Knox: The Dimensions of Duality

Click here to download the slides.

2013-09-28 18:19:17: Reassures us that this will *really* be non-technical.

Compares self to opinionated weekender in the land of String Theory (along lines of Nick’s earlier metaphor). [Laughter]

We will address a slightly more familiar case: gauge theories on fibre bundles.

2013-09-28 18:21:53: AdS/CFT is a generic name for the class of conjectures about dualities linking higher and lower dimensional theories.

Can we make sense of the idea that these two descriptions are of the *same* reality? Is there not a fact of the matter about how many dimensions there are?

But there questions are not so weird for philosophy, really.  Think of gauge theory arguments; configuration space realists in quantum theory.

2013-09-28 18:25:52: Intends to say many controversial things about spacetime. So leave objections til the end.

The debate we will be looking at:  holonomies (Belot, Healey) vs. fibre bundle substantivalism (Arntzenius, Nounou).

Aharonov-Bohm effect used to argue that holonomies about closed paths to be the fundamental physical quantities.

Arntzenius: Fibre bundles *really exist* and their parts.

Obviously dimensionality will be distinct here.

2013-09-28 18:27:31: Take it for granted that AdS/CFT correspondence holds.  What are the realist interpretational options?

1. Bulk theory is reality.

2. Boundary theory is reality

3. Both represent the same reality?

How do we make sense of 3?

2013-09-28 18:32:36: What is the arena in which physical reality unfolds? (From Albert)  That is what is at stake.

But EK will be questioning this approach.

1. Which is more fundamental?

Do they have different domains? Cope better with closely related cases?

2. Which objects represent space-time role?

EK: this is to be functionally determined (one of those controversial claims).

Contrast `functional kind’ with `compositional kind.’  We specify functional role and then look for what instantiates that role in the theory.

2013-09-28 18:34:57: We tend to have the view that spacetime and dynamical symmetries should match. (e.g. Earman).  But in GR it is not quite clear to say how this should go.

The roles we care about will be filled by an object if it provides (locally) preferred inertial co-ords. (Brown, more or less).

Thus we can interrogate a theory to see if it has such an object.

2013-09-28 18:37:27: Questions EK doesn’t understand (heading off objections to aforementioned controversial views?)

Q. Is the (e.g) fibre bundle space a *real* *physical* space?

A. It can represent physical d.o.f, sure. But what of it? It’s a mathematical space.

Q: Where does the stuff of the universe *really* *live*?!

A: You are pushing a metaphor too far, give it up.

2013-09-28 18:39:37: Which version of EM correctly describes spacetime?

They both seem to do a fairly good job.

Which offers a more fundamental picture?

By identifying gauge-related classes of bundles we arrive at the holonomies. Both descriptions describe the same (non-ST) degrees of freedom.

2013-09-28 18:45:38: Back to AdS/CFT. Which theory represents spacetime accurately? The target metric in string theory (or at least a non-compactified version) plays a pretty traditional role in spacetime.

CFTs are invariant under conformal transformations of the metric.  This means that the thing that might represent ST is the equivalence class of metric conformally equivalent to Minkowski.

Does a spacetime with only conformal structure even represent spacetime? No inertial structure. Geodesics, even, are not invariant under conformal transforms.  Thus the structure is not a candidate for representing phenomenological spacetime.

Giant insects!?  [any notion of being larger or smaller at stake]

Brian Pitts: Are these the same point? If nothing were massive, conformal structure would be all we need.

EK: Sure, but not exactly the same point.

2013-09-28 18:51:01: But what about fundamentality?  Can we use our previous criteria?

If equivalence is precise, hard to ground claims of superiority. But one could be more revealing w.r.t. structure of underlying [CW: Or overlying?] theory.  However, the usual claim is that duality implies equivalence.  The moral of this talk is that differences in dimensionality don’t need to have physical significance.

Distinguish between dimensionality of physical degrees of freedom and spacetime. Note that d.o.f. don’t disappear, they are just translated into a different form.

If we accept that we can have distinct representations of the same situation anyway then duality is less puzzling.

To the questions in good time!

2013-09-28 18:51:58


2013-09-28 19:02:39: D.O.: Isn’t a boundary less fundamental than something that’s not a boundary?

EK: No. This is the container view again.

MS: Are you happy for both to be equally fundamental? Are all grades of fundamentally equivalent?

EK. Yes. [Rest of answer lost]

JB: In EM, holonomies are non-local and vastly overcomplete. Fundamental meaning sub-ontic? If we resist your view, it is in part that we believe that dimension is background for everything else we do in physics.

EK. But need to cash out why that is difficult. Prima facie, we can cope with indeterminate dimensionality.  Why must it be determinate? Doesn’t quantum gravity do away with the container view entirely?

NH: One of the things relevant to T duality is that both of the duals predict that the space is large. States in either dual have an interpretation in terms of dual processes.  It doesn’t matter which you choose. Is this similar to AdS/CFT? What would beings who live in either of these spaces experience?

KH: If we define things functionally/structurally then plausible that we only live in one.

DB: Suppose the correct theory of our world is dual to boundary CFT.  Say I ostend to dimensions, then I’ve fixed what I mean and the theory on the boundary is just a notational variant.

EK: Any such theory will require us to identify my experiences within the theory. Are there principled reasons to prefer one description to the other?.


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