### The Role of Rigour

With apologies to David Corfield, this is going to be a little philosophical disgression on the role of mathematical rigour and “proving theorems” in theoretical physics.

Back in the old days, Math 55, the honours Freshman Mathematics course at Harvard, was infamous for its “True/False” exams. The typical question involved the statement of some theorem and — if the assumptions of theorem were not stated *precisely* correctly — then the correct answer was “False.” This was both a brutally difficult test of the students’ mathematical knowledge and a useful object lesson. A theorem is only as good as the assumptions underlying it.

This is particularly important in Physics, where we are typically *not* at liberty to “redefine the problem” so that the assumptions of the theorem are satisfied.

I was reminded of this lesson by two recent discussions in the blogosphere.

#### Rehren Duality

The first, of course, is our recent discussion of Rehren Duality, where a purported “Theorem,” establishing an isomorphism between a QFT in AdS_{$d+1$} and a conformal QFT on the $d$-dimensional boundary of AdS, rests on hopelessly flawed physical assumptions.

These flaws were evident after 5 minutes, flipping through the paper. I took far longer reading it carefully, trying to reassure myself that I hadn’t *somehow* misconstrued what Rehren was saying. And I spent *even longer*, trying to make my post as clear as possible. (From the ensuing discussion, I’m not sure how well I succeeded.)

I went through this effort, not out of any sudden rekindling of interest in an obscure 7 year old paper, nor because I have something against Professor Rehren (when we met in person, he seemed like a very nice guy). I did it because there continue to be people^{1} who go around claiming that Rehren’s “**Theorem**” implies that there must be something wrong with the Maldacena Conjecture. Even that would be have been pretty much ignorable (after all, there are plenty of wrong papers on the arXivs), were it not for the peculiar amplifying nature of the internet that has, apparently, given these ideas a certain currency among impressionable young students.

That there’s an abundance of bad information on the internet is not a surprise. The troubling aspect is the totemic power of the word “**Theorem**,” and its ability (in the minds of some) to trump sound physical arguments and abundant calculational evidence. And it’s applied in a particularly perverse way, here, because this “**Theorem**” is contrasted with alleged lack of a rigourous definition of “String Theory in AdS.”

At present, AdS/CFT provides a *definition* of nonperturbative String Theory in asymptotically anti-de Sitter spacetimes^{2}. The observables of the theory are *defined* to be the correlation functions of a certain QFT on the boundary. The *Conjecture* (supported by all those aforementioned calculations) is that, in appropriate limits, the resulting bulk theory reduces to semiclassical supergravity (or, in other circumstances, to a weakly-coupled string theory), and that other features that a theory of quantum gravity *ought* to possess do, in fact, emerge from this (somewhat unintuitive) definition.

Perhaps, someday, a better formulation of nonperturbative string theory will emerge, and we will then be able to demote the role of AdS/CFT from defining what we *mean* by nonperturbative string theory in AdS, to something *derivable* from this more fundamental formulation. But that alternative formulation will be preferred, not on the basis of its greater mathematical rigour, but on the basis of its greater explanatory power.

#### Helling-Policastro

Another example is discussed in a recent post by Robert Helling. Robert discusses the “polymer representation” of the spatial diffeomorphism constraints in Quantum Gravity. This odd-looking, non-separable, Hilbert space is what appears in Loop Quantum Gravity. And there is much highly technical analysis attached to it, along with many “rigourous results.” There’s even a theorem to the effect that the polymer state is unique.

It has been argued that, because of this uniqueness theorem, any background-independent quantization of gravity must proceed via this polymer representation on its “kinematical Hilbert space”.

Of course, the problem is that the assumptions of this theorem are vastly too strong. It requires that there be a state, $|0\rangle$, *invariant* under the entire group of spatial diffeomorphisms, whereas we usually assume only that the generators of spatial diffeomorphisms have vanishing *matrix elements* between physical states.

Moreover, we already know that there are counterexamples to this “**Theorem**”. 2+1 dimensional gravity can be quantized without invoking the polymer representation. And AdS/CFT sidesteps the whole procedure, by constructing directly the full, background-independent, quantum theory, with its set of observables, without recourse to the intermediate step of a “kinematical Hilbert Space”.

Moreover, as Robert shows in his paper, familiar systems quantized using this polymer representation seem to yield incorrect physical results.

Again, the problem is not that the **Theorem** is wrong, in some technical sense, but rather that its assumptions don’t (necessarily) hold in the physical systems of interest.

#### And Yet …

All of this is not to say that there is no place for mathematical rigour in Physics, or even in its more “speculative” areas, like String Theory and Quantum Gravity. By struggling to find a mathematically precise formulation, one often discovers facets of the subject at hand that were not apparent in a more casual treatment. And, when you succeed, rigourous results (“Theorems”) may flow from that effort.

But, particularly in more speculative subject, like Quantum Gravity, it’s simply a mistake to think that greater rigour can *substitute* for physical input. The idea that somehow, by formulating things very precisely and proving rigourous theorems, correct physics will eventually emerge simply misconstrues the role of rigour in Physics.

^{1} There’s much other drolly comment-worthy material in Schroer’s manifesto. But I’d like to ask readers to please *refrain* from indulging that temptation. The comment section of this post is likely to be wooly enough, as is, without degenerating completely into a fruitless discussion of Schroer’s missive.

^{2} Of course, this definition didn’t emerge out of thin air. It came from looking at the near-horizon geometry of a stack of D-branes in flat space. And the identification of which boundary field theory should correspond to which asymptotically-AdS string compactification comes from precisely such considerations. Still, we have no perturbative (let alone nonperturbative) formulation of string theory in most of these backgrounds. So, despite its origin in other, related, backgrounds that we *do* understand, AdS/CFT is, here, giving us a *definition* of string theory in AdS.

## Re: The Role of Rigour

All of this is not to say that there is no place for mathematical rigour in Physics, or even in its more “speculative” areas, like String Theory and Quantum Gravity.I would say it differently. In my view, if you do not have experiments, then rigor is all that you have. Without experiments or rigor, research reduces to an obscure kind of art, kind-of like Horgan’s pessimistic (and frankly offensive) characterization of “the end of science”. At the moment, the connection between string theory and experiment is inadequate; its real strength comes from partial rigor.

However (this one is my however), rigor does not have to a connected structure built out from axioms. That is what it should be eventually, but that is not what it has to be as a work in progress. To build a bridge, you do not have to start at one end and add planks to get to the other side. You can plant caissons in the middle, you can connect one tower to another, and you can even let a few things fall down.

If your philosophy is to only build from one end and never let anything significant fall down, you end up building a lot of bridges to nowhere. You will also build a lot of half-bridges that cannot be extended because they would fall down. Sometimes it seems to me that mathematicians make these mistakes too often out of loyalty to rigor. (But then, sometimes it seems defensible.)

You (Jacques) have argued that Rehren duality is an example of the former (at least as it has been misread by some people), while loop quantum gravity is an example of the latter. I am not expert on these points, but it seems likely that you are right.

Meanwhile string theory looks like a very large and important bridge between physical reality and mathematical axioms, even if it is grossly incomplete, inadequately anchored at both ends, and only connected in patches.