Musings

While we’re on a cosmological constant kick, I should mention a recent paper by Stephon Alexander. He claims to have found a mechanism by which the cosmological constant can be relaxed to a small value.

Which sounds pretty important. Anyone who’s been in the field for any length of time has tried and failed to find such a mechanism.

It’s well known that four-dimensional gravity has a CP-violating topological term,

$$S_\theta = \frac{\theta}{348\pi^2} \int \tr R\wedge R$$ which is very analogous to the $\theta$-term in QCD. If you have a Dirac fermion, chiral rotations are anomalous in a curved space background. A massless fermion makes the $\theta$-angle physically unobservable. If the fermion has a mass, then the linear combination $$\overline{\theta}= \theta +\arg(m)$$ is observable. In QCD, the apparent smallness of $\overline{\theta}$ is a problem, and the Peccei-Quinn mechanism was invented to solve it. $\overline{\theta}$ is replaced by a pseudoscalar field, the axion. Chiral symmetry breaking induces a potential for the axion which causes it to relax to zero.

Stephon points out that a very similar mechanism can be made to work in gravity, relaxing the gravitational $\theta$-angle.

“Who cares?” I hear you cry, “Gravity is so weak, gravitationally-induced CP-violating effects are unmeasurably small. Besides, I thought you were going to tell us about the cosmological constant.”

I sat down last night, with my two children, to watch the much-protested “Sugartime” episode of Postcards from Buster. A sweet, wholesome half-hour tour of life in rural Vermont (as seen through the video camera of the eponymous rabbit). Lots of maple sugar, dairy cows, and even a Shabbos dinner. Having now seen the show, I am more amazed than ever at the “controversy” it engendered.

We have a contingent of seriously twisted people in this country. And I don’t mean “Mom and Gillian,” the parents of Buster’s tour guides in this episode.

There’s been a lot of buzz about Kolb et al’s suggestion that superhorizon fluctuations can mock-up the effect of a cosmological constant (current observations suggest $\Omega_\Lambda=0.7$). I haven’t commented, because the calculations are a bit beyond me. They involve intricacies of second-order perturbation theory about FRW, and an infrared divergence which implies that — even though the amplitude of fluctuations at any individual wavelength is small, $\epsilon=\delta\rho/\rho\sim 10^{-4}$ — if there have been enough e-foldings of inflation, the contributions from all superhorizon modes may be large enough to actually dominate the energy density today.

Éanna Flanagan has a very interesting critique, which is simple enough that even I have a chance of understanding it.

Consider a gedanken-universe in which the initial spectrum of perturbations was such that there are no sub-horizon perturbations today. An observer in such a universe can measure the redshift, $z$ and luminosity distance, $\mathcal{L}$ of nearby events. In a conventional FRW universe, these are related by

But, since we won’t assume local isotropy, we have some more general angle-dependent relation,

and one reconstructs $H_0$ and $q_0$ as some angular averages of $A$ and $B$. The cosmological fluid has a stress tensor, $T_{\alpha\beta} = (p+\rho)u_\alpha u_\beta + p g_{\alpha\beta}$. One can expand the four-velocity in the usual way,

where $\theta$, $\sigma_{(\alpha\beta)}$, $\omega_{[\alpha\beta]}$ and $a_\alpha$ are the expansion, shear, vorticity and four-acceleration. Assuming matter domination and no dark energy, $p\sim 0$ and hence $\nabla_\alpha T^{\alpha\beta}=0$ implies $a_\alpha=0$.

At this point, Flanagan uses a local Taylor series expansion to compute $H_0$ and $q_0$ in terms of the density and the four-velocity and its gradients. The result is that the Hubble constant

measures the local expansion of the fluid and the deceleration parameter,

The first term is positive. In a spatially-flat, matter-dominated FRW universe, we would have $q_0=1/2$. Here, our ansatz allows for local spatial curvature, so $q_0\neq1/2$, but, in a spatially-curved, matter-dominated FRW universe, $q_0$ is nonetheless positive. The second and third terms involve the shear and vorticity of the cosmic fluid. Sure enough, we could get $q_0\lt 0$, provided the vorticity is large enough.

But, and this is Flanagan’s key observation, these are local observables of the cosmic fluid. We can estimate them, just knowing typical magnitude of peculiar velocities, and the fact that, in the absense of sub-horizon fluctuations, the scale over which the velocity varies, $l\gtrsim H_0$. The upshot is that these “second-order” contributions to the deceleration parameter, $\delta q_0\sim (\delta v)^2\sim\epsilon\sim 10^{-4}$.

That is, they’re tiny compared to the zeroth-order contribution, and can’t possibly give $q_0\sim -0.5$, to account for the observed cosmic acceleration.

Turning this around, Flanagan observes,

Thus, while an order-unity renormalization of $q_0$ from second order effects is possible in principle, our analysis implies that such a renormalization would also require second order contributions to the fluid velocity that violate observational bounds. (This also implies that the results of [Kolb et al] should yield an upper limit on the number of e-folds of inflation.)

Luboš has some more comments, but he ends somewhat glibly:

At any rate, Éanna assumes locality, and with this assumption, it seems clear that the paper of Kolb et al. cannot be correct without the need for complicated calculations such as those of Éanna.

That’s far from clear. Both papers today find that superhorizon fluctuations alter the expansion rate and deceleration parameter. The question is whether they alter the deceleration parameter enough to push it negative. This clearly can’t happen with spatial curvature alone (Geshnizjani et al). It is, however, technically possible, though it would require unphysically-large values of the vorticity (Flanagan).

WordPress 1.5 has been out for a month or so, and it finally makes using my itexToMML plugin practical. A certain amount of hacking of the WordPress source code is still required, but it’s pretty manageable.

The required patches can be applied with a simple

patch < WordPress1.5_math.patch

Here’s what they do:

• wp-blog-header.php is patched to send the correct MIME-type to MathML-capable browsers.
• wp-includes/functions-formatting.php is patched so that wp-texturize plays nice with MathML.
• wp-content/plugins/textile1.php is patched so that the Textile filter plays nice with MathML. Markdown works just fine as-is.
• wp-content/themes/default/header.php and wp-content/themes/classic/header.php are patched to send out an XML declaration and the correct XHTML+MathML DOCTYPE to capable browsers. If you use another theme, you’ll have to patch it on your own.

I also decided to dispense with the “deprecated” my-hacks.php file and set the allowed XHTML+MathML tags directly in the plugin. In fact, that’s the only change to the plugin itself.

If you’re running WordPress 1.5, you can

1. install the itex2MML commandline utility,
2. install the plugin,
3. apply the patches,

and be running a MathML-capable blog in minutes.

Having said that, you’re still probably better-off installing MovableType. After you’ve played around a bit with a basic MathML-capable installation, you’re going want to start converting XHTML+MathML named entities to NCRs on output. You’re going to want to start adding things like comment-validation, per-post and per-comment selectable text filters and so forth. MT is still far-and-away the better platform to build on for this sort of “advanced” weblog functionality. But competition is good. And WordPress is a very nimble competitor…

Sidney Coleman is my hero.

That statement requires a little bit of an explanation, as Sidney was my PhD thesis advisor. Truth be told, his direct influence on my thesis was negligible. Midway through my graduate career, string theory swept through high energy physics. As a sensible young man, I dropped everything I’d been doing and rode the wave. Sidney was not interested in string theory; he wasn’t even particularly interested in supersymmetry. I doubt he even read my thesis, though he did sit patiently through my numerous Family Meeting talks and occasionally asked penetrating questions.

But his indirect influence was profound. No one thought more clearly about quantum field theory. And no one has ever lectured or written more lucidly about the subject. If you haven’t read his Erice Lectures, you don’t know the heights that scientific writing can attain.

I’d like to hope that some little bit of his insight into physics and clarity of thinking rubbed off. One of the greatest compliments I ever received was, in a offhand way, a tribute to Sidney. When I first gave a seminar at Princeton, David Gross came up to me after my talk and asked, “Who was your advisor at Harvard?” “Sidney Coleman.” I replied. “Ah,” he said, “I could tell.”

At age 68, Sidney is far from well, ravaged by a Parkinson’s-like disease. So a two-day conference has been organized in his honour. The speakers are about as illustrious a bunch as you are likely to find (6 of 9 have Nobel prizes and the others — Paul Steinhardt, Erick Weinberg and Edward Witten are not exactly slouches).

I’m returning to Austin early tomorrow morning. But, at least, I can catch the first day’s proceedings.

Joanna Karczmarek points out to me that Mozilla/Firefox on Windows requires version 6 of the SVG plugin in order to view the figure in the previous post. Win/IE does fine with version 3.0.x of the plugin, as does Mozilla/Firefox on MacOSX and Linux.

If you don’t have the plugin installed, then you get a crufty old GIF image as a replacement. In addition to looking bad, it doesn’t rescale when you zoom (magnify or shrink) the text in your browser.

Imagine that 2 million light years from our current location is a collapsing shell of radiation, headed towards us. Two million years from now (minus a little bit) that shell will pass within its Schwarzschild radius and a blackhole will reside where we once stood.

But we don’t know that yet. We can’t know that yet. The shell of radiation is well outside our past lightcone. Still, despite our blissful ignorance of the doom that awaits us, an event horizon has already formed here on Earth. Light signal from our present location will never escape to infinity. The existence of this event horizon is a global geometrical statement. When, exactly, it formed depends on the total “mass” of that shell of radiation. And … we don’t know what that is.

Time to welcome another Harvard Physics faculty member to my blogroll. Lisa Randall’s an old friend. I’m sure she’ll have lots of interesting stuff to say.

Her first post is about the emergence of 11-dimensional M-theory from the strongly-coupled limit of Type IIA string theory. She manages to carry off a creditable explanation without ever using the phrase “threshold bound state.”

Over at the String Coffee Table, Eric Forgy asks:

Hanging out at Harvard

Do you ever think that maybe all this abstract mathematics is not how nature really operates? I think that physicists come to a turning point early in their careers where they need to decide on a philosophy. Are you going to try to develop theories that might describe some kind of phenomological aspect of nature, or are you going to really try to understand the true nature of the universe at the most fundamental level.

Are you pursuing this because you truly believe that nature operates according to the rules of gerbes and n-categories? It’s kind of a silly philosophical question, I suppose, but what is it that drives you?

Back in the early 1980s, vector bundles, the Atiyah-Singer Index Theorem, and basic ideas of homotopy theory were considered pretty cutting-edge¹ in high-energy theory. If you used them in a seminar, you generally had to make a few apologies to your audience before proceeding. There was, shall we say, a certain resistance to incorporating highbrow mathematics. When I started applying certain techniques of algebraic geometry (in particular, coherent sheaves) to problems in string compactification, I quickly became known as “Mr. Sheaf” among my graduate student colleagues.

It’s fair to say that two decades of string theory have broken down that resistance. All sorts of highbrow math have shown up (in, sometimes, surprising ways) in string theory. Indeed, many suspect that whole new mathematical frameworks may need to be invented, in order for us to formulate string theory properly.

Subtle questions require subtle mathematical techniques to study them. K-theory — to pick one example — is essential to the formulation of the Index Theorem. But, for the applications that were of interest in the early 1980s, you really didn’t need that extra baggage. Ordinary cohomological formulæ were sufficient. Only when you start dealing with more subtle matters, like mod-2 index theorems, does the K-theoretic formulation really come into its own. The converse is also true. Subtle physics often suggests interesting new mathematics. The past two decades provide numerous examples of new mathematical developments inspired by string theory constructs.

I’ve no idea whether n-categories will eventually play an important role in string theory. (Gerbes have already made an appearance, but not in the guise the Urs seems to want for them.) So far, there doesn’t seem to be any compelling reason to think that they do.

Urs is sniffing around various branches of string theory, looking for hints that they might be reformulated in a 2-categorical form. That’s a fair endeavour. It’s not clear he will succeed. But it’s not (I think) a philosophical statement about how he “truly believe[s] that nature operates.”

If there’s anything we ought to avoid, it’s approaching Mother Nature with such philosophical preconceptions.

Google AutoLink and Larry Summers tackled in the same post? … Priceless.

You go, girl!

This was in the days after Supernova 1987a, and Hans was having a ball, watching his theories of supernova formation being confirmed by observations. He might have been over 80, but he was working harder than any of us, a mere fraction of his age. I certainly felt like I was slacking off. Shouldn’t I, too, have been hunched over my desk, calculating? Hans, mercifully, was too busy to look up from his desk and notice my idleness.

He was a giant of theoretical physics, and an inspiration to anyone ever associated to Cornell Physics. He passed away last night, at age 98. The New York Times has an obituary. And there’s more from Matthew Nobes.

I recently wrote about Greg Moore and collaborators’ work computing the entropy of a certain class of $N=2$ blackholes which arise in the compactification of Type IIA on a Calabi-Yau.

Alex Maloney visited us this past week, and gave us a beautiful talk on his work on the geometry of the blackholes in question. The situation is really quite striking. Classically, these blackholes have vanishing horizon area. They have a null singularity, a pathology which, says Alex, is characterized as “naked with the lights off” in the GR literature.

Umh, whatever …

I recently declared the STIX Fonts Project “moribund”. Mark Doyle corrected me: it’s not dead, merely glacial.

Well, death twitch or melting, the web site has been recently updated:

Redesigned Web Site Coming in April 2005

STIX Fonts Status Update February 2005 – The STIX Fonts project is finally nearing completion. Only a few months more of design work remains. After the final character delivery is made, several additional months will be required to finalize the character parameters and package the glyphs into a font package. It appears likely that the STIX Fonts will be made available some time in the summer or early fall of 2005.

This web site will be re-launched in two months. The new design will provide information of interest to scientists, application designers, and publishers.

Who knows? Maybe there’s hope yet …

One of the unexpected side-benefits of being a sort of technological pioneer in the field of weblogs and MathML, is that I’ve made the acquaintance of a number of people from a field rather remote from physics and mathematics — that of web design, web standards, and web software. Each Spring, some large fraction of those folks converge on Austin for SXSW Interactive.

I’ve gotten several emails asking whether I’ll be around. Rather than break the “bad news” individually, I’ll post it here:

With my inimitable flair for scheduling, I have arranged to be at Harvard (up in sunny Cambridge MA), Mar 6-19. So, if you were hoping to meet up and share a beer, or some barbecue, it’ll have to be on some other occasion.

Désolé.