September 17, 2000

This Week's Finds in Mathematical Physics (Week 156)

John Baez

This week I want to catch you up on some of the experiments that have been going on lately. Mathematical physics is no fun without some experiments to think about now and then. So here's some news about black holes, superfluid hydrogen, T violation, the τ neutrino, and the Higgs boson.

I like black holes because they are a nice example of what general relativity can do. Once upon a time they seemed very exotic, but now it seems they're common. In particular, there appear to be black holes with masses between a million and several billion times that of the Sun at the centers of all galaxies with a "bulge". This includes galaxies like the Milky Way, which has a central bulge in addition to a flat spinning disk, and also elliptical galaxies, which consist solely of a bulge. Many of these supermassive black holes emit lots of X-rays as they swallow hapless stars. As I mentioned in week144, the X-ray telescope Chandra has seen evidence for about 70 million of these black holes!

Recently, two teams of researchers have found that the mass of these central black holes is correlated very closely to the dispersion of stellar velocities in the galaxy:

1) John Kormendy, Monsters at the heart of galaxy formation, Science 289 (2000), 1484-1485. Available online at

2) Laura Ferrarese and David Merritt, A fundamental relation between supermassive black holes and their host galaxies, Astrophys. J. Lett., 539, (2000) L9, preprint available as astro-ph/0006053.

3) Karl Gebhardt et al, A relationship between nuclear black hole mass and galaxy velocity dispersion, Astrophys. J. Lett. 539, (2000) L13, preprint available as astro-ph/0006289.

Tight correlations are a bit rare in astrophysics, so they tend to be important when they exist. If you look at a graph you'll see how nice this one is:

4) Supermassive Black Hole Group, Theory of black holes and galaxies,

Ferrarese and Merrit estimate that the black hole mass grows as roughly the 4.8th power of the stellar velocity dispersion, which they define as the standard deviation of the radial component of the velocities of stars in the galaxy.

But what does this correlation mean? Astrophysicists are still arguing about that. But at the very least, it suggests an intimate relation between supermassive black holes and the process of galaxy formation.

Part of the puzzle is that nobody knows how these supermassive black holes formed. You see, until very recently, all we've ever seen are small black holes formed by the collapse of a single star (between 3 and 20 solar masses), and these supermassive ones at the centers of galaxies. But last year, people started seeing middle- sized ones! Colbert and Mushotzky found black holes between 100 and 10,000 solar masses in about half of 30 nearby spiral and elliptical galaxies that they examined:

5) Ed Colbert's homepage,

E. J. M. Colbert and R. F. Mushotzky, The nature of accreting black holes in nearby galaxy nuclei, preprint available as astro-ph/9901023.

Ptak and Griffiths found a black hole of over 460 solar masses in an irregular galaxy called M82:

6) A. Ptak, R. Griffiths, Hard X-ray variability in M82: evidence for a nascent AGN?, preprint available as astro-ph/9903372.

This is a "starburst galaxy", meaning that it's full of supernovae going off like a big firework display. When a star dies in a supernova explosion, that's when a neutron star or black hole is formed - so it seems likely that this black hole in M82 was formed by the merger of several such black holes. Could we be seeing the gradual formation of a supermassive black hole?

Maybe someday we'll understand the complete ecology of black holes. I can't help but feel there's some important role they play which we don't understand yet. (For one theory about this, see the end of week33.)

Now: you've all heard how helium-4 becomes a superfluid below 2.18 kelvin and helium-3 does it below 2.4 millikelvin. But what about superfluid hydrogen? Unlike helium, hydrogen is not a snobbish loner: it's a friendly, sticky molecule. So usually it solidifies before it gets cold enough to go superfluid! But in 1997, some folks at the University of Illinois noticed a possible loophole: films of liquid hydrogen about one molecule thick on a silver substrate should form a 2d superfluid at a temperature of 1.2 kelvin. Here's a picture of a computer simulation:

7) David Ceperley et al, Prospective superfluid molecular hydrogen,

Since then, other people have cooked up other schemes.

Now it seems people have actually made the stuff. Tiny amounts of it! The way they do it is to take superfluid helium and put in a bit of carbonyl sulfide (OCS) and hydrogen. About 14 to 16 hydrogen molecules stick to the carbonyl sulfide molecule, and when the temperature drops to .15 kelvin, these molecules form a superfluid. The hard part is checking experimentally that this really happens - and even defining what it means for a cluster of so few molecules to be a superfluid. I can't explain the details; for that you'll have to read the paper:

8) Slava Grebenev, Boris Sartakov, J. Peter Toennies, and Andrei F. Vilesov, Evidence for superfluidity in para-hydrogen clusters inside helium-4 droplets at 0.15 Kelvin, Science 5484 (2000), 1532-1535, available online at

Here "para-hydrogen" refers to a molecule of hydrogen where the spins on the two nuclei are anti-parallel - as opposed to "ortho-hydrogen", where they're lined up. The two states have different properties and this matters a lot in delicate situations like these.

Next: T violation. Once people thought the laws of physics were symmetrical under exchanging either particles with their antiparticles, left with right, or future with past. These three symmetries are called C (for "charge conjugation"), P (for "parity") and T (for "time reversal"). The weak interaction is now believed to violate all of these.

Very briefly, the story goes like this: Yang and Lee won the Nobel prize for helping discover P violation in the β decay of radioactive cobalt back in 1956, though in retrospect it was only the sexism of the Nobel committee that prevented Wu from sharing this prize - she did the actual experiment. In β decay, a neutron turns into a proton, an electron and an electron anti-neutrino via the weak interaction. Since the electron anti-neutrino only comes in a right-handed form, this process violates P symmetry.

Cronin and Fitch won the Nobel prize for discovering in 1964 that neutral kaons decay in a way that violates CP symmetry - i.e., the symmetry where you switch particles with their antiparticles and switch left with right. I believe that neutral kaons are still the only system where CP violation has been seen.

Now there's something called the CPT theorem which says that various reasonable axioms for a quantum field theory imply symmetry under the combination of C, P and T. For the math of this, the obvious place to go is this classic text on axiomatic quantum field theory:

9) R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Addison-Wesley, Reading, Massachusetts, 1989.

In case you're worried, PCT is the same thing as CPT. I like this book a lot. The only thing I dislike is how it unleashed a flood of physics papers whose titles end with "and all that". For example:

"CFT, BCFT, ADE and all that"

"Quantum cohomology and all that"

"String theory, supersymmetry, unification, and all that"

"Anti-de Sitter space, branes, singletons, superconformal field theories and all that"

"The modified Bargmann-Wigner formalism: longitudinal fields, parity and all that"

"The Zamolodchikov C-Function, classical closed string field theory, the Duistermaat-Heckman theorem, the renormalization group, and all that"

Enough! Listen, guys: it was funny once, but now it's just lame. Stop it!

But I digress. Where was I? Oh yeah: given the CPT theorem, from CP violation we can conclude T violation. The future and the past are slightly different - but of all the known forces, only the weak force notices the difference! This is bizarre and fascinating. But the way we reached this conclusion was not completely satisfying, since we needed to assume the usual axioms of quantum field theory to get the CPT theorem. What if the axioms are wrong? It would be better to have more direct evidence of T violation, given how important this issue is.

So in the late 1990s, people in the CPLEAR collaboration at CERN did some precision experiments on neutral kaon decay, and found more direct evidence of T violation!

10) CPLEAR homepage,

11) CPLEAR collaboration, First direct observation of time-reversal non-invariance in the neutral kaon system, Phys. Lett. B 444 (1998) 43, available online with all other papers by this collaboration at

Now we can all sit back and rack our brains even harder about what T violation really means. So far, all we know is that it arises from the darkest corner of the Standard Model: the Kobayashi-Maskawa matrix. This is a matrix describing quarks' couplings to the Higgs. The fact that it's not diagonal means that the "flavor eigenstates" of the quarks - up and down, strange and charmed, bottom and top - are not the "mass eigenstates". Why does the Kobayashi-Maskawa matrix equal what it equals? Why is it of a form that violates T symmetry? Nobody knows.

Another nice confirmation of what we already believed was the recent discovery of direct evidence for the τ neutrino. If you don't remember the particles in the Standard Model, try week119: you'll see that it has 3 generations of quarks (listed above) and 3 generations of leptons: the electron, muon and τ and their corresponding neutrinos. Of the leptons, the τ is the heaviest and thus hardest to produce. Tau neutrinos are produced by the decay of τ particles, but since it's hard to make these particles and hard to catch neutrinos, until recently nobody had ever done the clinching experiment: creating a beam of a τ neutrinos and letting it collide with some stuff to form τ particles again.

On July 21st, 2000, the DONUT collaboration at Fermilab announced that they had successfully done this experiment:

12) Christina Hebert, Phyisicists find first direct evidence for τ neutrino at Fermilab,

In case you're wondering, "DONUT" stands for "Direct Observation of the Nu Tau", where ντ is the standard abbrevation for τ neutrino.

In short, the final details of the Standard Model are all falling into place just as expected - except for the fact that neutrinos are doing lots of weird stuff they shouldn't be doing! As I explained in week130, neutrino physics is the big place for surprises in particle physics these days. This is yet another reason why it was good to directly observe the τ neutrino.

And then, of course, there's the Higgs - the final particle in the Standard Model. As you've probably heard, we're getting awfully close to seeing it - or at least definitively not seeing it. Right now they're looking for it at LEP - the big particle accelerator at CERN, in Geneva. They're just about to shut LEP down, since it's done pretty much all it can do, and they need to deactivate it to build an even more powerful accelerator - LHC, the Large Hadron Collider. But at the last minute they decided to extend its life to November 2nd, 2000:

13) LEP shutdown postponed by one month,

They're going for broke, boosting its power to the utter max, so that they can see hints of the Higgs as long as its mass is 114 GeV or so. In fact they have already seen a couple of events that suggest a Higgs of about this mass.

Whether or not LEP sees the Higgs the folks at the Tevatron at Fermilab should see it when they start Run II in a while, as long as its mass below 130 GeV. And if they don't see it, folks at CERN should see it with the LHC accelerator by around 2005, as long as its mass is below 180 GeV. A Higgs more massive than that would mean the Standard Model is seriously screwed up, so at that point, even not seeing the Higgs would be an important discovery.

The folks getting ready to analyze the Run II data at the Tevatron are doing so with a few theories in mind: the Standard Model, the minimal supersymmetric extension of the Standard Model, and a "next-to-minimal" supersymmetric extension. This is a major project; you can find lots of details here:

14) Higgs Working Group webpage,

That's basically it for this week. I just have a couple of questions about CPT. A while back on sci.physics.research I emphasized a little theorem that says: any self-dual irreducible unitary group representation H must admit an antiunitary intertwiner J: H → H with either J2 = 1 or J2 = -1. In the first case H comes from a real representation; in the second case it comes from a quaternionic representation. For more details, try this:

15) John Baez, Symplectic, quaternionic, fermionic,

Now, after I mentioned this, someone who goes by the name of "squark" suggested that the CPT operator for massive spin-1/2 particles was a an antiunitary intertwiner with (CPT)2 = -1. I'm not sure this is true, but it's definitely antiunitary, so we have an intesting question: which unitary irreducible representations of the Poincare group are self-dual? Of these, which come from real representations and which come from quaternionic ones? My hunch is that the bosonic (i.e. integral-spin) reps are real and the fermionic (i.e. half-integral-spin) reps are quaternionic. And then the question is: is the operator J just the the CPT operator? This would certainly shed some nice mathematical light on the meaning of CPT symmetry.

By the way, This Week's Finds has a nice new feature, courtesy of Laurent Bartholdi: now you can search all the old issues for a keyword or phrase! This is very useful, at least for me. Check it out on my website.


Squark found in Volume 1 of Weinberg's "Quantum Field Theory" that the CPT operator on the Hilbert space of a spin-j representation of the Poincare group is an antiunitary operator with (CPT)2 = -12j. So indeed we do have (CPT)2 = 1 in the bosonic case, making these representations real, and (CPT)2 = -1 in the fermionic case, making these representations quaternionic.

Allen Knutson points out that Streater and Wightman's title "PCT, Spin and Statistics, and All That" was itself modelled after that of Sellar and Yeatman's humorous history: "1066 and all that; a memorable history of England, comprising all the parts you can remember including one hundred and three good things, five bad kings and two genuine dates."

Martin Hardcastle wonders if Streater and Wightman were inspired by the similarity of their names to those of Sellar and Yeatman!

© 2000 John Baez