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Low-functioning pinhead...
January 29, 2007
itex2MML 1.1.9
I fixed some minor bugs in the \array command and the aligned environment. And I added a gathered and a split environment.
So here’s itex2MML 1.1.9 and an updated list of itex commands.
Note that, because of this bug, the spacing on all of the array-like environments is a bit screwed-up in Mozilla. If Roger ever gets around to fixing that bug, the spacing will be much improved, and I’ll be able to fine-tune it to look really sharp.
Update (2/7/2007):
Per Lieven le Bruyn’s suggestion, the MacOSX binary of itex2MML is now a Universal Binary.January 27, 2007
Barbarians at the PAC
I was a student at Harvard, when the first recording of Einstein on the Beach was released. Philip Glass did a record signing at the Harvard COOP, and I decided to go down to check it out. Naturally, they played the opera on the stereo, as the composer sat at a table, signing records and chatting amiably with the customers. After about an hour of this, one of the COOP staffers cracked. Shaking visibly, she strode over to the stereo, said “I just can’t take this any more!” and — right in front of the composer — took Einstein on the Beach off the stereo and put something else on in its stead.
Everyone looked slightly embarrassed, at what had just transpired. Except for Philip Glass, who kept right on chatting and signing records, as if nothing out-of-the-ordinary had taken place.
It was around this time that I read J.M. Coetzee’s novella, Waiting for the Barbarians. It was, or so it appeared, an allegory for the brutalities of South African apartheid regime’s battle with the black nationalists.
The libretto of the Philip Glass opera, whose American premier was performed, here, by the Austin Lyric Opera is quite faithful to Coetzee’s novella. The set is stunning, and the music sweeps us along with the Magistrate, and his town’s, descent into the abyss.
The barbarians, whose threat, Colonel Joll assures the Magistrate, requires the “strictly temporary” suspension of various civil liberties (and the rather graphic torture of as many “prisoners” as he can scoop up) seem much closer to home than they did back then, not so easily dismissed as an allegory about the troubles in some faraway land. And this, too, lends a certain urgency to the unfolding drama.
If there are still tickets to the final performance, Monday night, … go.
January 25, 2007
Gravitational Leptogenesis?
I was very excited when I saw a recent paper by Alexander, Peskin and Sheikh-Jabbari (which, I guess, is based on this older paper). They claimed to produce an acceptable lepton-asymmetry during inflation if the inflaton is a (CP-odd) pseudoscalar field, , with a coupling
to the curvature, where is an odd function of .
Fluctuations generated during inflation drive an expectation-value for , which generates a lepton-asymmetry via the anomaly,
For reasons, that will become apparent shortly, however, they proceed to say something very unconventional about the anomaly coefficient, :
In general, when heavy right-handed neutrinos are also added to the Standard Model, as is done in the seesaw mechanism for explaining the smallness of the neutrino mass, (2) will be correct in an effective theory valid below a scale , of order of the right-handed neutrino mass. More concretely, can in general be a function of energy. At low energies, below the right-handed neutrino mass scale . At higher energies, could be anywhere between zero to three, depending on the details of the particle physics invoked.
Umh … no. That’s not how anomalies work. The anomaly depends only on the massless spectrum and the anomaly coefficient does not run with energy.
Now, it’s true that, above the scale , one can treat the right-handed neutrino, , as massless. In that theory, there’s a new current which is non-anomalous.
However, once one includes the Majorana mass for , is not conserved, even classically. The correct, classically-conserved, current in the theory which includes the mass term, is . And the anomaly in that current is the same, whether one calculates it in the high- or low-energy theory.
The motivation for the seemingly bizarre statement about the anomaly (2) becomes apparent a little later on. One would naïvely expect the gravitational contribution to the lepton asymmetry, during inflation, to be tiny, suppressed by at least a factor of . They, however, claim that there’s an enhancement over the naïve answer by a factor of . This enhancement comes about because, in their calculation, is UV-divergent, dominated by short-distance effects near the cutoff, .
I don’t really understand their calculation. But, if they are correct that, in the effective theory, , then what this means is that, in the full theory, where one includes loops of , the contribution of those extra diagrams cancels the erstwhile UV divergence, yielding a finite (but -dependent) answer. I don’t see why there should be such a cancellation, here.