## June 25, 2007

### Why Theoretical Physics is Hard…

#### Posted by Urs Schreiber

… while mathematical physics is easy, but much slower? Because for the former you need an oracle, without being able to strictly confirm what counts as an oracle.

In his latest piece #, E. Witten comments on the role of these oracles as follows:

We make at each stage the most optimistic possible assumption. Decisive arguments in favor of the proposals made here are still lacking. The literature on three-dimensional gravity is filled with claims (including some by the present author) that in hindsight seem less than fully satisfactory. Hopefully, future work will clarify things.

The proposal in question is about

[…] the problem of identifying the [2-dimensional quantum field theories] that may be dual to pure gravity in three dimensions with negative cosmological constant.

Jacques Distler has a summary here.

It is hard to find a precise definition even of what the word “dual” here – meant is holographic duality – is supposed to refer to.

Last time somebody guessed what the axiomatics behind this might be, he was told that this is the wrong axiomatics for what physicsists had in mind. Which may be true. But then it would be good to try to find the right axiomatics.

So I have my own proposal for what the holographic principle really is.

If a $d$-dimensional QFT is a $d$-functor from $d$-dimensional extended cobordisms to $d$-vector spaces, then a $\left(d-1\right)$-dimensional QFT $\mathrm{tra}$ and a $d$-dimensional QFT $\mathrm{curv}$ are “holographically related” if the former is the component map of a pseudonatural transformation “trivializing” the latter : $I\stackrel{\mathrm{tra}}{\to }\mathrm{curv}\phantom{\rule{thinmathspace}{0ex}}.$

Looks pretty through the binocular like this, but turns out to be an intimidating mountain height as one approaches it.

With a little luck Jens Fjelstad will be vising Hamburg in a month, and we’ll continue to scale that rock.

Posted at June 25, 2007 9:01 PM UTC

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### Re: Why theoretical physics is hard…

That principle is maths, not physics.

Posted by: Kea on June 25, 2007 10:38 PM | Permalink | Reply to this

### Re: Why theoretical physics is hard…

That principle is maths, not physics.

The holographic principle? Remember that it was motivated originally by t’Hooft in an attempt to understand black holes.

It is being used to understand rational conformal field theories (and is in fact currently the only way to completely understand these), which, among other things, models critical behaviour in 2-dimensional statistical mechanics of systems that people study in the laboratory.

It is also being used in an attempt to understand puzzling recent measurements in QCD – though there is disagreement about how useful the principle is, in this context.

In as far as they have been well defined, QFTs, and the space of all QFTs, can be studied by pure mathematics, but it would be overly pedantic not to call the study of QFTs part of theoretical physics, I think. That’s what fundamental physics is, QFT.

In the large realm of all these beasts, there may be many pheonomena (holography, supersymmetry, conformal invariance, topological invariance, etc.) which have little direct application in practical collider physics. But still, since practical collider physics needs certain QFTs, and since these aren’t fully understood yet, it behooves us to try to understand the space of all QFTs as much as possible – and call that endeavour (theoretical) physics. I think.

Posted by: urs on June 26, 2007 8:54 AM | Permalink | Reply to this

### Re: Why theoretical physics is hard…

Firstly, no one would debate that QFT is physics, but it has its own principles, quite separate from any QG principles. Now I am quite fond of the Holographic Principle, but your formulation of it is just maths. A physics formulation would discuss, for instance, Machian-Peircean philosophy and the meaning of inertia in a background independent theory. I am also quite fond of SUSY - but not the usual kind - because that is not a principle either, but rather a mathematical consequence of one.

Posted by: Kea on June 26, 2007 10:41 PM | Permalink | Reply to this

### Re: Why theoretical physics is hard…

Now I am quite fond of the Holographic Principle, but your formulation of it is just maths

But isn’t that like saying

“I am quite fond of Maxwell’s theory of electromagnetism, but saying that it is a theory of line bundles with connection is just math.”

?

How does physics work, since Newton? We see a phenomenon in nature and are looking for the math modelling it. The math by itself is always “just math”, but that’s why physics is successful.

A physics formulation would discuss […] the meaning of inertia

Right. But before I can do so, I have to understand what is actually going on. Before I can really understand the role of inertia, or anything else for that matter, in the context of the holographic principle, I need to know exactly what that principle actually is.

Right at the moment, I don’t see that this is the case. From the standpoint of mathematical physics (which is not at all the standpoint of most practicing physicists) the holographic principle is a mystery at the moment. Try to explain to a mathematician what the statement of the principle is, to quickly see that there is really no precise statement yet.

For instance, one reads sentences about how the principle is supposed to relate “observables” of one theory to correlators of another. But the observables here are that of a gravitational theory. It’s hard to find precise statement about what is actually meant by “observables” here. I don’t think people really know.

Just look at Witten’s latest paper, whose comments triggered my above entry. He quite openly admits at several places how at least one side of the supposed duality is by itself quite a mystery, conceptually. Without much ado, he seems to accept the point of view that “observable” on the side of the gravitational theory has to mean “S-matrix”.

It seems clear that in order to make progress beyond this proverbial handwaving (nonwithstanding the dramatic success this handwaving may have, in particular when it’s Witten’s hands that are being waved) we need to

- first understand what QFT actually is

- and then identitfy that property of the space of all QFTs which we want to identity as the “holographic principle”.

There are essentially two approaches for giving QFT a rigorous underpinning:

A) what is called “AQFT”: this is built on the Heisenberg picture of quantum mechanics, the basic building blocks are certain cosheaves of “local observable” algebras.

B) functorial QFT (as conceived originally by Atiyah and Segal): this is built on the Schrödinger picture of quantum mechanics, the basic building blocks are representations of cobordism categories.

The problem with looking for axiomatics is that one is likely to fall short of the goal. Maybe both these approaches are insufficient for capturing whatever is meant by the holographic principle. But maybe not. We can at least try.

Karl-Henning Rehren once tried in the context of A). While it seems he found some kind of relation between higher and lower dimensional QFTs which depends on AdS structures, my impression is that his axiomatics does not really capture the idea of the holographic principle.

So here I am trying to understand what the axiomatics of the holographic principle would be in the context of B).

And I am saying: there is i) a natural candidate mechanism there which relates $d$-dimensional to $\left(d-1\right)$-dimensional QFTs (namely pseudonatural transformation), and, ii), there is a bit of evidence that this actually does describe the holographic principle in the situation which is best understood, namely in $d=2$/$d=3$.

Posted by: urs on June 27, 2007 10:41 AM | Permalink | Reply to this

### Re: Why theoretical physics is hard…

This is incorrect. The right way to understand a new principle is to think about the new physics - not to focus on translating AQFT etc, which is established physics, into category theory - although I don’t for one moment deny that this will probably end up being useful.

Secondly, you seem to have missed my point that the holographic principle is incomplete. By trying to reformulate it in terms of categories (again - probably a useful exercise) you do not in any way address the real problem, which is to understand its physical underpinnings. The maths comes second, not first.

Posted by: Kea on June 27, 2007 11:01 PM | Permalink | Reply to this

### Re: Why theoretical physics is hard…

This may be a cultural difference between mathematical and theoretical physicists - one (in my view) tries to take physics and put it on a firm mathematical footing, whilst the other considers phenomena and cooks up an abstract reason why this works, pulling some quantitative and predictive power into the mix.

I don’t know if anyone has made a physical breakthough by considering the mathematics behind existing theory (I suppose Lorentz came close) but it is often the case a physical insight is hard to recognise until powerful enough mathematics has come onto the scene to express it (think Minkowski and spacetime). Again, with the eightfold wayand the quark picture - it seemed mystical why the particles arranged themselves into such patterns but saying `particles are related to representations of the gauge group’ (in the physics sense) makes such things obvious.

This raises an interesting point - how much mathematics does one have to use in a physical principle before it becomes mathematical? By saying we should use topos theory to do QM, because it encodes the right sort of logic, have we strayed beyond the physical? I don’t know, myself. I would use any tools that came my way. If one provides a precise dictionary between the physics and the maths, can we then just say it in maths? By saying the Poincare group describes the symmetries of space in the absence of matter and then talking about invariant quantities for that group, and consider relativity in this manner (providing more details of course), have we left physics? What about Noether’s theorem about conserved quantities? Depending on the perspective this could be either a deep physical statement or something to do with symplectic reduction.

If we state the holographic principal as the following : “we cannot tell which of these two physical theories of widely differing natures is the “real” one, because they both return the same predictions as to what we will measure”, does it make it physical? (let’s talk modulo what a physical theory *is*, here). Can we apply a Euclidean notion of transitivity here: physics is the same as this mathematical structure, which is in turn the same as a second mathematical struture, so physics is the same as the second mathematical structure, and then use that to infer things about the physics

Is is reasonable to expect there is a unique mathematical structure which underpins reality? I don’t know. If it exists it will necessarily be expressed in mathematics to be a theory of any use, and what if there is an equivalent mathematical structure we discover a century after finding a theory of everything? Why do we use this topos for working in and not that one which is equivalent? Indeed, which of the Schrodinger or Heisenberg pictures do you pick as the “right one”? Mu!

That’s all I’ve got.

Posted by: David Roberts on June 28, 2007 4:05 AM | Permalink | Reply to this

### Re: Why theoretical physics is hard…

Hi David

“but it is often the case a physical insight is hard to recognise until powerful enough mathematics has come onto the scene to express it.”

Yes, this is a good point, but using the Minkowski example: SR was physically very well developed before it was ‘simplified’ by Minkowski. The holographic principle, at least as I have heard it stated, is nowhere near that level of physical sophistication.

“By saying we should use topos theory to do QM”

Actually, we can’t use classical topos theory to do QM. And this comes back to your point that one usually ends up needing sophisticated mathematics, even NEW mathematics. But one can only really know what the right maths is once one has already expended a great effort finding a physical picture that makes some sort of sense. This physical picture must be much more developed than a statement along the lines of ‘the speed of light is constant’ because, as we have seen in the past, the simple statement that works best is most likely to be very far from obvious within the context of existing theory. In order to develop the picture one needs to consider EXPERIMENTAL results in some detail, and this is not a trivial amount of work.

“I would use any tools that came my way.”

Exactly! I’m not objecting to Urs’ category theory at all - it looks like a handy tool set - but this should not be confused with foundational physics.

“Is it reasonable to expect there is a unique mathematical structure which underpins reality?”

My view on this is that it will always appear so from our limited (2007) perspective. Sure, we’ll probably find a better way of writing things down in 100 yrs time (if we’re still here) but that’s not the point. At present, there doesn’t seem to be much choice. Having to invent more category theory is difficult enough.

Posted by: Kea on June 28, 2007 5:04 AM | Permalink | Reply to this

### Re: Why theoretical physics is hard…

Actually, we can’t use classical topos theory to do QM.

Probably David was referring to the work by Döring and Isham.

They use (ordinary, so I guess “classical”) topos theory to describe at least some aspects of QM.

Posted by: urs on June 28, 2007 7:44 PM | Permalink | Reply to this

### Re: Why theoretical physics is hard…

one (in my view) tries to take physics and put it on a firm mathematical footing

Yes. And even before this yields a fully rigorous mathematical framework, we need a process here which organizes vague heuristic ideas in something coherent.

if anyone has made a physical breakthough by considering the mathematics behind existing theory

Somebody noticed that special relativity seems to be governed but what has to be regarded as a flat pseudo-Riemannian metric. He then took this mathematical structure seriously and worked out what happens to the physics when one allows non-flat metrics. A trivial step from the point of the math. The most miraculous breakthrough from the point of physics.

which of the Schrödinger or Heisenberg pictures do you pick as the “right one”? Mu!

Mu indeed! I wasn’t saying that one (AQFT) is better than the other (Segal functorics). I was just saying that in the latter the answer to the holographic question may be more obvious.

And by the way, I am disconcerted by the fact that there are no real attempts to relate the AQFT school with Atiyah-Segal functorial QFT. They must be two sides of the same medal. It was my humble proposal to catalyze establishing a connection between the two by following the slogan that they are like Heisenberg and Schrödinger picture to each other.

But right now this statement is a little bit like the holographic principle: it looks right, but we need to work out what it really means in detail.

Posted by: urs on June 28, 2007 3:16 PM | Permalink | Reply to this