## This Week's Finds in Mathematical Physics (Week 131)

#### John Baez

I've been thinking more about neutrinos and their significance for grand unified theories. The term "grand unified theory" sounds rather pompous, but in its technical meaning it refers to something with limited goals: a quantum field theory that attempts to unify all the forces except gravity. This limitation lets you pretend spacetime is flat.

The heyday of grand unified theories began in the mid-1970s, shortly after the triumph of the Standard Model. As you probably know, the Standard Model is a quantum field theory describing all known particles and all the known forces except gravity: the electromagnetic, weak and strong forces. The Standard Model treats the electromagnetic and weak forces in a unified way - so one speaks of the "electroweak" force - but it treats the strong force seperately.

In 1975, Georgi and Glashow invented a theory which fit all the known particles of each generation into two irreducible representations of SU(5). Their theory had some very nice features: for example, it unified the strong force with the electroweak force, and it explained why quark charges come in multiples of 1/3. It also made some new predictions, most notably that protons decay with a halflife of something like 1029 or 1030 years. Of course, it's slightly inelegant that one needs two irreducible representations of SU(5) to account for all the particles of each generation. Luckily SU(5) fits inside SO(10) in a nice way, and Georgi used this to concoct a slightly bigger theory where all 15 particles of each generation, AND ONE MORE, fit into a single irreducible representation of SO(10). I described the mathematics of all this in "week119", so I won't do so again here.

What's the extra particle? Well, when you look at the math, one obvious possibility is a right-handed neutrino. As I explained last week, the existence of a right-handed neutrino would make it easier for neutrinos to have mass. And this in turn would allow "oscillations" between neutrinos of different generations - possibly explaining the mysterious shortage of electron neutrinos that we see coming from the sun.

This "solar neutrino deficit" had already been seen by 1975, so everyone got very excited about grand unified theories. The order of the day was: look for neutrino oscillations and proton decay!

A nice illustration of the mood of the time can be found in a talk Glashow gave in 1980:

1) Sheldon Lee Glashow, The new frontier, in First Workshop on Grand Unification, eds. Paul H. Frampton, Sheldon L. Glashow and Asim Yildiz, Math Sci Press, Brookline Massachusetts, 1980, pp. 3-8.

I'd like to quote some of his remarks because it's interesting to reflect on what has happened in the intervening two decades:

Pions, muons, positrons, neutrons and strange particles were found without the use of accelerators. More recently, most developments in elementary particle physics depended upon these expensive artificial aids. Science changes quickly. A time may come when accelerators no longer dominate our field: not yet, but perhaps sooner than some may think.

Important discoveries await the next generation of accelerators. QCD and the electroweak theory need further confirmation. We need know how b quarks decay. The weak interaction intermediaries must be seen to be believed. The top quark (or perversions needed by topless theories) lurks just out of range. Higgs may wait to be found. There could well be a fourth family of quarks and leptons. There may even be unanticipated surprises. We need the new machines.

Of course we now know how the b (or "bottom") quark decays, we've seen the t (or "top") quark, we've seen the weak interaction intermediaries, and we're quite sure there is not a fourth generation of quarks and leptons. There have been no unanticipated surprises. Accelerators grew ever more expensive until the U.S. Congress withdrew funding for the Superconducting Supercollider in 1993. The Higgs is still waiting to be found or proved nonexistent. Experiments at CERN should settle that issue by 2003 or so.

On the other hand, we have for the first time an apparently correct theory of elementary particle physics. It may be, in a sense, phenomenologically complete. It suggests the possibility that there are no more surprises at higher energies, at least for energies that are remotely accessible. Indeed, PETRA and ISR have produced no surprises. The same may be true for PEP, ISABELLE, and the TEVATRON. Theorists do expect novel higher-energy phenomena, but only at absurdly inacessible energies. Proton decay, if it is found, will reinforce the belief in the great desert extending from 100 GeV to the unification mass of 1014 GeV. Perhaps the desert is a blessing in disguise. Ever larger and more costly machines conflict with dwindling finances and energy reserves. All frontiers come to an end.

You may like this scenario or not; it may be true or false. But, it is neither impossible, implausible, or unlikely. And, do not despair nor prematurely lament the death of particle physics. We have a ways to go to reach the desert, with exotic fauna along the way, and even the desolation of a desert can be interesting. The end of the high-energy frontier in no ways implies the end of particle physics. There are many ways to skin a cat. In this talk I will indicate several exciting lines of research that are well away from the high-energy frontier. Important results, perhaps even extraordinary surprises, await us. But, there is danger on the way.

The passive frontier of which I shall speak has suffered years of benign neglect. It needs money and manpower, and it must compete for this with the accelerator establishment. There is no labor union of physicists who work at accelerators, but sometimes it seems there is. It has been argued that plans for accelerator construction must depend on the "needs" of the working force: several thousands of dedicated high-energy experimenters. This is nonsense. Future accelerators must be built in accordance with scientific, not demographic, prioriries. The new machines are not labor-intensive, must not be forced to be so. Not all high energy physicsts can be accomodated at the new machines. The high-energy physicist has no guaranteed right to work at an accelerator, he has not that kind of job security. He must respond to the challenge of the passive frontier .

Of course, the collapse of the high-energy physics job market and the death of the Superconducting Supercollider give these words a certain poignancy. But what is this "passive frontier" Glashow mentions? It means particle physics that doesn't depend on very high energy particle accelerators. He lists a number of options:

A) CP phenomenology. The Standard Model is not symmetrical under switching particles and their antiparticles - called "charge conjugation", or "C". Nor is it symmetrical under switching left and right - called "parity", or "P". It's almost, but not quite, symmetrical under the combination of both operations, called "CP". Violation of CP symmetry is evident in the behavior of the neutral kaon. Glashow suggests looking for CP violation in the form of a nonzero magnetic dipole moment for the neutron. As far as I know, this has still not been seen.

B) New kinds of stable matter. Glashow proposes the search for new stable particles as "an ambitious and risky field of scientific endeavor". People have looked and haven't found anything.

C) Neutrino masses and neutrino oscilllations. Glashow claims that "neutrinos should have masses, and should mix". He now appears to be right. It took almost 20 years for the trickle of experimental results to become the lively stream we see today, but it happened. He urges "Let us not miss the next nearby supernova!" Luckily we did not.

D) Astrophysical neutrino physics. In addition to solar neutrinos and neutrinos from supernovae, there are other interesting connections between neutrinos and astrophysics. The background radiation from the big bang should contain neutrinos as well as the easier-to-see photons. More precisely, there should be about 100 neutrinos of each generation per cubic centimeter of space, thanks to this effect. These "relic neutrinos" have not been seen, but that's okay: by now they would be too low in energy to be easily detected. Glashow notes that if neutrinos had a nonzero mass, relic neutrinos could contribute substantially to the total density of the universe. The heaviest generation weighing 30 eV or so might be enough to make the universe eventually recollapse! On the other hand, for neutrinos to be gravitationally bound to galaxies, they'd need to be at least 20 eV or so.

E) Magnetic monopoles. Most grand unified theories predict the existence of magnetic monopoles due to "topological defects" in the Higgs fields. Glashow urges people to look for these. This has been done, and they haven't been seen.

F) Proton decay. As Glashow notes, proton decay would be the "king of the new frontier". Reflecting the optimism of 1980, he notes that "to some, it is a foregone conclusion that proton decay is about to be seen by experiments now abuilding". But alas, people looked very hard and did not find it! This killed the SU(5) theory. Many people switched to supersymmetric theories, which are more compatible with very slow proton decay. But with the continuing lack of new experiments to explain, enthusiasm for grand unified theories gradually evaporated, and theoretical particle physics took refuge in the elegant abstractions of string theory.

But now, 20 years later, interest in grand unified theories seems to be coming back. We have a rich body of mysterious experimental results about neutrino oscillations. Somebody should explain them!

On a slightly different note, one of my little side hobbies is to study the octonions and dream about how they might be useful in physics. One place they show up is in the E6 grand unified theory - the next theory up from the SO(10) theory. I said a bit about this in "week119", but I just bumped into another paper on it in the same conference proceedings that Glashow's paper appears is:

2) Feza Gursey, Symmetry breaking patterns in E_6, in First Workshop on Grand Unification, eds. Paul H. Frampton, Sheldon L. Glashow and Asim Yildiz, Math Sci Press, Brookline Massachusetts, 1980, pp. 39-55.

He says something interesting that I want to understand someday - maybe someone can explain why it's true. He says that E6 is a "complex" group, E7 is a "pseudoreal" group, and E8 is a "real" group. This use of terminology may be nonstandard, but what he means is that E6 admits complex representations that are not their own conjugates, E7 admits complex reps that are their own conjugates, and that all complex reps of E8 are complexifications of real ones (and hence their own conjugates). This should have something to do with the symmetry of the Dynkin diagram of E6.

Octonions are also prominent in string theory and in the grand unified theories proposed by my friends Geoffrey Dixon and Tony Smith - see "week59", "week91", and "week104". I'll probably say more about this someday....

The reason I'm interested in neutrinos is that I want to learn what evidence there is for laws of physics going beyond the Standard Model and general relativity. This is also why I'm trying to learn a bit of astrophysics. The new hints of evidence for a nonzero cosmological constant, the missing mass problem, the large-scale structure of the universe, and even the puzzling γ-ray bursters - they're all food for thought along these lines.

The following book caught my eye since it looked like just what I need - an easy tutorial in the latest developments in cosmology:

3) Greg Bothun, Modern Cosmological Observations and Problems, Taylor & Francis, London, 1998.

On reading it, some of the remarks about particle physics made me unhappy. For example, Bothun says "the observed entropy S of the universe, as measured by the ratio of baryons to photons, is ~ 5 x 10-10." But as Ted Bunn explained to me, the entropy is actually correlated to the ratio of photons to baryons - the reciprocal of this number. Bothun also calls the kinetic energy density of the field postulated in inflationary cosmology, "essentially an entropy field that currently drives the uniform expansion and cooling of the universe". This makes no sense to me. There are also a large number of typos, the most embarrassing being "virilizing" for "virializing".

But there's a lot of good stuff in this book! The author's specialty is large-scale structure, and I learned a lot about that. Just to set the stage, recall that the Milky Way has a disc about 30 kiloparsecs in diameter and contains roughly 100 or 200 billion stars. But our galaxy is one of a cluster of about 20 galaxies, called the Local Group. In addition to our galaxy and the Large and Small Magellanic Clouds which orbit it, this contains the Andromeda Galaxy (also known as M31), another spiral galaxy called M33, and a bunch of dwarf irregular galaxies. The Local Group is about a megaparsec in radius.

This is typical. Galaxies often lie in clusters which are a few megaparsecs in radius, containing from a handful to hundreds of big galaxies. Some famous nearby clusters include the Virgo cluster (about 20 megaparsecs away) and the Coma cluster (about 120 megaparsecs away). Thousands of clusters have been cataloged by Abell and collaborators.

And then there are superclusters, each typically containing 3-10 clusters in an elongated "filament" about 50 megaparsecs in diameter. I don't mean to make this sound more neat than it actually is, because nobody is very sure about intergalactic distances, and the structures themselves are rather messy. But there are various discernible patterns. For example, superclusters tend to occur at the edges of larger roundish "voids" which have few galaxies in them. These voids are very large, about 100 or 200 megaparsecs across. In general, galaxies tend to be falling into denser regions and moving away from the voids. For example, the Milky Way is falling towards the center of the Local Supercluster at about 300 kilometers per second, and the Local Supercluster is also falling towards the next nearest one - the Hydra-Centaurus Supercluster - at about 300 kilometers per second.

Now, if the big bang theory is right, all this stuff was once very small, and the universe was much more homogeneous. Obviously gravity tends to amplify inhomogeneities. The problem is to understand in a quantitative way how these inhomogeneities formed as the universe grew.

Here are a couple of other books that I'm finding useful - they're a bit more mathematical than Bothun's. I'm trying to stick to new books because this subject is evolving so rapidly:

4) Jayant V. Narlikar, Introduction to Cosmology, Cambridge U. Press, Cambridge, 1993.

5) Peter Coles and Francesco Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure, Wiley, New York, 1995.

While I was looking around, I also bumped into the following book on black holes:

6) Sandip K. Chakrabarti, ed., Observational Evidence for Black Holes in the Universe, Kluwer, Dordrecht, 1998.

It mentioned some objects I'd never heard of before. I want to tell you about them, just because they're so cool!

X-ray novae: First, what's a nova? Well, remember that a white dwarf is a small, dense, mostly burnt-out star. When one member of a binary star is a white dwarf, and the other dumps some of its gas on this white dwarf, the gas can undergo fusion and emit a huge burst of energy - as much as 10,000 times what the sun emits in a year. To astronomers it may look like a new star is suddenly born - hence the term "nova". But not all novae emit mostly visible light - some emit X-rays or even γ rays. A "X-ray nova" is an X-ray source that suddenly appears in a few days and then gradually fades away in a year or less. Many of these are probably neutron stars rather than white dwarfs. But a bunch are probably black holes!

Blazars: A "blazar" is a galactic nucleus that's shooting out a jet of hot plasma almost directly towards us, exhibiting rapid variations in power. Like quasars and other active galactic nuclei, these are probably black holes sucking in stars and other stuff and forming big accretion disks that shoot out jets of plasma from both poles.

Mega masers: A laser is a source of coherent light caused by stimulated emission - a very quantum-mechanical gadget. A maser is the same sort of thing but with microwaves. In fact, masers were invented before lasers - they are easier to make because the wavelength is longer. In galaxies, clouds of water vapor, hydroxyl, silicon monoxide, methanol and other molecules can form enormous natural masers. In our galaxy the most powerful such maser is W49N, which has a power equal to that of the Sun. But recently, still more powerful masers have been bound in other galaxies, usually associated with active galactic nuclei. These are called "mega masers" and they are probably powered by black holes. The first mega maser was discovered in 1982; it is a hydroxyl ion maser in the galaxy IC4553, with a luminosity about 1000 times that of our sun. Subsequently people have found a bunch of water mega masers. The most powerful so far is in TXFS2226-184 - it has a luminosity of about 6100 times that of the Sun!

Addendum: Here is something from Allen Knutson in response to my remark that E6 has complex representations that aren't their own conjugates. I hoped that this is related to the symmetry of the Dynkin diagram of E6, and Allen replied:

```It does. The automorphism G→G that exchanges representations with their
duals, the Cartan involution, may or may not be an inner automorphism.
The group of outer automorphisms of G (G simple) is iso to the diagram
automorphism group. So no diagram auts, means the Cartan involution is inner,
means all reps are iso to their duals, i.e. possess invariant bilinear forms.

(Unfortunately it's not iff - the Dn's alternate between whether the
Cartan involution is inner, much as their centers alternate between
Z4 and Z22.)

Any rep either has complex trace sometimes, or a real, or a quaternionic
structure, morally because of Artin-Wedderburn applied to the real
group algebra of G. Given a rep one can find out which by looking at
the integral over G of Tr(g2), which comes out 0, 1, or -1 (respectively).
This is the "Schur indicator" and can be found in Serre's LinReps of
Finite Groups.
Allen K.
```