December 24, 2007

This Week's Finds in Mathematical Physics (Week 260)

John Baez

Since it's Christmas Eve, I thought I'd list some free books you can download. I'm a big fan of giving the world presents... and I'm not the only one.

But first, this week's nebulae! Here's one called the Retina:

1) Retina Nebula, Hubble Heritage Project,

This is actually a tube of ionized gas about a quarter of a light-year across and one light-year long. It's a planetary nebula produced by a dying star. If you zoom in and look closely, you can see this star lurking in the middle, now a mere white dwarf.

The blue light is the most energetic, so it's really hot where you see blue. This blue light comes from singly ionized helium - helium where one electron has been knocked off. The green light is a bit less energetic: that's from doubly ionized oxygen. The red light comes from even cooler regions: that's from singly ionized nitrogen.

You can also see a lot of "dust lanes" in this photo. They're beautiful. And they're big! The width of each one is about 160 times the distance between the Sun and the Earth. The gas and dust in these lanes is about 1000 times higher than elsewhere. But what creates them?

Apparently, when the fast-moving glowing hot gas from the star crashes into the invisible gas in the surrounding interstellar space, the boundary gets sort of crumpled, and these dust lanes form. It's vaguely similar to the puffy surface of a cumulus cloud. But here the mechanism is different, because it involves a "shock wave": the hot gas is moving faster than the speed of sound as it hits the cold gas!

This effect is called a "Vishniac instability", since in 1983, the astrophysicist Ethan Vishniac showed that a shock wave moving in a sufficiently compressible medium would be subject to an instability of this sort, growing as the square root of time. I've never seen how Vishniac's calculations work, so the mathematics underlying this beautiful phenomenon will have to wait for another day.

Note that this planetary nebula, like the others I've shown you, is far from spherically symmetric. Astrophysicists used to pretend stars were spherically symmetric. But, that's a bad approximation whenever anything really exciting happens... just like in the old joke where the punchline is "consider a spherical cow".

As I said, the Retina Nebula is actually shaped like a tube. Viewed from either end, this tube would look very different - probably like the Ring Nebula:

2) Ring Nebula, Hubble Heritage Project,

This is one light-year across. Again we see He II blue light with a wavelength of 4686 angstroms, then O III green light at 5007 angstroms, then N II red light at 6584 angstroms. You can also see the white dwarf as a tiny dot in the center; it's about 100,000 kelvin in temperature.

(In case you're wondering, an "angstrom" is an obsolete but popular unit of distance, equal to 10-10 meters. Just like the "parsec", it's a sign that astronomy is an old science. Anders Jonas Ångström was one of the founders of spectroscopy, back around 1860. Archaic conventions may also explain why singly ionized helium is called "He II", and so on. Maybe the number zero hadn't fully caught on.)

Next: free books!

At least around here, Christmas seems to be all about buying stuff and giving it away. Giving is good. But I think gifts have more soul if you make them yourself. This is one of the great things about the internet: it lets us create things and give them to everyone in the world - or more precisely: everybody who wants them, and nobody who doesn't.

In this spirit, here's a roundup of free books on math and physics: gifts from their authors to you. There are lots out there. I'll only list a few. For more, try these sites:

3) George Cain, Online Mathematics Textbooks,

4) Free Online Mathematics Books,

5) Alex Stefanov, Textbooks in Mathematics, or (with annoying ads, but more permanent)

Despite its title, Stefanov's excellent site includes a lot of books on physics. I can't find lists specifically devoted to free physics books, but there are a lot out there - including a lot on the arXiv.

Anyway, let's dive in!

What if you're dying to learn physics, but don't know where to start? Start here:

6) Physics Books Online,

You'll find plenty of free online books, starting from the basics and working up to advanced topics. But to dig deeper into these mysteries, you'll eventually need to learn a bunch of math. Do you remember what Victor Weisskopf said when a student asked how much math a physicist needs to know?


This can be scary when you're just getting started. What if you don't know calculus, for example?

Simple: learn calculus! This book is a classic - and it's free:

7) Gilbert Strang, Calculus, Wellesley-Cambridge Press, Cambridge, 1991. Also available at

It really explains things clearly. I may use it the next time I teach calculus. We professors need to quit making our students buy expensive textbooks, and switch to free online books! We could join forces and make wiki textbooks that are a lot better and more flexible than the budget-busting, back-breaking mammoths we currently inflict on our kids. But there are already a lot of good texts available free online.

Or: what if you know calculus, but you're still swimming through the undergraduate sea of differential equations, Fourier transforms, matrices, vectors and tensors? Then this should be really helpful:

8) James Nearing, Mathematical Tools for Physics, available at

Unlike the usual dry and formal textbook, it reads like a friendly uncle explaining things in plain English, trying to cut through the red tape and tell you how to actually think about this stuff.

For example, on page 3 he introduces the hyperbolic trig functions:

Where do hyperbolic functions come from? If you have a mass in equilibrium, the total force on it is zero. If it's in stable equilibrium then if you push it a little to one side and release it, the force will push it back to the center. If it is unstable then when it's a bit to one side it will be pushed farther away from the equilibrium point. In the first case, it will oscillate about the equilibrium position and the function of time will be a circular trigonometric function - the common sines or cosines of time, Acos(ωt). If the point is unstable, the motion will be described by hyperbolic functions of time, sinh(ωt) instead of sin(ωt). An ordinary ruler held at one end will swing back and forth, but if you try to balance it at the other end it will fall over. That's the difference between cos and cosh.
He goes into more detail later, after introducing the complex numbers. This book also features some great animations of Taylor series and Fourier series, like this movie of the Taylor series of the sine function:

There are free online books at all levels... so let's soar a bit higher. How about if you're a more advanced student trying to learn general relativity? Here you go:

9) Sean M. Carroll, Lecture Notes on General Relativity, available as arXiv:gr-qc/9712019

How about quantum field theory? Then you're in luck - there are two detailed books available online:

10) Warren Siegel, Fields, available as arXiv:hep-th/9912205

10) Mark Srednicki, Quantum Field Theory, Cambridge U. Press, Cambridge, 2007. Also available at

Or what about algebraic topology? Again you're in luck, since you can read both Allen Hatcher's gentle introduction and Peter May's high-powered "concise course":

11) Allen Hatcher, Algebraic Topology, Cambridge U. Press, Cambridge, 2002. Also available at

12) Peter May, A Concise Course in Algebraic Topology, U. of Chicago Press, Chicago, 1999. Also available at

May has a lot of more advanced topology books available at his website, too - like this classic, where he used operads to solve important problems involving loop spaces:

13) Peter May, The Geometry of Iterated Loop Spaces, Lecture Notes in Mathematics 271, Springer, Berlin, 1972. Also available at

Or say you want to learn about vector bundles and how they show up in physics, from the basics all the way to fancy stuff like D-branes and K-theory? Try this - it's a great sequel to Husemoller's classic intro to fiber bundles:

14) Dale Husemoller, Michael Joachim, Branislav Jurco and Martin Schottenloher, Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics 726, Springer, Berlin, 2008. Also available at

The list goes on and on! The American Mathematical Society will give you books for free if you prove that you're not a robot by solving a little puzzle:

15) American Mathematical Society, Books Online By Subject,

Apparently they don't want robots learning advanced math and putting us professors out of business by teaching with more charisma and flair. (By the way: make sure to let them put cookies on your web browser, or they'll send you an endless succession of these puzzles, without explaining why!)

Since James Dolan and I plan to explain symmetric groups and their Hecke algebras in our online seminar, this particular book from the AMS caught my eye:

16) David M. Goldschmidt, Group Characters, Symmetric Functions, and the Hecke Algebra, AMS, Providence, Rhode Island, 1993. Also available as

Since we're also struggling to understand the Langlands program, this looks good too:

17) Armand Borel, Automorphic Forms, Representations, and L-functions, AMS, 2 volumes, Providence, Rhode Island, 1979. Also available at and

It's a serious collection of expository papers by bigshots like Borel, Cartier, Deligne, Jacquet, Knapp, Langlands, Lusztig, Tate, Tits, Zuckerman, and many more.

"Motives" are the mysterious virtual building blocks that algebraic varieties are built from. If you're ready to learn about motives - I'm not sure I am - try this:

18) Marc Levine, Mixed Motives, AMS, Providence, Rhode Island, 1998. Also available at

Or, if you're interested in using category theory to make analysis clearer and more beautiful, try this:

19) Andreas Kriegl and Peter W. Michor, The Convenient Setting of Global Analysis, AMS, Providence, Rhode Island, 1997. Also available at

The focus is on getting and working with a "convenient category" of infinite-dimensional manifolds. The idea of a "convenient category" goes back to topology: at some point, people realized they wanted this property to hold:

C(X × Y, Z) ≅ C(X, C(Y, Z))

Here C(X,Y) is the space of maps from X to Y. So, the isomorphism above says that a map from X × Y to Z should correspond to a map from X to C(Y,Z). A category with this property is called "cartesian closed". While it may not be obvious why, this property is so wonderful that people threw out the category of topological spaces and continuous maps and replaced it with a slightly different one, just to get this to hold.

Another sort of "convenient category" for differential geometry uses infinitesimals. Again, you can learn about this in a free book:

20) Anders Kock, Synthetic Differential Geometry, Cambridge U. Press, Cambridge, 2006. Also available at

This category is not just cartesian closed - it's a topos!

If you don't know what a topos is, never fear - more free books are coming to your rescue:

21) Robert Goldblatt, Topoi, the Categorial Analysis of Logic, Dover, 1983. Also available at

22) Michael Barr and Charles Wells, Toposes, Triples and Theories, Springer, Berlin, 1983. Also available at

The first one is so gentle it makes a good introduction to category theory as a whole. The second scared the bejeezus out of me for a decade, but now I like it.

I like Jordan algebras, so I was also pleased to see this classic offered for free at the AMS website:

23) Nathan Jacobson, Structure and Representations of Jordan Algebras, AMS, Providence, Rhode Island, 1968. Also available at

Fans of exceptional Lie algebras will like the last two chapters, on "connections with Lie algebras" and "exceptional Jordan algebras".

Speaking of Lie algebras, I'd never seen this textbook before:

24) Shlomo Sternberg, Lie Algebras,

It's a somewhat quirky introduction, not for beginners I think, but it features some nice special topics: character formulas, the Kostant Dirac operator, and a detailed study of the center of the universal enveloping algebra.

This intro to Lie groups is also a bit quirky, but if you like Feynman diagrams or spin networks, it's irreplaceable:

25) Predrag Cvitanovic, Birdtracks, Lie's, and Exceptional Groups, available at

One of the great things about this book is that it classifies simple Lie groups according to their "skein relations" - properties of their representations, written out diagrammatically. In so doing, Cvitanovic realized that there's a "magic triangle" containing all the exceptional Lie groups. This subsumes the "magic square" of Freudenthal and Tits, which I discussed in "week145" and my octonion webpages.

This idea of Cvitanovic is closely related to the "exceptional series" of Lie groups - a pattern whose existence was conjectured by Deligne. I love the term "exceptional series". It's an oxymoron, since the exceptional groups were defined as those that don't fit into any series. But, it makes sense!

To see the exceptional series, it helps to do a mental backflip called "Tannaka-Krein duality", where you focus on the category of representations of the Lie group, instead of the group itself. Then, draw the morphisms in that category as diagrams, like Feynman diagrams! Then see what identities they satisfy. New patterns leap out: new series unify what had been "exceptions".

Very briefly, the idea goes like this. Suppose we have a Lie group G with Lie algebra L. The Lie bracket takes two elements x and y and spits out one element [x,y], and it's linear in each variable, so it gives a linear operator

L ⊗ L &rarr L

which is actually a morphism in the category of representations of G.

So, following the philosophy of Feynman diagrams, we can draw the bracket operation like this:

                \     /
                 \   /
                  \ /
We can even use this to state the definition of a Lie algebra using diagrams! To say the bracket is antisymmetric:

[y,x] = -[x,y]

we just draw this:

                 \   /              |       |
                  \ /               |       |
                   /                |       |
                  / \               |       |
                 /   \               \     /
                 \   /     =    -     \   /
                  \ /                  \ /
                   |                    |
                   |                    |
                   |                    |
To say the Jacobi identity:

[x,[y,z]] = [[x,y],z] + [y,[x,z]]

we just draw this:

  \     \     /          \     /     /            \    /      /   
   \     \   /            \   /     /              \  /      /
    \     \ /              \ /     /                \       / 
     \     /                \     /                  \     /
      \   /        =         \   /         +       /  \   /
       \ /                    \ /                 /    \ /
        |                      |                  \     /
        |                      |                   \   /
        |                      |                    \ /
        |                      |                     |
If that's too cryptic, maybe this will explain what I'm doing:
 x     y       z        x       y     z          x      y      z
  \     \     /          \     /     /            \    /      /   
   \     \   /            \   /     /              \  /      /
    \     \ /              \ /     /                \       / 
     \     /                \     /                  \     /
      \   /        =         \   /         +       /  \   /
       \ /                    \ /                 /    \ /
        |                      |                  \     /
        |                      |                   \   /
        |                      |                    \ /
        |                      |                     |
   [x,[y,z]]              [[x,y],z]              [y,[x,z]]
But in fact, people usually massage this picture to make it even more cryptic, and call it the "IHX" identity - since the three terms look like the letters I, H, and X by the time they're done twisting them around. For a good explanation, with pretty pictures, see:

26) Greg Muller, Chord diagrams and Lie algebras,

It then turns out that the exceptional Lie algebras F4, E6, E7 and E8 satisfy yet another identity:

 \        /                
  \      /                
   |    |                       
   |    |           = 
  /      \          
 /        \                
  \        /           \   /
   \      /             \ /   
    \    /               |   
A    ----     +     A    |       +
    /    \               |  
   /      \             / \  
  /        \           /   \  

  \     /       \        /         \        /
   \   /         \      /           \      /
    \ /           \    /             \____/ 
B    /     +  B    |  |       +  B    ____    
    / \           /    \             /    \
   /   \         /      \           /      \
  /     \       /        \         /        \
for various choices of the constants A and B. So, they fit into a "series"!

I believe the main point of this identity, going back to Vogel's paper "Algebraic structures on modules of diagrams", is that for these Lie algebras, the square of the quadratic Casimir is the only degree-4 Casimir.

I think there's a lot more to be discovered here, in part by taking the gnarly computations people have done so far and making them more beautiful and conceptual. So, I urge all fans of exceptional mathematics, diagrams, and categories to look at these:

27) Pierre Deligne, La serie exceptionnelle des groupes de Lie, C. R. Acad. Sci. Paris Ser. I Math 322 (1996), 321-326.

Pierre Deligne and R. de Man, The exceptional series of Lie groups II, C. R. Acad. Sci. Paris Ser. I Math 323 (1996), 577-582.

Pierre Deligne and Benedict Gross, On the exceptional series, and its descendants, C. R. Acad. Sci. Paris Ser. I Math 335 (2002), 877-881. Also available as

28) Pierre Vogel, Algebraic structures on modules of diagrams, 1995. Available at or

The universal Lie algebra, 1999. Available at

Vassiliev theory and the universal Lie algebra, 2000. Available at

For a good overview, try this:

28) J. M. Landsberg and L. Manivel, Representation theory and projective geometry, 2002. Available at arXiv:math/0203260.

Alas, they avoid drawing Feynman diagrams, though they talk about them in section 4. They prefer to use ideas from algebraic geometry:

29) J. M. Landsberg and L. Manivel, The projective geometry of Freudenthal's magic square, J. Algebra 239 (2001), 477-512. Also available as arXiv:math/9908039.

J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math. 171 (2002), 59-85. Also available as arXiv:math/0107032.

J. M. Landsberg and L. Manivel, Series of Lie groups, available as arXiv:math/0203241.

Bruce Westbury, whom longtime readers of This Week's Finds will remember as John Barrett's collaborator, has also worked on this subject. He has pointed out that both the magic square and the magic triangle can be given an extra row and column if we introduce a 6-dimensional algebra halfway between the quaternions and the octonions:

30) Bruce Westbury, Sextonions and the magic square, available as arXiv:math/0411428.

For even more references, try this:

31) Bruce Westbury, References on series of Lie groups,

This stuff has been on my mind recently, since I've been working on exceptional groups and grand unified theories with my student John Huerta. Also, my friend Tevian Dray has a student who just finished a thesis on a related topic:

32) Aaron Wangberg, The structure of E6, available as arXiv:0711.3447.

In a nutshell: E6 is secretly SL(3,O). Octonions rock!

Happy holidays. Keep learning cool stuff.

Addenda: Thomas Riepe listed some more free online math books. Tony Smith pointed out something I already knew, but didn't make clear above: the idea that E6 is secretly SL(3,O) is far from new.

Thomas wrote:

Some more links:

Milne's great collection (incl. the famous LNM 900), leading the reader from basic algebra through algebraic number theory, class fields, modular forms, arithmetic groups,... up to etale cohomology, Shimura varieties etc.

Friedhelm Waldhausen's lectures on algebraic topology and K-theory.

DML: Digital Mathematics Library

G. Harder's math-links

MSRI online books


"Nearly three and a half centuries of scientific study and achievement is now available online in the Royal Society Journals Digital Archive. This is the longest-running and arguably most influential journal archive in Science, including all the back articles of both Philosophical Transactions and Proceedings."

Tony Smith wrote:

Thanks for an interesting list of stuff in week 260, but I have some questions about this:
32) Aaron Wangberg, The structure of E6, available as arXiv:0711.3447.

In a nutshell: E6 is secretly SL(3,O). Octonions rock!

Not only from your brief list descrption, but also from reading the paper at pages 96 ff I get the impression that Wangberg is claiming the result E6 = SL(3,O). Do you get the same impression? I hope not, and I hope that my impression is somehow mistaken, because the result E6 = SL(3,O) is (and has been for some time) well known and in the literature. For example, in hep-th/9309030 Martin Cederwall and Christian R. Preitschopf said:
... It should be possible to realize E6 = SL(3;O) [18,24] on them in a "spinor-like" manner, much like SO(10) = SL(2;O) acts on its 16-dimensional spinor representations that play the role of homogeneous coordinates for OP1 ...

18. H. Freudenthal, Adv. Math. 1 (1964) 145.
24. A. Sudbery, J. Phys. A17 (1984) 939. ....

Although that Freudenthal Adv. Math. is listed as a reference in Wangberg's paper (as reference 5), I did not see the Sudbery paper listed, and I did not see the Freudenthal reference on page 96.

Please don't misunderstand this message. I think that Wangberg's thesis is very interesting. I am just trying to get a correct historical record.


PS - In Sudbery's 1984 paper, he not only says (at page 950) "... sl(3,K) ... When K = O, this Lie algebra is a n on-compact form of the exceptional Lie algebra E6, the maximal compact subalgebra being F4 ..." but he goes on to say "... sp(6,K) ... when K = O it is a non-compact form of E7, the maximal compact subalgebra being E6 ⊕ so(2). ...".

For more discussion, go to the n-Category Café.

If nature has made any one thing less susceptible than all others of exclusive property, it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of every one, and the receiver cannot dispossess himself of it. Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it. - Thomas Jefferson

© 2007 John Baez