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December 15, 2008

This Week’s Finds in Mathematical Physics (Week 273)

Posted by John Baez

In week273 of This Week’s Finds, read more about the geysers on Enceladus. Hear the story of the Earth, with an emphasis on mineral evolution — from chondrites to the Big Splat, the Late Heavy Bombardment, the Great Oxidation Event, Snowball Earth… to now.

Then, learn about Pontryagin duality!

Here’s a cool chart from this paper:

  • Robert M. Hazen, Dominic Papineau, Wouter Bleeker, Robert T. Downs, John M. Ferry, Timothy J. McCoy, Dmitri A. Sverjensky and Henxiong Yang, Mineral evolution, American Mineralogist 91 (2008), 1693-1720.




Posted at December 15, 2008 2:34 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 273)

Just before the big splat “The ‘first atmosphere’ was mainly hydrogen and helium”, but was there a first ocean? I ask this question because I am wondering whether comets that we see nowadays could be these very energetic tiny particles in Mackenzie’s animation that correspond to the impact point of Theia with Earth.

Posted by: yael on December 15, 2008 5:04 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

I’m no expert, and of course there’s a lot of controversy about these matters, but I get the impression that the first oceans formed after the Big Splat. If you look at the timeline further up on this page, that’s what you’ll see — and you’ll see a similar story on the Palaeos webpage about the Hadean Eon.

However, I don’t know why there couldn’t have been a ‘first ocean’ before the Big Splat. I had the impression that before then, most oxygen around Earth orbit was bound up in silica, feldspars and so on — and that water was only produced later, by volcanism. But I could be wrong.

Most of the spectacular comets we see come from the outer Solar System, either the Kuiper Belt or the more distant Oort cloud.



I don’t think comets last long once they spend a lot of time in the inner Solar System — it’s too warm. Their tails demonstrate how they’re falling apart. Any cometary material produced by the Big Splat would probably by now be a dried-up rocky core.

There’s a lot of asteroids at Earth’s Lagrange points that could, for all I know, be leftovers of the formation of Theia.

There are also some asteroids like 3752 Cruithne and 2002 AA29 with complicated near-Earth orbits. Could these be remnants of the Big Splat, or would any junk from that collision have drifted away or hit Earth by now? I don’t know. But you’ve got to check out this movie of Cruithne’s truly amazing orbit, viewed in a rotating frame of reference so the Earth looks like it’s standing still. 2002 AA29 is even more tricky: it oscillates between being a quasi-satellite of the Earth and having a horseshoe orbit!

Now you’ve got me wondering about the distribution of water in the early Solar System. How did Mars get its water?

There’s a theory that Earth got lots of water from comet impacts. This theory is called the Big Splash – not to be confused with the Big Splat! But, I don’t think it’s widely accepted.

Posted by: John Baez on December 15, 2008 9:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

thanks for the illuminating answer. If one looks at this webpage one sees two interesting facts: the Halley comet seems to spend quite a lot of time of its 76 years of revolution rather far away from the sun, so not much subject to its radiation. As far as I remember (I was fortunately old enough to watch tv last time Halley visited us!) the tail starts to exist when it arrives near the sun. The second observation is that its orbit seems to be tangent to earth’s orbit!. Do they run through their orbits in the same direction? Is there a way to know Halley’s age?

Maybe there exist more energetic particles created by the big splash having much longer periods of revolution, spending a lot of time in the outer space, hence keeping safely their water? do we know which is the biggest possible “radius” of a body in orbit around the sun?

Posted by: yael on December 16, 2008 8:55 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Yael writes:

… the Halley comet seems to spend quite a lot of time of its 76 years of revolution rather far away from the sun, so not much subject to its radiation.

I’m pretty sure that’s typical of all comets that remain bright enough for us to see for a long time. Comets lose mass when they get near the sun, so I just don’t think there are comets that spend their whole time as close to the Sun as, say, the Earth does. I can imagine a comet getting knocked into such a small orbit, but then it would lose all its volatile material — the stuff that makes the ‘tail’ — fairly soon.

Its orbit seems to be tangent to earth’s orbit! Do they run through their orbits in the same direction?

No, according to Wikipedia the orbit of Halley’s comet is retrograde: it goes around the Sun the opposite way from Earth and all the planets.

Is there a way to know Halley’s age?

Sure, you look it up in Wikipedia.

Just kidding — this doesn’t actually work. But, it was worth a try. Here’s what they say:

Halley’s comet may have been recorded in China as early as 467 BC, but this is uncertain. The first certain observation dates from 240 BC, and subsequent appearances were recorded by Chinese, Babylonian, Persian, and other Mesopotamian texts.

Halley is classified as a short period comet (a descriptor for comets with orbits lasting 200 years or less). However, its orbit is such that it is believed to have been originally a long period comet whose orbit was perturbed by the gravity of the giant planets and sent into the inner Solar System. It gives its name to the ‘Halley group’ of comets, which share these orbital characteristics.

If Halley was once a long period comet, it is likely to have originated in the Oort Cloud, a sphere of cometary bodies which has its inner edge at 50,000 AU. This distinguishes it from most other short period comets, which originate instead from the Kuiper Belt, a flat disc of icy debris between 38 AU (Pluto’s orbit) and 50 AU from the Sun.

In 1989, Boris Chirikov and Vitaly Vecheslavov performed an analysis of 46 apparitions of Halley’s Comet taken from historical records and computer simulations. These studies showed that the comet’s dynamics follow a simple area-preserving map similar to the standard map. Its dynamics were shown to be chaotic and unpredictable on long timescales. Halley’s projected lifetime, as determined by differential escape, is roughly 10 million years.

So, I guess nobody knows how old it is, but on very hand-wavy grounds I’d suspect no more than 10 million years. That’s a very short time, as far as astronomy goes.

A famous example of a comet with a much shorter period than Halley’s comet is Comet Encke. It goes around the Sun every 3 years. I wonder how long it will last.

I don’t know any comet that stays closer to the Sun than Encke.

do we know which is the biggest possible “radius” of a body in orbit around the sun?

By radius, do you mean the radius of the body or the radius of the orbit? The second option is more interesting to me — in week222 I wrote about ‘Transneptunian objects’, which orbit the Sun at very huge distances. The most distant ones seems to live in the Oort cloud. This consists of lots of comets in orbit up to one light-year away from the Sun. That’s 50,000 times farther from Sun than the Earth is! Perturbations from nearby stars sometimes push these comets towards the Sun, and they become the comets we see. Then they lose their volatile materials and fizzle out, or get tossed back out.

Posted by: John Baez on December 17, 2008 5:00 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

John quoted Wikipedia:

These studies showed that the comet’s dynamics follow a simple area-preserving map similar to the standard map.

At first I thought that this meant that it obeyed Kepler's Laws (particularly the Third Law: equal areas in equal times). But no, actually it means that the orbit is chaotic (as the next, unquoted sentence says), only chaotic in a certain simple and nearly standard way.

Posted by: Toby Bartels on December 17, 2008 6:29 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

That’s Kepler’s second law. The third law has to do with orbital periods.

Posted by: Todd Trimble on December 17, 2008 1:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

I guessed that the ‘standard map’ was Arnold’s famous ‘cat map’, but it’s this area-preserving map from the square [0,2π]×[0,2π][0,2 \pi] \times [0,2\pi] to itself:

(x,y)(x+ksin(y),y+x+ksin(y))(x,y) \mapsto (x+k sin(y), y+x+k sin(y))

(addition mod 2π2 \pi). It describes the time evolution of a ‘kicked rotator’. As we turn up the constant kk, we eventually get ‘global chaos’ — that is, a dense orbit. You can play with an applet that demonstrates this.

Posted by: John Baez on December 17, 2008 5:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

More of a comment than a question. My space encyclopedia (1987ish, edited by Sir Patrick Moore), says that the Oort Cloud is hypothetical (one of my kids asked about comets recently - I don’t usually spend my evenings reading encyclopediæ). I was going to ask if it had been discovered since then as you seem to refer to it as an existent object. However, Wikipedia, Oort Cloud, agrees with my encyclopedia so I’ve answered my own question and can go back to sleep … I mean, grading papers.

Posted by: Andrew Stacey on December 17, 2008 3:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Good point! People are starting to see candidates for Oort cloud objects, like Sedna — but naturally, most of the ones they’ve seen so far are rather huge, very different than your average comet. Plus, they could be living in the outskirts of the scattered disk rather than the inner Oort cloud. But that could be a subtle distinction, a bit like the distant outskirts of Los Angeles versus the housing developments of western Riverside County.

In week222, I wrote:

The Oort cloud is a hypothesized spherical cloud of comets, perhaps between 50,000 and 100,000 AU from the Sun. The idea is this cloud consists of leftovers from the original nebula that collapsed to form our Solar system, and comets come from this region when they are perturbed from their orbits by the gravity of other stars.

Nobody has seen a certified Oort cloud object. The best candidate so far is Sedna, an object roughly 1500 kilometers in diameter with a wildly eccentric orbit taking it between 80 to 930 AU from the Sun.

Sedna was discovered in 2004 when it was 90 AU from the Sun. It’s redder than Mars, its temperature never rises above 23 Kelvin, and its year lasts 11,250 years. It’s the farthest known object in our Solar System, but still much closer than the Oort cloud was supposed to be. Maybe it’s a drastic example of a scattered disc object, maybe it’s part of an “inner Oort cloud”… or maybe the Oort cloud isn’t as far out as people thought.

The closest people have come to seeing the Oort cloud is seeing a “Bok globule”:

A Bok globule is a cloud of dust and gas that’s collapsing to form a star. This one is about 12,500 AU across. The scientists who observed it say it’s about the size of the Oort cloud. This just goes to show how little we know about the Oort cloud!

Comets have got to come from somewhere, so while seeing a comet a light-year away may be tough, I believe in the Oort cloud. Plus, Lisa used to own an apartment in Cambridge Massachusetts, and she rented it to Oort’s son, the number theorist and algebraic geometer Frans Oort. It would be rude to cast doubt on my wife’s tenant’s father’s cloud.
Posted by: John Baez on December 17, 2008 5:25 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

The most distant ones seems to live in the Oort cloud. This consists of lots of comets in orbit up to one light-year away from the Sun. That’s 50,000 times farther from Sun than the Earth is!

More like 65,000. You just said before that the Oort cloud starts at about 50,000 AU.

Remember that 1AU is about 8 light-minutes, and Rent tells us that there are 525,600 minutes in a year.

Posted by: John Armstrong on December 17, 2008 4:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Thanks. I can never remember numbers like how many AU’s there are in a lightyear. The reason I pack This Week’s Finds with numbers is so I can look ‘em up!

Posted by: John Baez on December 17, 2008 6:01 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

I thought google made remembering these things unnecessary, it’s really really good at unit conversion. A search for One light year in AU gets exactly what you want. It also does various obscure units, say you want to turn feet into rods, or whatever.

Similarly google can do arithmetic in roman numerals. Search for XX*VII or similar things. It’s the best calculator ever.

Posted by: Noah Snyder on December 18, 2008 8:14 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

A search for One light year in AU gets exactly what you want.

Actually, you need 1 light year in AU instead.

Posted by: Toby Bartels on December 18, 2008 10:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

From Wikipedia:

1 light-year ≈ 63.241 × 103 AU

Posted by: Russ Van Rooy on December 18, 2008 10:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Noah Snyder wrote:

I thought google made remembering these things unnecessary, it’s really really good at unit conversion.

I know I can instantaneously look up how many AU there are in a light-year. What I really want is not the exact number, but a visceral sense of how big things are: for example, how big the Earth’s orbit is compared to the orbit of Jupiter, and then the size of the Oort cloud — which even if hypothetical I’ll take as the size of the realm that’s strongly under the Sun’s gravitational influence — and then the distance to the nearest star… and so on.

So, I really don’t care if a light-year is 50,000 or 63,239.6717 AU. What I’d like to remember is the general order of magnitude. But I’m really bad at even that.

The reason is that most of my actual work is in mathematics, where these issues don’t come up. Getting to know the physical universe is really just a part-time hobby for me. I know the cohomology groups of P \mathbb{C}\mathrm{P}^\infty deep in my bones, but ask me the mass of the Earth and I’d have to really struggle to come with a sensible answer.

This is why I made my chart of distances and my timeline — in a feeble attempt to familiarize myself with the scale of the universe we inhabit.

Posted by: John Baez on December 19, 2008 12:18 AM | Permalink | Reply to this

Star Maker; AU per LY and primality; Re: This Week’s Finds in Mathematical Physics (Week 273)

Of course I like the Baez Timeline, and I’d made a similar one interwoven with “Star Maker” [1937] at TIMELINE COSMIC FUTURE.

As to AU in a Light Year:

63239 = 11 x 5749

632396717 = 233 x 1303 x 2083

Posted by: Jonathan Vos Post on December 20, 2008 5:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Mike Stay points to an interesting blog entry about this book on the future of mineral evolution:

  • Jan Zalasiewicz, The Earth After Us: What Legacy Will Humans Leave in the Rocks?, Oxford University Press, Oxford, 2009.

(It’s not 2009 yet, but the best books about the future are actually published in the future.)

Posted by: John Baez on December 16, 2008 12:58 AM | Permalink | Reply to this

Robert M. Hazen: good in prose and verse; Re: This Week’s Finds in Mathematical Physics (Week 273)

Robert M. Hazen rocks! He is a spectacularly good “big picture” thinker and writer. I admire the textbook that he coauthored:

The Sciences: An Integrated Approach, 2nd Edition, by James Trefil and Robert M. Hazen.

If you look at the review of the first edition, you’ll find “The 20 Great Ideas of Science” [“Sciences Top 20 Hits”, Science, 18 Jan 1991, p.267] A poem of the same name, with a sentence by Trefil and Hazen in each verse, is by myself and my wife, Professor Christine M. Carmichael. Four of these 20 verses themselves appeared in that book review follow-up in Science.

A good scientist, educator, and writer inspires others to help spread his/her message. And here’s more proof, with John Baez spreading another of the grand syntheses of Robert M. Hazen. Kudos!

Posted by: Jonathan Vos Post on December 15, 2008 5:23 AM | Permalink | Reply to this

273 weeks above absolute zero; Re: This Week’s Finds in Mathematical Physics (Week 273)

Forgot to say: congratulations, John Baez, on reaching the freezing point of water at last. Week 273, I mean.

273 Kelvin = 0 degrees Centigrade…

Posted by: Jonathan Vos Post on December 15, 2008 5:30 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

John, my condolences. Take care!

Posted by: lievenlb on December 15, 2008 7:02 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Thanks very much.

Posted by: John Baez on December 15, 2008 10:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

My condolences on your father.

Posted by: Tim Silverman on December 15, 2008 1:44 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Thanks, Tim. That’s very kind.

(I hope this sad aspect of week273 doesn’t squelch discussion of the subjects it covers. In particular, I always wish more people would talk about the fun astronomy stuff in This Week’s Finds! It’s so much easier to ask interesting questions about the early universe than, say, Pontryagin duality. But I guess the n-Café draws an audience of hardcore mathematicians.)

Posted by: John Baez on December 16, 2008 12:56 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Actually, I did try quite hard to think of a maths question, but I couldn’t come up with anything more intelligent than: dualities are very interesting, I wish I understood them better. And I didn’t wish to put off paying my respects pending a mathematical insight!

One thing I have observed: it seems to me that a lot of dualities can be classified broadly as Poincaré-type dualities, and a lot of others as Pontryagin-type dualities. In fact, most dualities seem to fall into one or the other class. But I don’t understand what relationship, if any, exists between these two classes.

On the plus side, at the prompting of David’s comment below, I just read and understood the Wikipedia article on Tannaka-Krein duality, so either the article or my brain has improved dramatically since I last read it.

The geology and astronomy are interesting, but I don’t have anything to say about them. I don’t think this is because I’m a hardcore mathematician; I have about the same dilettantish interest in geology and astronomy as in mathematics. In fact, it’s somewhat the other way round—I think your discussion of geology and astronomy would have to be more hardcore before I could think of questions. (This isn’t a criticism: this isn’t an astronomy blog, and I’d go elsewhere if I wanted hardcore astronomy. The current level is fine. And other people seem to have comments …. )

Just one other remark: the oceans \rightarrow life thing. As far as I can tell, we have almost no idea what conditions give rise to life or how common they are are. (In fact, except at the most abstract level, I’d say we haven’t the remotest, foggiest notion, except that “not very common at all” seems to be the way the current evidence is pointing.) So whenever anybody suggests liquid water, or liquid generally, as a necessary condition (or, more bizarrely, but more frequently, a sufficient condition!) for life, I get kind of jumpy. Really discovering an independently evolved life would certainly be very interesting, but speculating about it in the absence of evidence … not so much. I always feel—What, liquid nitrogen geysers shooting material into orbit around Saturn aren’t amazing enough for you? This just comes up over and over again in every discussion of the planets, and it annoys me every time.

Posted by: Tim Silverman on December 16, 2008 11:41 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Tim wrote:

In fact, it’s somewhat the other way round—I think your discussion of geology and astronomy would have to be more hardcore before I could think of questions.

I hope this means the stuff I’m writing makes sense, so you don’t feel the need to ask ‘huh?’ type questions.

But there are also questions of another sort, where one reads something and it makes one start wondering… I find I can easily generate huge wads of fairly sensible questions of this sort about astronomy. They may be questions nobody can answer, but they’re not completely idiotic.

Like: “Which moons of other planets were probably formed by collisions? And if not that way, how did they form? Is the fact that our Moon is especially large compared to its planet related to the fact that it formed from a collision? And how much evidence is there for this collision theory, anyway? What are the other main theories?”

I could keep generating these for pages, and I don’t think it takes any vast knowledge of astronomy. It seems to require more specialized knowledge to generate equally promising questions about, say, Pontryagin duality. For subjects like that, it seems one has to start by fighting through several layers of ‘huh?’ type questions.

Just one other remark: the oceans → life thing. As far as I can tell, we have almost no idea what conditions give rise to life or how common they are are.

That’s true. But Carolyn Porco’s remarks on this subject are pretty sensible, given our overall stunning lack of knowledge of xenobiology. She writes:

But in all, it is almost unavoidable that liquid water is present somewhere below its surface. If so, we face the thrilling possibility that within this little moon is an environment where life, or at least its precursor steps, may be stirring. Everything life needs appears to be available: liquid water, the requisite chemical elements and excess energy. The best analogues for an Enceladan ecosystem are terrestrial subsurface volcanic strata where liquid water circulates around hot rocks, in the complete absence of sunlight and anything produced by sunlight. Here are found organisms that consume either hydrogen and carbon dioxide, creating methane, or hydrogen and sulfate — all powered not by the sun but by Earth’s own internal heat.

As far as I can tell, she’s not saying that water is necessary for life. She’s mainly saying that an Enceladus with a buried ocean of liquid water would be an environment a lot like some places on Earth that support life — especially since we already know it has organic chemicals. In particular, it seems more promising than Europa, because there’s more energy flowing through the system.

But, I agree that emphasizing the search for water and life in our Solar System is likely to be a recipe for disappointment, since it makes otherwise interesting places seem like ‘failed Earths’.

It’s a bit like the approach to the history of mathematics in China that concentrates on the question why didn’t the Chinese come up with the Greek concept of ‘proof’? You wind up ignoring the cool stuff that’s actually there.

Posted by: John Baez on December 17, 2008 4:31 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

My condolences on your father.

Posted by: nad on December 22, 2008 3:35 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

From the copy-editing department:

In “” we looked back into the deep past, all the way to the electroweak phase transition 10 picoseconds after the Big Bang.

I believe the quotation marks should contain a pointer to week196.

Posted by: Blake Stacey on December 15, 2008 5:44 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

You’re right. Thanks — fixed!

Posted by: John Baez on December 16, 2008 1:01 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

What’s known about nonabelian Pontryagin duality? From page 2 of this paper by Mukul S. Patel, it looks as though several different attempts have been made.

Patel’s own approach leads to the dual of a locally compact group being an abelian quantum group.

Or is the true path Tannaka-Krein duality? If Joyal and Street say so, it must be.

Now what happens if you throw qualities like metrizable, second countable, torsion-free, etc. into the mix? I.e., what properties would be possessed by the category of representations of groups with these properties?

Posted by: David Corfield on December 16, 2008 9:13 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Warning: IIRC Patel has claimed a proof of the invariant subspace problem for Hilbert spaces. You might want to do some googling on his affiliation/status and “previous”, and run that through your own favourite Bayesian method for prior –> posterior.

I’d be a little (well, very) surprised if what Patel is doing has trumped the work of Effros, Rieffel, Rosenberg et al. If he obtains abelian quantum groups from duals of locally compact groups this smells all wrong to me.

As far as I understand, which isn’t very far, the descendant of Tannaka-Krein from an analysts’ point of view is either Kac algebras or the Kustermans-Vaes framework which I’ve seen alluded to on some cafe threads.

Posted by: Yemon Choi on December 16, 2008 8:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

One true path to generalizing Pontryagin duality is the Tannaka–Krein reconstruction theorem: we can reconstruct any Hopf algebra HH from its symmetric monoidal category Comod(H)Comod(H) of finite-dimensional comodules, together with the forgetful functor Comod(H)VectComod(H) \to Vect.

(A Hopf algebra is both an algebra and a coalgebra, so it has both modules and comodules. The beautiful version of Tannaka–Krein uses the subtle advantage of comodules over modules which I mentioned in your coalgebra thread.)

Another true path to generalizing Pontryagin duality is the Doplicher–Roberts reconstruction theorem: we can reconstruct any compact Hausdorff group from its symmetric monoidal C*-category of finite-dimensional continuous unitary representations.

These two paths are not wholly distinct.

I suspect there’s a lot more very good very abstract stuff to do on this subject.

For example, I generalized Doplicher–Roberts to ‘compact supergroupoids’ using 2-Hilbert spaces. But what about the locally compact case? Here we may want to use infinite-dimensional 2-Hilbert spaces.

This direction involves a lot of analysis. But there are many other directions to generalize, too. Lawvere’s work on reconstructing an algebraic theory from its category of models is closely akin to Tannaka-Krein reconstruction. And in a very different direction, only last year Hendryk Pfeiffer discovered the kind of gadget whose representation category is precisely a modular tensor category!

Furthermore: most of the theorems I’m talking about now say “the category of representations of an XX is a YY category” — where XYX \ne Y. So, these results aren’t quite as charming as Pontryagin duality, which says “the group of isomorphism classes of 1-dimensional continuous unitary representations of an LCA group is an LCA group”. Can we broaden our concept of XX and YY to the point where we get another theorem that applies to nonabelian groups but has X=YX = Y?

Posted by: John Baez on December 16, 2008 11:56 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

But isn’t this charm (of Pontryagin duality) bought at the expense of decategorification? If a group is a category, shouldn’t we expect an equivalence with any dual, rather than an isomorphism?

And if abelian groups are really 2-categories of a kind…

Posted by: David Corfield on December 17, 2008 9:38 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Good point, David. Does the charm of Pontryagin duality really come at the expense of decategorification?

Doplicher–Roberts duality tells us how to recover a compact group GG from its symmetric monoidal C*-category of continuous unitary representations Rep(G)Rep(G). But what happens when this group is abelian?

The special thing about abelian groups is that all their irreducible representations are 1-dimensional. So, we don’t really need all of Rep(G)Rep(G) to recover GG; the 1-dimensional reps are enough.

The interesting thing is that if someone hands you Rep(G)Rep(G) as an abstract symmetric monoidal category, you can pick out the 1-dimensional reps without knowing anything of the fact that objects of Rep(G)Rep(G) are secretly vector spaces! The 1-dimensional objects are the invertible objects: the objects XX that have objects YY with

XY1X \otimes Y \cong 1

where 11 is the unit for the tensor product (namely, the trivial rep of GG).

Let’s call the full subcategory of invertible objects Rep(G) ×Rep(G)^\times. It’s easy to see that the tensor product of invertible objects is again invertible, so Rep(G) ×Rep(G)^\times is again a monoidal category — but one of a special sort, since every object has an inverse! Ulbrich and Laplaza call such a thing a category with group structure.

Rep(G) ×Rep(G)^\times not a 2-group, since not every morphism is invertible. But that’s easily fixed, if we want to fix it: just throw out all the noninvertible morphisms! We get a 2-group — and indeed a symmetric 2-group: that is, a symmetric monoidal category where all objects and all morphisms are invertible.

Now, what you’re complaining about, quite rightly, is that I’m taking this symmetric 2-group and decategorifying it to obtain an abelian group, which is the Pontryagin dual G *G^* of our compact abelian group GG.

Somehow this last step — the decategorification — is not losing any essential information, since we’re still able to recover GG from G *G^*.

How does it work? Rather roughly, it must go like this. We start with a continuous 1-dimensional unitary representation of GG and we say “Oh, this is isomorphic to one where GG is acting on our favorite 1d Hilbert space, namely \mathbb{C}. So, it might as well be a continuous homomorphism ρ:GU(1)\rho: G \to \mathrm{U}(1). But that’s an element of G *G^*.”

The question is why we don’t lose any interesting information when we engage in this decategorification. Some symmetric 2-groups do have information not contained in their decategorifications! In general a symmetric 2-group is classified by:

  • an abelian group AA (the group of isomorphism classes of objects),
  • an abelian group BB (the group of automorphisms of any object),
  • a stable 3-cocycle: an element of H 5(K(A,3),B)H^5(K(A,3),B) which records the associator and braiding.

I think that in the case at hand we have A=G *A = G^*, B= ×B = \mathbb{C}^\times, and the trivial stable 3-cocycle.

Why the trivial one? Well, I can’t see what else it would be, and still have no information lost when we decategorify and get G *G^*.

Posted by: John Baez on December 17, 2008 5:02 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

A beautiful article on dualities is

MR1084537 (92b:01048)
Shafarevich, I. R.(2-AOS)
Abelian and nonabelian mathematics.
Translated from the Russian by Smilka Zdravkovska.
Math. Intelligencer 13 (1991), no. 1, 67–75.
01A60 (11-01)

The Math Review says the following about it:

The philosophy of this paper is that the most beautiful parts of 19th- and 20th-century mathematics have as a common algebraic nucleus the duality of abelian groups, and that basic difficulties in mathematics have their origin in the lack of generalization to the nonabelian case. The goal of the present stimulating survey is to give a general idea of the intertwining of these notions. In abelian mathematics the author considers the duality between vector spaces and their duals, between differential forms and curves (especially Maxwell’s equations), character theory for abelian groups, Galois theory of abelian extensions (Kummer theory) and, as an analogue to the rational function field as ground field, the theory of abelian functions, and class field theory in the language of id` eles. In nonabelian mathematics the natural analogues of characters are the irreducible representations, and there is a duality between the set of irreducible representations and the set of conjugacy classes; one gets algebras, but not groups (examples: Clebsch-Gordan formula, special functions). Generalization of the theory of abelian functions and complex tori is performed, replacing numbers by matrices thus leading to vector bundles and the Yang-Mills equations as a “nonabelian” generalization of the Maxwell equations; the generalization of class field theory leads to the Langlands conjectures. For π1(S) (S an algebraic curve) and its analogue Gal(\overline{K}, K) for number fields, something is known about nilpotent quotients of these groups; in the function field case the study of coverings with nilpotent Galois group leads to a “nonabelian” generalization of integration.

Posted by: Maarten Bergvelt on December 17, 2008 2:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

I just want to de-lurk long enough to say that I love this blog! I am just an amateur in mathematics and astronomy but I always get something from the dialog no matter how abstract the discussion. The recent (week 273) post about the formation of the moon was marvelous . It just really captures my imagination! Thank you to all the scientists/mathematicians who contribute to the high quality here! I also want to extend my sincere condolences to Dr. Baez and his family.

Posted by: Russ Van Rooy on December 16, 2008 3:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Thanks! I hope you delurk a bit more often… maybe you could ask questions about astronomy. I’m just an amateur in that subject myself, but I enjoy reading about it, and questions are a great excuse to look stuff up. The professional mathematicians here are either unconcerned with petty things like stars, planets and galaxies, or perhaps embarrassed to ask the questions they have.

Posted by: John Baez on December 17, 2008 3:56 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

I’d love to see you try your hand at an explanation of the Hanbury-Brown and Twiss effect!

Posted by: Mike Stay on December 17, 2008 7:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

I haven’t thought about this effect for a long time. I see the Wikipedia article pushes the classical explanation as being more intuitive than the quantum one; that’s a nice slant, but I’ve only even thought about the quantum explanation, so I find that more intuitive than the classical one! The quantum explanation also handles fermions, like electrons.

At the bottom of this all is the simple fact: identical bosons like to ‘bunch’, identical fermions like to ‘antibunch’.

Maybe you know how that works, but I’ll just say it again. There are really three cases: boltzons, bosons, and fermions. “Boltzons” — one of Jim Dolan’s charming terms — satisfy Maxwell–Boltzmann statistics, rather than Bose–Einstein or Fermi–Dirac. Boltzons are actually the most intuitive of the three cases, since this case happens a lot in everyday life.

Suppose you have two identical balls and two urns, and you randomly put each ball in an urn, with 50-50 odds. What happens? More precisely: what’s the maximum entropy state of two balls in two urns?

Someone guess the answer in in each of the three cases: boltzons, bosons and fermions! Boltzons are the ‘intuitive’ case.

Posted by: John Baez on December 20, 2008 10:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Fermions must end up with one ball in each urn.

Bosons may have one ball in each urn, two balls in one urn, or two balls in the other urn, and there’s a 1/3 chance of each.

Boltzons have the same possibilities as for bosons, but there’s a 1/2 chance for one ball in each urn, and only a 1/4 chance for each of the two-balls-one-urn possibilities.

Posted by: John Armstrong on December 20, 2008 10:56 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Right! So, here’s the moral about ‘bunching’:

Boltzons: both balls wind up in the same urn with probability 1/2.

Bosons: both balls wind up in the same urn with probability 2/3.

Fermions: both balls wind up in the same urn with probability 0.

As compared with boltzons, bunching is enhanced for bosons, strongly discouraged for fermions.

Posted by: John Baez on December 21, 2008 12:39 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

I was much taken with Kevin Buzzard’s letter at the end of TWF 273 on the relations between Pontryagin duality and number theory, specifically how they enter into Tate’s thesis. After thinking a little, I feel like I had a small epiphany about zeta-like functions, but before getting into any of that, I had a couple of simple Fourier analysis questions which I’m sure someone reading this could answer.

(1) The Pontryagin self-duality of \mathbb{R} given by the pairing

×quotient(/)×(/)+/exp(2πi)S 1\mathbb{R} \times \mathbb{R} \stackrel{quotient}{\to} (\mathbb{R}/\mathbb{Z}) \times (\mathbb{R}/\mathbb{Z}) \stackrel{+}{\to} \mathbb{R}/\mathbb{Z} \stackrel{\exp(2\pi i-)}{\to} S^1

underlies the L 2L^2-duality given by Fourier transform:

f^(y)= e 2πixyf(x)dx\hat{f}(y) = \int_{\mathbb{R}} e^{-2\pi i x y} f(x) d x

and it is well-known that the function f(x)=e πx 2f(x) = e^{-\pi x^2} is an eigenfunction of the Fourier transform, with eigenvalue 1. Is the 1-eigenspace one-dimensional?

(2) That was the archimedean case; now for the non-archimedean case. Let p\mathbb{Q}_p denote the pp-adic completion of \mathbb{Q}. Again there is Pontryagin self-duality of p\mathbb{Q}_p, this time given by a pairing

p× pquotient( p/ p)×( p/ p)(/)×(/)+/exp(2πi)S 1\mathbb{Q}_p \times \mathbb{Q}_p \stackrel{quotient}{\to} (\mathbb{Q}_p/\mathbb{Z}_p) \times (\mathbb{Q}_p/\mathbb{Z}_p) \cong (\mathbb{Q}/\mathbb{Z}) \times (\mathbb{Q}/\mathbb{Z}) \stackrel{+}{\to} \mathbb{Q}/\mathbb{Z} \stackrel{\exp(2\pi i -)}{\to} S^1

and again this underlies an L 2L^2-duality over p\mathbb{Q}_p given by Fourier transform

f^(y)= pe 2πix py pf(x)dx\hat{f}(y) = \int_{\mathbb{Q}_p} e^{-2\pi i x_p y_p} f(x) d x

(where x px_p denotes the pp-adic valuation) and now the characteristic function for the subset of pp-adic integers p\mathbb{Z}_p, taking the value 1 on this subset and the value 0 outside this subset, is [under a suitable normalization of the Haar measure] an eigenfunction for the Fourier transform with eigenvalue 1. Same question: is the 1-eigenspace one-dimensional?

(3) Same question as (2), but for local completions at primes in general number fields.

Posted by: Todd Trimble on December 22, 2008 1:41 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Bah. Dumb mistake: I wrote

p/ p/\mathbb{Q}_p/\mathbb{Z}_p \cong \mathbb{Q}/\mathbb{Z}

which is of course false; the right side should be [1p]/\mathbb{Z}[\frac1{p}]/\mathbb{Z}. Corrections should be inserted accordingly.

Posted by: Todd Trimble on December 22, 2008 1:51 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Bad form to reply to myself, but I see that I was way off in asking the question (1) above. I’ve since learned that the eigenspace for the eigenvalue 1 is huge (and in fact there are only four eigenvalues of the Fourier transform over \mathbb{R}, given by the four fourth roots of unity).

Question (2) is still open, but I now have little faith that that eigenspace is one-dimensional, either.

Posted by: Todd Trimble on December 22, 2008 2:39 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

No fair — you posted your reply while I was writing up mine, below.

Posted by: John Baez on December 22, 2008 3:03 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Todd wrote:

It is well-known that the function f(x)=e πx 2f(x)= e^{-\pi x^2} is an eigenfunction of the Fourier transform, with eigenvalue 1. Is the 1-eigenspace one-dimensional?

Ooh, ooh — I know! I know!

No, this space is infinite-dimensional.

You’re using different normalizations than the physicists — they use this Fourier transform:

f^(k)=12πe ikxf(x)dx \hat{f}(k) = \frac{1}{\sqrt{2 \pi}} \, \int e^{-i k x} f(x) d x

and of course being physicists they use the variable kk to denote ‘frequency’. This Fourier transform is still unitary, and in these conventions your Gaussian becomes

ψ 0(x)=e x 2/2 \psi_0(x) = e^{-x^2/2}

with

ψ^ 0=ψ 0 \hat{\psi}_0 = \psi_0

We get all the other eigenfunctions from this one by repeatedly applying the creation operator:

a *=qip2a^* = \frac{q - i p}{\sqrt{2}}

Here qq is the position operator, just multiplication by xx:

(qψ)(x)=xψ(x) (q\psi)(x) = x \psi(x)

and pp is the momentum operator, just differentiation tweaked a bit to make it self-adjoint:

(pψ)(x)=idψdx(x) (p\psi)(x) = -i \frac{d \psi}{d x}(x)

If we write the Fourier transform as FF:

Fψ=ψ^ F \psi = \hat{\psi}

then a fun calculation reveals:

Fp=qF F p = q F

Fq=pF F q = -p F

and thus

Fa *=ia *F F a^* = - i a^* F

As a result, the creation operator maps eigenfunctions of the Fourier transform to eigenfunctions!

In particular, if we let

ψ n=(a *) nψ 0 \psi_n = (a^*)^n \psi_0

then the functions ψ n\psi_n are all eigenfunctions of the Fourier transform:

Fψ n=(i) nψ n F \psi_n = (-i)^n \psi_n

Even better, with a little work one can show these functions form an orthogonal basis of L 2()L^2(\mathbb{R}).

So, the Fourier transform has only four eigenspaces, with eigenvalues 1,i,1,i1, i, -1, -i. The 1-eigenspace is spanned by these:

ψ 0,ψ 4,ψ 8,\psi_0, \psi_4, \psi_8, \dots

There’s probably some much simpler characterization of the 1-eigenspace, but I’m forgetting it. The functions ψ 2n\psi_{2n} are a basis for the even functions in L 2()L^2(\mathbb{R}), while the functions ψ 2n+1\psi_{2n + 1} are a basis for the odd functions. The functions ψ 4n\psi_{4n} are a basis of some space that’s ‘half as big’ as the even functions. So I need to dream up some additional condition, similar to ‘even’, so I can say that the 1-eigenspace of the Fourier transform consists of ‘functions that are even and ????’.

If you want to see details of any of the calculations I skipped, try week4 and week5 of my notes on Quantization and Categorification. This sort of math is the bread and butter of quantum mechanics; Jim and I categorified some of it. I would love to see pp-adic analogues of all this stuff. Maybe we could categorify that too!

Now I’ll read the rest of your comment.

Posted by: John Baez on December 22, 2008 2:59 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Yeah, basic quantum mechanics. I’m pretty embarrassed by the questions now.

For the pp-adics, I think we can again say that the fourth iteration of the Fourier transform brings us back to the identity, so I think again there are only four eigenvalues given by the fourth roots of unity, and it probably works out pretty much the same way as for the reals. I don’t know the pp-adic analogues of the ψ n\psi_n you wrote down, but I think it ought to be a simple calculation (but it could still be fun!).

The small epiphany I mentioned at the start of all this was seeing that these “Gaussians” for the reals and pp-adics were closely connected with the Euler factors for the zeta function. You often see the functional equation for the zeta function written in the form

Z(s)=Z(1s)Z(s) = Z(1-s)

where

Z(s)=π s/2Γ(s/2)ζ(s)Z(s) = \pi^{-s/2}\Gamma(s/2)\zeta(s)

Let’s write, according to the Euler product identity,

Z(s)=π s/2Γ(s/2) p(1p s) 1Z(s) = \pi^{-s/2}\Gamma(s/2)\prod_p (1 - p^{-s})^{-1}

About a year ago I read that these factors (1p s) 1(1 - p^{-s})^{-1} were called “Euler factors” at the places pp, and that that funny fudge factor

π s/2Γ(s/2)\pi^{-s/2}\Gamma(s/2)

should be regarded as the Euler factor at the archimedean place. That’s sort of made me go, “huh?” – didn’t see where that was coming from. But it occurred to me [this is the “epiphany”] that the fudge factor is actually

e πx 2|x| sdx\int_{\mathbb{R}} e^{-\pi x^2} |x|^s d x

and that the other Euler factors arose also in this way, by multiplying the analogous “Gaussian” at the non-archimedean place by a multiplicative homomorphism and integrating. So that gave me the happy feeling that I have a little toehold on getting at the functional equation (not just for the classical zeta, but fancier things like LL-functions attached to characters) by means of Fourier analysis on local completions. And I was hoping that these “Gaussians” had particularly pleasant characterizations which would make everything feel very forced and canonical (and that’s what motivated the stupid questions in the first place). Hm… could we see the pp-adic Gaussian [that is, the characteristic function of the pp-adic integers] as the “vacuum state” in some sense??

The deeper mathematics underlying the small epiphany I had is without question very well understood, but it’s always pleasant to notice such little factoids on one’s own.

Posted by: Todd Trimble on December 22, 2008 3:41 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Sorry, I meant

π s/2Γ(s/2)= e πx 2|x| sdxx\pi^{-s/2}\Gamma(s/2) = \int_{\mathbb{R}} e^{-\pi x^2} |x|^s \frac{d x}{x}

and even then I don’t guarantee that an arithmetic mistake hasn’t crept in. :-P

Posted by: Todd Trimble on December 22, 2008 3:48 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

John said:

‘functions that are even and ????’

Erm, even and zero? According to my sums qψ 0=xψ 0q\psi_0=x\psi_0 and pψ 0=ixψ 0p\psi_0=i x\psi_0 so (q+ip)ψ 0=0(q+i p)\psi_0 =0.

Posted by: Simon Willerton on December 22, 2008 3:33 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

Sorry, I wrote

a *=q+ip2a^* = \frac{q + i p}{\sqrt{2}}

but I meant

a *=qip2a^* = \frac{q - i p}{\sqrt{2}}

I’ll take the liberty of retroactively correcting this typo.

As you probably know, ψ n=(a *) nψ 0\psi_n = (a^*)^n \psi_0 is supposed to be a degree-nn polynomial times the Gaussian ψ 0\psi_0. By a friend of Weierstrass’s theorem, polynomials times a Gaussian are dense in L 2L^2, so the ψ n\psi_n span L 2L^2.

Posted by: John Baez on December 22, 2008 7:45 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

So, Todd: it sounds like you’re saying the ‘improved zeta function’

Z(s)=π s/2Γ(s/2)ζ(s)Z(s) = \pi^{-s/2}\Gamma(s/2)\zeta(s)

is a product over completions of \mathbb{Q} of factors that are roughly Mellin transforms of Gaussians — or pp-adic analogues thereof.

That sounds cool! But do you — or anyone reading this — know the conceptual importance of taking a Mellin transform of a Gaussian?

Up to annoying fudge factors, the Mellin transform is basically a Fourier transform not for the locally compact abelian group (,+)(\mathbb{R}, +) but its multiplicative version ( +,×)(\mathbb{R}^+, \times).

(It’s actually more like a Laplace transform, but I count that among the ‘annoying fudge factors’.)

On the other hand, the Gaussian plays a fundamental role in the Fourier theory of (,+)(\mathbb{R}, +). You were overoptimistic in guessing that it was the only 1-eigenvector of the Fourier transform, but something very similar is true. L 2( n)L^2(\mathbb{R}^n) comes with a god-given projective representation of the symplectic group Sp(n)Sp(n), which contains the unitary group U(n)\mathrm{U}(n), and the Gaussian is, up to a constant, the unique vector that’s fixed up to phase by all of U(n)\mathrm{U}(n). This is a mathematical way of saying it’s the ‘ground state’ of the nn-variable harmonic oscillator.

I’m sure someone understands all this stuff and how it’s related to number theory. For example, I know André Weil was interested in the projective representation of the symplectic group on L 2( n)L^2(\mathbb{R}^n) — the so-called ‘metaplectic representation’. So, now I’ll wildly guess that he was also interested in the pp-adic and adelic analogues of this thing.

But, I’d like to hear a simple story about what’s going on.

Of course, what I’m calling Sp(1)Sp(1) is none other than SL(2)SL(2), a group greatly beloved by number theorists. This has got to be part of the story.

Posted by: John Baez on December 22, 2008 8:12 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

In physics, what’s special about that Gaussian I called ψ 0\psi_0 is not that it’s the only eigenvector of the Fourier transform with eigenvalue 1 (there are tons), but that it’s the ‘ground state of the harmonic oscillator Hamiltonian’. In other words, it’s the eigenvector of

H=p 2+q 22=aa *+a *a2H = \frac{p^2 + q^2}{2} = \frac{a a^* + a^* a}{2}

with the lowest possible eigenvalue. Repeatedly hitting ψ 0\psi_0 with the creation operator, we get all the other eigenvectors.

Posted by: John Baez on December 22, 2008 3:10 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 273)

I’ve attached a comment by Michael Barr on Pontryagin duality and *-autonomous categories to the end of the Addenda to week273.

Posted by: John Baez on December 31, 2008 7:45 PM | Permalink | Reply to this

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