Musings

Yes, tonight was our annual neighbourhood pumpkin carving. Some years, my pumpkins are drawn from popular culture (at our very first event, I did Kenny and Kyle from South Park). Other years, they are more overtly political. And some years, they are both …

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.

Recently, there have been some quite lively discussions of Sean’s review of Lee Smolin’s book and Clifford’s synopsis of Lee’s radio appearance with Jeff Harvey. One of the things that one discovers about such discussions is that the same issues keep cropping up over and over, and one gets the sense of very little progress being made, as the participants don’t seem to have assimilated the lessons of previous discussions.

So, I found myself bashing my head against the keyboard when I saw the following comment

Bert Schroer, in the paper here says, beginning on page 18:

A profound mathematical theorem reveals that there is even a unique correspondence between Local Quantum Physics {QFT both Lagrangian and non-Lagrangian} models in n+1 AdS spacetime with a n-dimensional conformal invariant Local Quantum Physics model?..I have tried all possibilities of what Maldacena could have meant and none of them seem to be consistent with the above structural theorem.

As I understand it, most string theorists regard Maldacena?s AdS/CFT as a done deal. Will they continue to believe this in the face of a rigorous proof of the contrary by a non-string theorist?

The commenter, here, is referring to Rehren Duality, a proposal, in the context of AQFT, that a conformal field theory on the boundary of AdS is isomorphic to an ordinary, (non-gravitational) QFT in the bulk of AdS. If true, this would, presumably, be incompatible with the AdS/CFT, which posits that the theory in the bulk is a gravitational one.

Since it’s tiresome to have to explain what’s wrong with this proposal over and over again, every time the subject comes up anew, I decided to grit my teeth, and try to write a post so that, in future, one can simply get away with linking back here.

The folks at Design Science are working on a new release of the MathPlayer plugin for Internet Explorer. Since many of you, who read this blog, are users of the current version, they asked me to pass along this announcement

We are close to releasing a MathPlayer 2.1 beta, and are looking for beta testers. MathPlayer 2.1 will feature IE7 compatibility, accessibility (math to speech) enhancements, and lots of small bug fixes and enhancements. A more complete list of fixes will be provided to beta testers later.

If you create web pages using MathML, and are interested in being a beta tester, please e-mail us at beta@dessci.com.

This past week, in the Geometry and String Theory Seminar, as part of our series on differential cohomology, Dan gave a talk about his derivation of the Wess-Zumino term in the Chiral Lagrangian. It’s a pretty little application of generalized differential cohomology theories and it has several advantages over, say, Witten’s derivation

1. For a general 4-manifold, $X$, the signature, $\sigma(X)$ is an obstruction to finding a 5-manifold, $B$, such that $X=\partial B$. Witten, of course, worked on $X=S^4$, which has vanishing signature, but this construction clearly doesn’t make sense for general $X$.
2. To define integration in the generalized differential cohomology theory in question (which Dan, unimaginatively, dubs differential “E-theory”) requires a choice of spin structure on $X$. In particular, the requirement that $X$ be a spin manifold, which was clearly present in QCD, but not apparent in previous derivations of the Wess-Zumino term from low-energy considerations, re-emerges here.
3. When quantizing on $X=M_{(3)}\times \mathbb{R}$, the class in differential $E$-theory, whose integral over $X$ is the Wess-Zumino term, gives a natural $\mathbb{Z}/2$ grading to Hilbert space (for $N_c$ odd). This is what we call baryon number (compare the Skyrme model).

Anyway, that wasn’t what I really wanted to talk about.

In the course of a subsequent discussion, the following point arose. The 't Hooft Anomaly Matching Conditions ensure that there must be some light degrees of freedom in the low energy theory on which the chiral symmetries are realized. If we assume that the theory confines, then there are really only two possibilities¹:

1. There are massless composite fermions which transform in appropriate representations to saturate the 't Hooft Anomaly Matching Conditions. 2. The chiral symmetry is spontaneously broken, and is realized nonlinearly on some Goldstone boson fields, whose Wess-Zumino term provides the anomalous variation satisfying the 't Hooft Anomaly Matching constraints.

Clearly, possibility 1 is impossible for $N_c$ even. But, for $N_c$ odd, there are all sorts of oddball special cases to check. John Terning found some oddball solutions. Surely there are others.

There’s a broad class of theories, however, where one can essentially prove that chiral symmetry breaking must take place (as it’s the only way to satisfy the Anomaly Matching constraints). Does anyone know of a systematic statement of what’s known?

Roger Sidje (the moving force behind Mozilla’s MathML support) is experimenting with MathML in HTML. There’s an interesting discussion in the Mozilla newsgroup. The idea is to relax the requirement that MathML be embedded in XHTML¹, with an eye, ultimately, to using MathML as the mathematical markup for HTML5.

I think this, if done right, is a wonderful idea. But I have a few comments

1. Today, XHTML is a huge barrier. I know of lots of people who would be all too happy to use MathML (generated by tools like itex2MML), but are stymied by the exigencies of XHTML. CMS’s that support XHTML are nearly nonexistent. Few people are crazy enough to either write their own, or to hack an existing one into producing XHTML. And, even if they did have an XHTML-capable CMS, many users don’t have access to their server configuration to do the required content-negotiation, to sent application/xhtml+xml to capable browsers. So MathML-in-HTML would lower the barrier tremendously.
2. MathPlayer 2.0 already allows IE/6 to consume MathML in tag soup. If I understand correctly, Mozilla’s implementation will be better similar to MathPlayer’s (except that the latter actually processes the MathML as XML), in that the MathML nodes will be part of the DOM. That’s very important if, like me, you’d like to apply CSS styling or Javascript to MathML content.
3. On the other hand, I think we want to avoid having two incompatible markup languages, both named MathML. Rather than creating some incompatible markup language, confusingly called MathML, I think the Mozilla people should create a profile of MathML 2.0 (along the lines of XHTML’s Appendix C) which is safe to be consumed by the HTML5 parser.
4. Roger Sidje and Ian Hickson disagree about whether to support the 2000+ MathML named entities. I side with Hickson. Sending named entites over the web is unsafe in an XML context. Hence the existence of tools to convert them to NCR’s or utf-8, before sending them over the wire. Allowing MathML named entities in HTML5 will break interop with existing HTML documents, and cause problems if the same MathML content is served as XML. Not worth the trouble.

If there are stumbling blocks to creating a suitable profile of MathML 2.0, now is the time to find out. Then you can pass on suggested changes to the MathML 3.0 Working group.