his talk.  It started out as a talk for philosophers who have only been exposed to pop sci presentations of string theory, to get them to not be afraid of it.  It’s a bit like a travel log from an explorer to a strange country…

There will be 3 parts to the talk:

1. The formalism
2. Duality
3. General Relativity from String theory

For physicists, the hope is that this talk will help understand how philosophers are likely to think about ST (string theory, from here on), what they find interesting, etc.

Now we’ll hear from the other main organizer of the conference, Nick Huggett.  He will give a primer on string theory for philosophers (of physics). This should nicely complement Harvey’s talk, which gave a nice historical overview of string theory and some of its motivations, and the background and motivations of its practitioners.

Q&A:

Audience: There’s a part of the LQG community that doesn’t take the dynamics of GR as fundamental. Also the fundamental problems faced by researchers in LQG don’t have to do with the dynamics of GR.

Harvey: Point taken.

Harvey: In QFT the problem is connecting the UV to the IR. The real problem with a theory that doesn’t exist mathematically is whether you can start where you want in the UV and how to get to the IR. 

String theory doesn’t have a full description of what the fundamental degrees of freedom are. 

Audience: What about the other jeopardy game where you answer fundamental questions about the structure of the black hole by looking at the SYM theory. 

Harvey: In principle that can be done, but in practice it’s hard to have control over it. 

Audience: What role does SUSY play in the AdS/CFT correspondence?

Harvey: It’s not essential, but it’s a useful tool for checking it.

Audience: A worry about string theory: one uses concepts that don’t make sense without defining the metric. Doesn’t that make sense?

Harvey: That’s a valid critique; we don’t have the fundamental formulation. It only makes sense in certain limits

Audience: You said that you’re inclined to regard the two sides of a duality as surprisingly different descriptions of a single reality. Is that the most widespread view amongst practitioners? Also, if something is “gauge” at one point in the moduli space, does that really imply that it’s gauge in general?

Harvey: I think the majority do. 

Mathematical consistency:

It’s been claimed that superstring theory provides a UV finite theory. There have been some nagging wories about defining amplitudes on higher genus Riemann surfaces, but these have been laid to rest by Witten and collaborators.

Audience: does UV finite mean that there’s no need for renormalization?

Harvey: Yes.

As in any interesting theory, the perturbation theory is not convergent, and non-perturbative effects must be included. This is difficult, but there’s no reason to think that it can’t be done.

Finally, on philosophy:

Historically, people in GR have been more keen to engage with philosophy. String theorists believe that GR is just another non-renormalizable effective field theory. So they take this attitude to LQG. It doesn’t exist, in the same way that QED doesn’t exist (not well-defined to all energy scales).

Audience: Do you think string theory is well-defined at all scales?

Harvey: Yes, the strings get fuzzier below the Planck length, but they’re still well-defined.

Some particle theorists have been hostile towards philosophy. But that doesn’t reflect on the extent to which string theory could use philosophical work!

Background independence:

In many cases, we write classical part of a field as background and consider small fluctuations around it. String theory as formulated in the mid 80s involves this sort of separation. But with no classical background, it’s hard to interpret what the theory means! What does GR describe when g_{\mu\nu} = 0?

AdS/CFT sheds light on this– in AdS space, we need boundary data in order to get a well-posed initial value problem (for massless particles). Harvey says that a formalism that is independent of boundary data may or may not exist, but it’s fine if we only have a formalism that depends on boundary data, because finite energy excitations will never affect the asymptotic boundary.