# Huggett: Philosophical Paths Into String Theory

Abstract: This paper sets out some of the features of string theory that are most salient for philosophers interested in the theory. One goal is to convince philosophers of physics that the formalism is not to be feared – that familiar physical intuitions, and knowledge of general relativity and quantum field theory will carry you a long way. In particular, the first section attempts to give an overview of the formalism of string theory, a kind of large scale road map of what you will find in textbooks; something to help you keep the big picture in mind as you work through the details. Another goal is to pose a series of question that I believe should be addressed by philosophers of physics – my hope is that these will convince the reader that string theory is ripe with philosophical issues, that make an investment in learning the formalism well worth it.

2013-09-28 14:59:56: Now we’ll hear from the 2nd main organizer of the conference, Nick Huggett, who will give a primer on string theory for philosophers (of physics).  This will nicely complement Harvey’s presentation which gave a nice overview of string theory’s history and motivations.

2013-09-28 15:09:41: 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.

2013-09-28 15:12:00:

What!?!

2013-09-28 15:16:40: 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.

2013-09-28 15:31:17: 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.

2013-09-28 15:40:30: 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.

Nick Huggett
The two main points up to now are: (a) that the sigma action (equivalent
to the N-G action mentioned below) involves a 1+1-D metric, h, over the
string worldsheet – but its absence from the N-G action indicates that it is
introduced as a trick, so that there is conformal symmetry wrt h – worldsheet
distances have no physical significance. (Unlike target space distances –
only Lorentz symmetry there.) And (b) the action corresponds to a basic wave
equation, so that everything you’d expect goes through: the solution is
$X = X_0 + V + \sum \mathrm{modes}$, and when you quantize the modes become raising and lowering operators. In short, the math, and perhaps the ontology, it just that of a QFT on the worldsheet.

2013-09-28 15:44:58: 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?

2013-09-28 15:56:27: 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?

Nick Huggett
In brief, given the strategy described by Carl, both winding and momentum are represented by wavefunctions living in on circles – momentum in physical space, with radius R, and winding in a ‘new’ space of radius 1/R. If you now consider a dual system in which physical space has radius 1/R, it follows that the winding wavefunction lives in a space of radius 1/(1/R) = R. If you also exchange the momentum and winding wavefunctions, then nothing changes – except which space is physical space. But is that any change at all?

2013-09-28 16:06:25: Huggett offers 2 alternative interpretations of T-duality, in connection with the 3rd question.   Two interpretive decisions need to be made:

•  First interpretive decision: either the T-duals agree on the physical world, or they do not. If they do . . .
• Second interpretive decision: do both say that strings literally live in a space of radius R (but represent that fact dierently); or do they represent facts the same way, and so say nothing beyond their shared consequences  (so that the radius is indeterminate between R and 1=R.)

Either way, both interpretations agree that the target space’s radius (i.e., the phenomenal space we live in) is very very big.  Whew!

Nick Huggett
Disagree with that! There are three interpretations here – first the duals could disagree, then there are two ways for them to agree. On the first, target space is phenomenal space; on the second, target space has indeterminate radius and so cannot be phenomenal space, since that does have determinate radius. (So what I think I’m disagreeing with is the claim that target space is always big – what I said is that phenomenal space is big. The point being to drive a wedge between them.)

2013-09-28 16:17:09: Having to move much more quickly now, Huggett briefly goes over how ST implies General Relativity, first in empty space and then adding in some matter.  Question 6:  What is the significance of these results?

Huggett claims that one possible answer is: String theory is actually background independent.

Assume (i) that ST is an adequate TOE, in sense that the string spectrum includes quanta for all the background fields you like; and (ii) that the terms in the action accurately capture the effective behavior of those coherent states.  Then by (i) the G_mu_nu field is composed of stringy excitations, and by (ii) it satises the Einstein Field Equations (making it the gravitational field, and the excitations gravitons). In other words, in the most literal sense the general relativistic theory of spacetime is a low energy effective theory of strings.

Which ought to get philosophers excited!

No time for questions 😦

Nick Huggett
The results are that to low order in perturbation theory, conformal
symmetry of string theory requires that background fields satisfy the
Einstein Field Equations. For example, if you replace the Minkowski metric in
the action with a general metric – so it is the only background field – then
it must be Ricci flat. Vistarini’s talk went into more detail, and also
addressed the issue of why quantized string theory should be conformally
symmetric (as I noted above, the classical action just is). But if background
fields just represent the effect of coherent states – so the quantum theory
is just an expansion about a coherent state – then we have learned something
very important about the collective behaviour of string quanta describing
gravitons.
Nick Huggett
Thanks Carl – great job!Dear reader – my paper will soon be available on the website.