Also available at http://math.ucr.edu/home/baez/week271.html
October 26, 2008
This Week's Finds in Mathematical Physics (Week 271)
John Baez
This week I'll talk about quasicrystals and how they arise from
the interplay between crystallographic and noncrystallographic
Coxeter groups. I'll describe Jeffrey Morton's new paper on
groupoids and 2-vector spaces, and Stephen Summers' review of
new work on constructive quantum field theory. But first - more
pictures of Jupiter's moon Io!
Here's a great photo of volcanic activity on Io - the "Masubi plume":
1) NASA Photojournal, Masubi plume on Io,
http://photojournal.jpl.nasa.gov/catalog/PIA02502
You can see hot gas and dust shooting 100 kilometers up into the
atmosphere!
Here's another:
2) Solarviews, Pele volcano and Pillan Patera,
http://www.solarviews.com/eng/iopele.htm
In front we see a volcanic feature called Pillan Patera. Over the
horizon we see an enormous eruption 300 kilometers high coming from
the most intense persistent hot spot on Io: Pele. This seems to be
an active lava lake inside a volcanic depression, or "patera", about
20 x 30 kilometers in size.
But Pillan Patera is no slouch either when it comes to eruptions.
Look at these "before and after" pictures taken 5 months apart in 1997:
3) NASA Photojournal, Arizona-sized Io eruption,
http://photojournal.jpl.nasa.gov/catalog/PIA00744
The big red ring is sulfur spewed out by Pele. But the exciting new
feature in the "after" picture is the dark blotch centered at Pillan
Patera. It's 400 kilometers in diameter, roughly the size of Arizona.
It consists of about 50 cubic kilometers of lava laid down by a big
eruption. At the peak of the activity, 10,000 cubic meters of lava
were spewing out each second. This was the largest volcanic eruption
ever seen, anywhere!
For more, try these:
4) A.G. Davies et al, Thermal signature, eruption style and eruption
evolution at Pele and Pillan on Io, Jour. Geophys. Research 106
(2001), 33,079-33,103. Also available at
http://europa.la.asu.edu/pgg/associates/members/williams/gw/pdf/2001Daviesetal.pdf
5) Jani Radebaugh et al, Observations and temperatures of Io's
Pele Patera from Cassini and Galileo spacecraft images,
Icarus 169 (2004), 65-79.
In case you're wondering about the red sulfur around Pele versus
the yellow sulfur you saw last week, let me say a bit about that.
Sulfur comes in an incredible number of forms, or allotropes:
6) Wikipedia, Allotropes of sulfur,
http://en.wikipedia.org/wiki/Allotropes_of_sulfur
It can form different molecules consisting of 2 to 20 atoms. The most
common form on Earth is alpha-sulfur: rhombic crystals made of
ring-shaped molecules containing 8 atoms each. Alpha-sulfur is lemon
yellow, but above 95 degrees Celsius it gradually turns into paler
yellow beta-sulfur: the ring-shaped molecules reorganize to form
crystals with less symmetry - monoclinic crystals, to be precise.
Sulfur melts around 115 Celsius. But when you heat it above 160
Celsius, something weird happens: contrary to the usual pattern for
liquids, it gets more viscous as it gets hotter! The reason: the
atoms start forming long chain polymers, called "catena sulfur".
As these predominate, the stuff gets darker in color: first orange,
then red, then dark red, and finally almost black. Blecch! If you
then cool it suddenly, it can form a red amorphous solid. And that,
presumably, is what we see in the ring around Pele.
Now, from crystals to quasicrystals...
Last week I asked if quasicrystals with approximate 5-fold symmetry
could be obtained by slicing lattices in higher dimensions. Greg
Egan answered - yes! He even has a beautiful Java applet that
demonstrates it:
7) Greg Egan, deBruijn,
http://www.gregegan.net/APPLETS/12/12.html
It shows some nice quasiperiodic tilings of the plane with approximate
n-fold symmetry, made by cleverly slicing a cubical lattice in
n-dimensional space. The idea comes from this paper:
8) N. G. deBruijn, Algebraic theory of Penrose's nonperiodic tilings
of the plane, I, II, Nederl. Akad. Wetensch. Indag. Math. 43 (1981),
39-52, 53-66.
When n is odd, we can also get deBruijn's tiling by slicing the
A_{n-1} lattice in (n-1)-dimensional space. You're probably most
familiar with the A_3 lattice, which shows up when you stack oranges.
You'll notice this pattern has tetrahedral symmetry. The symmetry
group of the tetrahedron is also called the A_3 Coxeter group.
It's the group of all permutations of 4 things (the corners of the
tetrahedron). This contains the symmetry group of the square, since
that group contains some but not all permutations of the 4 corners of
the square. Indeed, if you view a regular tetrahedron from the
correct angle, it looks like a square!
This pattern goes on for higher n. Last week I spoke about the A_4
lattice, whose symmetry group consists of all permutations of 5 things
- namely the 5 corners of a 4-simplex, which is the 4d analogue of a
tetrahedron. I explained how this group contains the symmetry group
of the pentagon. Indeed, if you view a 4-simplex from the correct angle,
it looks like a pentagon!
So, it's not surprising that we can get a quasiperiodic tiling of the
plane with approximate 5-fold symmetry by taking a 2d slice of the
A_5 lattice and doing a few other tricks.
But this generalizes: the symmetries of an (n-1)-simplex include the
symmetries of a regular n-gon. Just as this Coxeter group, the
symmetry group of the pentagon:
5
o---o
sits inside the A_4 Coxeter group:
o---o---o---o
similarly the symmetries of a hexagon:
6
o---o
sit inside the A_5 Coxeter group:
o---o---o---o---o
and so on: the noncrystallographic Coxeter groups I_2(n)
sit nicely inside the Coxeter groups A_{n+1}. But the really
cool part is how deBruijn uses these to get quasiperiodic tilings
of the plane! You can see details on Greg Egan's page above.
And this idea generalizes to the *other* two noncrystallographic
Coxeter groups. Remember, there are just two more:
H_3, the symmetry group of the dodecahedron, with 120 elements;
H_4, the symmetry group of the 120-cell, with 120 x 120 elements.
We can get 3d quasicrystals with approximate dodecahedral symmetry by
cleverly slicing the 6-dimensional D_6 lattice. This is actually
practical, since there really *are* such quasicrystals in nature. And
we can get 4d quasicrystals with approximate 120-cell symmetry by
cleverly slicing the E_8 lattice! This is just incredibly cool as
pure mathematics:
9) Veit Elser and Neil Sloane, A highly symmetric four-dimensional
quasicrystal, J. Phys. A 20 (1987), 6161-6168.
Also available at http://akpublic.research.att.com/~njas/doc/Elser.ps
10) J. F. Sadoc and R. Mosseri, The E8 lattice and quasicrystals:
geometry, number theory and quasicrystals, J. Phys. A 26 (1993),
1789-1809.
11) Robert V. Moody and J. Patera, Quasicrystals and icosians,
J. Phys. A. 26 (1993), 2829-2853.
Yes, this is the same Moody who helped invent Kac-Moody algebras! For
the last decade or so he's been working on quasicrystals. In "week20"
I explained the "icosians" - a subring of the quaternions built from
the symmetries of a dodecahedron - and how Conway and Sloane used them
to construct the E8 lattice. Moody's article uses the icosians to
study the 4d quasicrystals that we get by slicing the E8 lattice.
While they may seem remote from the real world, these 4d quasicrystals
can be further sliced to give 3d quasicrytals with approximate
dodecahedral symmetry. So in some sense, the quasicrystals we find
in nature are "shadows" of the E_8 lattice... trying their best to
have a symmetry that can only exist in 8 dimensions, but never quite
succeeding.
I love this idea, because it's gotten me over my fear of quasicrystals.
They look unruly and complicated, but now I see that some of them
have close ties to the beautiful, perfectly symmetrical world of
Dynkin diagrams. The "noncrystallographic" Coxeter groups are really
"quasicrystallographic"!
Next let me discuss this paper by Jeffrey Morton:
12) Jeffrey Morton, 2-vector spaces and groupoids, available as
arXiv:0810.2361.
It's an important new twist in the Tale of Groupoidification! As
part of this tale, in "week256" I described a functor from the
category with
finite groupoids as objects,
equivalence classes of spans of finite groupoids as morphisms
to the category with
finite-dimensional vector spaces as objects,
linear operators as morphisms.
I called this "degroupoidification". The idea is that a lot of linear
algebra has an underlying purely combinatorial "skeleton" that doesn't
involve the complex numbers - just symmetry in its purest form.
Groupoidification is quest to strip the fat off linear algebra and
do it using groupoids.
Jeffrey boosts this idea up one notch, getting a 2-functor from the
2-category with:
finite groupoids as objects,
spans of finite groupoids as morphisms,
equivalence classes of spans of spans of finite groupoids as 2-morphisms
to the 2-category with
finite-dimensional 2-vector spaces as objects,
linear functors as morphisms,
natural transformations as 2-morphisms.
Here by "finite-dimensional 2-vector space" I really mean a
"Kapranov-Voevodsky 2-vector space". That's a category equivalent to
Vect^n for some n, where Vect is the category of finite-dimensional
vector spaces. A "linear functor" is one that's linear on each
homset. More concretely, we can describe a linear functor
F: Vect^n -> Vect^m
as an m x n matrix of finite-dimensional vector spaces, just as we
can describe a linear operator
F: C^n -> C^m
as a m x n matrix of complex numbers.
This suggests that Jeffrey is secretly talking about a categorified
version of Heisenberg's "matrix mechanics" - and that's true.
I want to explain that. But I'm getting really sick of saying
"finite" and "finite-dimensional". So, henceforth I'll leave out
those adjectives... but they're really always there. Okay?
Degroupoidification turns each groupoid into a vector space... but
in fact it gives more: a Hilbert space! Similarly, Jeffrey's process
actually turns each groupoid into a 2-Hilbert space. I proved that
a long time ago:
12) John Baez, Higher-dimensional algebra II: 2-Hilbert spaces,
Adv. Math. 127 (1997), 125-189. Also available as arXiv:q-alg/9609018.
So, just as degroupoidification reveals that a fair amount of
quantum mechanics can be done with groupoids instead of vector spaces,
Jeffrey's process reveals that a fair amount of *categorified*
quantum mechanics can also be done with groupoids!
Categorified quantum mechanics becomes important when we go from the
physics of particles (which is really field theory on 1d spacetimes)
to the physics of strings (which is really field theory on 2d
spacetimes). The simplest case is "topological string theory",
also known as "extended 2d topological quantum field theory". And
the simplest example of such a theory is the "Dijkgraaf-Witten model":
a gauge theory with a finite gauge group.
In his thesis:
13) Jeffrey Morton, Extended TQFT's and Quantum Gravity, Ph.D.
thesis, U. C. Riverside, 2007. Available at arXiv:0710.0032.
Jeffrey showed that a special case, the "untwisted" Dijkgraaf-Witten
model, gives a 2-functor from the 2-category with
0d manifolds as objects,
1d cobordisms between these as morphisms,
equivalence classes of 2d cobordisms between these as 2-morphisms
to the 2-category with
finite groupoids as objects,
spans of finite groupoids as morphisms,
equivalence classes of spans of spans finite groupoids as morphisms.
Composing this 2-functor with the 2-functor I just described,
he gets the untwisted Dijkgraaf-Witten model as an extended TQFT!
And in fact, he does it in all dimensions, not just dimension 2.
By the way, most of the 2-categories and 2-functors here are "weak".
Also by the way, Jeffrey constructed the above cobordism 2-category in
an earlier paper, which I discussed in "week242". He recently
polished up this paper, changing the title to make it focus on
the algebraic essence of his construction:
14) Jeffrey Morton, Double bicategories and double cospans,
available as arXiv:math/0611930.
There's a lot more I could say about this, but not a lot more time.
So, let me wrap up with a pointer to Stephen Summers' review of new
work on constructive quantum field theory.
Constructive quantum field theory is the branch of mathematical
physics where you try to rigorously construct examples of quantum
field theories. I did my Ph.D. thesis on this subject under Irving
Segal, but it was too hard for me, and my heart was never really in
it, so I soon fled - first to classical field theory, and then further.
I recently met Stephen Summers at a conference in honor of von
Neumann, and he tried to call me back to my roots. It turns out
there's been a lot of interesting progress in constructive quantum
field theory! I'll probably keep working on topological quantum field
theory and other wimpy subjects - but it's great to hear someone out
there is doing the hard work of getting physically realistic quantum
field theories to make rigorous mathematical sense.
Here's some of what he has to say:
The development of the tools and techniques of algebraic quantum
field theory (AQFT) has reached the point where they can be turned
upon the question of existence of quantum field models. Although
the program of constructing models via AQFT is still in its infancy
and only a few researchers are working in the field, already some
encouraging successes can be displayed. I personally find it
stimulating that the ideas employed go well beyond the range of the
semiclassical ideas which were mathematically developed by
researchers in constructive quantum field theory in the 70's and
80's. There is no appeal to Lagrangians, actions and perturbation
theory, nor does one "work in the Euclidean realm", and one
generally avoids a direct construction of strictly local quantum
field operators (as these either do not exist or are prohibitively
difficult to construct), preferring to construct more physically
relevant quantities such as the scattering amplitudes and local
"observables". Some of the constructed models are local and free,
some are local and have nontrivial S-matrices, and yet others
manifest only certain remnants of locality, although these remnants
suffice to enable the computation of nontrivial two-particle
S-matrix elements. This includes models with nontrivial scattering
in four spacetime dimensions.
This is just the beginning of a fascinating review. Check it out:
15) Stephen J. Summers, Constructive AQFT,
http://www.math.ufl.edu/~sjs/construction.html
Also check out his big AQFT page, which lists textbooks and
many more references:
16) Stephen J. Summers, Algebraic quantum field theory,
http://www.math.ufl.edu/~sjs/aqft.html
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Quote of the Week:
During the journey we commonly forget its goal. Almost every
profession is chosen as a means to an end but continued as an
end in itself. Forgetting our objectives is the most frequent
act of stupidity. - Friedrich Nietzsche
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Addenda: For more discussion visit the n-Category Cafe:
http://golem.ph.utexas.edu/category/2008/10/this_weeks_finds_in_mathematic_32.html
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