- #1
John Baez
Also available at http://math.ucr.edu/home/baez/week262.html
March 29, 2008
This Week's Finds in Mathematical Physics (Week 262)
John Baez
I'm done with teaching until fall, and now I'll be travelling
a lot. I just got back from Singapore. It's an incredibly
diverse place. I actually had to buy a book to understand
all the foods! I'm now acquainted with the charms of appam,
kaya toast, and babi buah keluak. But I didn't get around to
trying a chendol, a bandung, or a Milo dinosaur, even though
they're all available in every hawker center.
Today I'll talk about quantum technology in Singapore, atom
chips, nitrogen-vacancy pairs in diamonds, graphene transistors,
a new construction of E8, and a categorification of sl(2).
But first - the astronomy pictures of the week!
First another planetary nebula - the "Southern Ring Nebula":
1) Hubble Heritage Project, Planetary Nebula NGC 3132,
http://heritage.stsci.edu/1998/39/index.html
This bubble of hot gas is .4 light years in diameter. You
can see *two* stars near its center. The faint one is the
white dwarf remnant of the star that actually threw off the
gas forming this nebula. The gas is expanding outwards at
about 20 kilometers per second. The intense ultraviolet
radiation from the white dwarf is ionizing this gas and making
it glow.
The Southern Ring Nebula is 2000 light years from us. Much
closer to home, here's a new shot of the frosty dunes of Mars:
2) HiRISE (High Resolution Imaging Science Experiment),
Defrosting polar sand dunes, http://hirise.lpl.arizona.edu/PSP_007043_2650
These horn-shaped dunes are called "barchans"; you can read
more about them at "week228". The frost is carbon dioxide,
evaporating as the springtime sun warms the north polar
region. Here's another photo, taken in February:
3) HiRISE (High Resolution Imaging Science Experiment),
Defrosting northern dunes, http://hirise.lpl.arizona.edu/PSP_007193_2640
The dark stuff pouring down the steep slopes reminds me of
water, but they say it's dust!
Meanwhile, down here on Earth, I had some good conversations with
mathematicians and physicists at the National University of Singapore
(NUS), and also with Artur Ekert and Valerio Scarani, who work here:
4) Centre for Quantum Technologies, http://www.quantumlah.org/
I like the name "quantumlah". "Lah" is perhaps the most famous
word in Singlish: you put it at the end of a sentence for
emphasis, to convey "acceptance, understanding, lightness,
jest, and a medley of other positive feelings". Unfortunately
I didn't get to hear much Singlish during my visit.
The Centre for Quantum Technologies is hosted by NUS but
is somewhat independent. It reminds me a bit of the Institute
for Quantum Computing - see "week235" - but it's smaller, and
still getting started. They hope to take advantage of the
nearby semiconductor fabrication plants, or "fabs", to build
stuff.
They've got theorists and experimentalists. Being overly
theoretical myself, I asked: what are the most interesting
real-life working devices we're likely to see soon? Ekert
mentioned "quantum repeaters" - gadgets that boost the power
of a beam of entangled photons while still maintaining
quantum coherence, as needed for long-distance quantum
cryptography. He also mentioned "atom chips", which use tiny
wires embedded in a silicon chip to trap and manipulate cold
atoms on the chip's surface:
5) Atomchip Group, http://www.atomchip.org/
6) Atom Optics Group, Laboratoire Charles Fabry, Atom-chip
experiment,
http://atomoptic.iota.u-psud.fr/research/chip/chip.html
There's also a nanotech group at NUS:
7) Nanoscience and Nanotechnology Initiative, National
University of Singapore, http://www.nusnni.nus.edu.sg/
who are doing cool stuff with "graphene" - hexagonal sheets
of carbon atoms, like individual layers of a graphite crystal.
Graphene is closely related to buckyballs (see "week79") and
polycyclic aromatic hydrocarbons (see "week258").
Some researchers believe that graphene transistors could
operate in the terahertz range, about 1000 times faster than
conventional silicon ones. The reason is that electrons move
much faster through graphene. Unfortunately the difference
in conductivity between the "on" and "off" states is less for
graphene. This makes it harder to work with. People think
they can solve this problem, though:
8) Kevin Bullis, Graphene transistors, Technology Review,
January 28, 2008, http://www.technologyreview.com/Nanotech/20119/
Duncan Graham-Rowe, Better graphene transistors, Technology
Review, March 17, 2008, http://www.technologyreview.com/Nanotech/20424/
Ekert also told me about another idea for carbon-based computers:
"nitrogen-vacancy centers". These are very elegant entities.
To understand them, it helps to know a bit about diamonds.
You really just need to know that diamonds are crystals made
of carbon. But I can't resist saying more, because the geometry
of these crystals is fascinating.
A diamond is made of carbon atoms arranged in tetrahedra, which
then form a cubical structure, like this:
9) Steve Sque, Structure of diamond,
http://Newton.ex.ac.uk/research/qsystems/people/sque/diamond/structure/
Here you see 4 tetrahedra of carbon atoms inside a cube.
Note that there's one carbon at each corner of the cube, and
also one in the middle of each face. If that was all, we'd
have a "face-centered cubic". But there are also 4 more
carbons inside the cube - one at the center of each tetrahedron!
If you look really carefully, you can see that the full
pattern consists of two interpenetrating face-centered
cubic lattices, one offset relative to the other along the
cube's main diagonal!
While the math of the diamond crystal is perfectly beautiful,
nature doesn't always get it quite right. Sometimes a carbon
atom will be missing. In fact, sometimes a cosmic ray will
knock a carbon out of the lattice! You can also do it yourself
with a beam of neutrons or electrons. The resulting hole is
called a "vacancy". If you heat a diamond to about 900
kelvin, these vacancies start to move around like particles.
Diamonds also have impurities. The most common is nitrogen,
which can form up 1% of a diamond. Nitrogen atoms can take
the place of carbon atoms in the crystal. Sometimes these
nitrogen atoms are isolated, sometimes they come in pairs.
When a lone nitrogen encounters a vacancy, they stick together!
We then have a "nitrogen-vacancy center". It's also common for
4 nitrogens to surround a vacancy. Many other combinations are
also possible - and when we get enough of these nitrogen-vacancy
combinations around, they form larger structures called
"platelets".
10) R. Jones and J. P. Goss, Theory of aggregation of nitrogen
in diamond, in Properties, Growth and Application of Diamond,
eds. Maria Helena Nazare and A. J. Neves, EMIS Datareviews
Series, 2001, 127-130.
A nice thing about nitrogen-vacancy centers is that they act
like spin-1 particles. In fact, these spins interact very
little with their environment, thanks to the remarkable properties
of diamond. So, they might be a good way to store quantum
information: they can last 50 microseconds before losing
coherence, even at room temperature. If we could couple them
to each other in interesting ways, maybe we could do some
"spintronics", or even quantum computation:
11) Sankar das Sarma, Spintronics, American Scientist
89 (2001), 516-523. Also available at
http://www.physics.umd.edu/cmtc/earlier_papers/AmSci.pdf
Lone nitrogens are even more robust carriers of quantum
information: their time to decoherence can be as much as a
millisecond! The reason is that, unlike nitrogen-vacancy
centers, lone nitrogens have "dark spins" - their spin
doesn't interact much with light. But this can also makes
them harder to manipulate. So, it may be easier to use
nitrogen-vacancy centers. People are busy studying the options:
12) R. J. Epstein, F. M. Mendoza, Y. K. Kato and D. D.
Awschalom, Anisotropic interactions of a single spin and
dark-spin spectroscopy in diamond, Nature Physics 1 (2005),
94-98. Also available as arXiv:cond-mat/0507706.
13) Ph. Tamarat et al, The excited state structure of the
nitrogen-vacancy center in diamond, available as
arXiv:cond-mat/0610357.
14) R. Hanson, O. Gywat and D. D. Awschalom, Room-temperature
manipulation and decoherence of a single spin in diamond,
Phys. Rev. B74 (2006) 161203. Also available as
arXiv:quant-ph/0608233
But regardless of whether anyone can coax them into quantum
computation, I like diamonds. Not to own - just to contemplate!
I told you about the diamond rain on Neptune back in "week162".
And in "week193", I explained how diamonds are the closest thing
to the E8 lattice you're likely to see in this 3-dimensional world.
The reason is that in any dimension you can define a checkerboard
lattice called Dn, consisting of all n-tuples of integers that
sum to an even integer. Then you can define a set called Dn+ by
taking two copies of the Dn lattice: the original and another
shifted by the vector (1/2,...,1/2). D8+ is the E8 lattice,
but D3 is the face-centered cubic, and D3+ is the pattern formed
by carbons in a diamond!
In case you're wondering: in math, a "lattice" is technically
a discrete subgroup of R^n that's isomorphic to Z^n. Dn+ is
only a lattice when n is even. So, the carbons in a diamond
don't form a lattice in the strict mathematical sense. On the
other hand, the D3 lattice is secretly the same as the A3
lattice, familiar from stacking oranges. It's one of the
densest ways to pack spheres, with a density of
pi/3 sqrt(2) ~ .74
The D3+ pattern, on the other hand, has a density of just
pi sqrt(3)/16 ~ .34
This is why ice becomes denser when it melts: like diamond,
it's arranged in a D3+ pattern.
(Do diamonds become denser when they melt? Or do they always
turn into graphite when they get hot enough, regardless of
the pressure? Inquiring minds want to know. These days
inquiring minds use search engines to answer questions like
this... but right now I'd rather talk about E8.)
As you probably noticed, Garrett Lisi stirred up quite a media
sensation with his attempt to pack all known forces and particles
into a theory based on the exceptional Lie group E8:
15) Garrett Lisi, An exceptionally simple theory of everything,
available as arXiv:0711.0770
Part of his idea was to use Kostant's triality-based description
of E8 to explain the three generations of leptons - see "week253"
for more. Unfortunately this part of the idea doesn't work, for
purely group-theoretical reasons:
16) Jacques Distler, A little group theory,
http://golem.ph.utexas.edu/~distler/blog/archives/001505.html
A little more group theory,
http://golem.ph.utexas.edu/~distler/blog/archives/001532.html
There would also be vast problems trying get all the dimensionless
constants in the Standard Model to pop out of such a scheme - or
to stick them in somehow.
Meanwhile, Kostant has been doing new things with E8. He's mainly
been using the complex form of E8, while Lisi needs a noncompact
real form to get gravity into the game. So, the connection between
their work is somewhat limited. Nonetheless, Kostant enjoys the
idea of a theory of everything based on E8.
He recently gave a talk here at UCR:
17) Bertram Kostant, On some mathematics in Garrett Lisi's
"E8 theory of everything", February 12, 2008, UCR. Video and
lecture notes at http://math.ucr.edu/home/baez/kostant/
He did some amazing things, like chop the 248-dimensional Lie
algebra of E8 into 31 Cartan subalgebras in a nice way, thus
categorifying the factorization
248 = 8 x 31
To do this, he used a copy of the 32-element group (Z/2)^5
sitting in E8, and the 31 nontrivial characters of this group.
Even more remarkably, this copy of (Z/2)^5 sits inside a copy
of SL(2,F_{32}) inside E8, and the centralizer of a certain
element of SL(2,F_{32}) is a product of two copies of the gauge
group of the Standard Model! What this means - if anything -
remains a mystery.
Indeed, pretty much everything about E8 seems mysterious to me,
since nobody has exhibited it as the symmetry group of anything
more comprehensible than E8 itself. This paper sheds some
new light this puzzle:
17) Jose Miguel Figueroa-O'Farrill, A geometric construction
of the exceptional Lie algebras F4 and E8, available as
arXiv:0706.2829.
The idea here is to build the Lie algebra of E8 using Killing
spinors on the unit sphere in 16 dimensions.
Okay - what's a Killing spinor?
Well, first I need to remind you about Killing vectors. Given
a Riemannian manifold, a "Killing vector" is a vector field that
generates a flow that preserves the metric! A transformation
that preserves the metric is called an "isometry", and these
form a Lie group. Killing vector fields form a Lie algebra
if we use the ordinary Lie bracket of vector fields, and this
is the Lie algebra of the group of isometries.
Now, if our manifold has a spin structure, a "Killing spinor" is
a spinor field psi such that
D_v psi = k v psi
for some constant k for every vector field v. Here D_v psi
is the covariant derivative of psi in the v direction, while
v psi is defined using the action of vectors on spinors.
Only the sign of the constant k really matters, since rescaling
the metric rescales this constant.
It's a cute equation, but what's the point of it? Part
of the point is this: the action of vectors on spinors
V tensor S -> S
has a kind of adjoint
S tensor S -> V
This let's us take a pair of spinor fields and form a vector
field. This is what people mean when they say spinors are
like the "square root" of vectors. And, if we do this to
two *Killing* spinors, we get a *Killing* vector! You can
prove this using that cute equation - and that's the main point
of that equation, as far as I'm concerned.
Under good conditions, this fact let's us define a "Killing
superalgebra" which has the Lie algebra of Killing vectors
as its even part, and the Killing spinors as its odd part.
In this superalgebra, the bracket of two Killing vectors
is just their ordinary Lie bracket. The bracket of a Killing
vector and a Killing spinor is defined using a fairly obvious
notion of the "Lie derivative of a spinor field". And, the
bracket of two Killing spinors is defined using the map
S tensor S -> V
which, as explained, gives a Killing vector.
Now, you might think our "Killing superalgebra" should be a
Lie superalgebra. But in some dimensions, the map
S tensor S -> V
is skew-symmetric. Then our Killing superalgebra has a chance
at being a plain old Lie algebra! We still need to check
the Jacobi identity. And this only works in certain special cases:
If you take S^7 with its usual round metric, the isometry group is
SO(8), so the Lie algebra of Killing vectors is so(8). There's
an 8-dimensional space of Killing spinors, and the action of so(8)
on this gives the real left-handed spinor representation S_8^+.
The Jacobi identity holds, and you get a Lie algebra
so(8) + S_8^+
But then, thanks to triality, you knock ourself on the head and
say "I could have had a V_8!" After all, up to an outer
automorphism of so(8), the spinor representation S_8^+ is the
same as the 8-dimensional vector representation V_8. So, your
Lie algebra is just the same as
so(8) + V_8
with its obvious Lie algebra structure. This is just so(9).
So, it's nothing exceptional, though you arrived at it by a
devious route.
If you take S^8 with its usual round metric, the Lie algebra of
Killing vector fields is so(9). Now there's a 16-dimensional
space of Killing spinor fields, and the action of so(9) on this
gives the real (non-chiral) spinor representation S_9. The Jacobi
identity holds, and you get a Lie algebra structure on
so(9) + S_9
This gives the exceptional Lie algebra f4!
Finally, if you take S^{15} with its usual round metric, the Lie
algebra of Killing vector fields is so(16). Now there's a
128-dimensional space of Killing spinor fields, and the action of
so(16) on this gives the left-handed real spinor representation
S_{16}^+. The Jacobi identity holds, and you get a Lie algebra
so(16) + S_{16}^+
This gives the exceptional Lie algebra e8!
In short, what Figueroa-O'Farrill has done is found a nice
geometrical interpretation for some previously known algebraic
constructions of f4 and e8. Unfortunately, he still needs to
verify the Jacobi identity in the same brute-force way. It
would be nice to find a slicker proof. But his new interpretation
is suggestive: it raises a lot of new questions. He lists some
of these at the end of the paper, and mentions a really big one
at the beginning. Namely: the spheres S^7, S^8 and S^{15} all
show up in the Hopf fibration associated to the octonionic projective
line:
S^7 -> S^{15} -> S^8
Does this give a nice relation between so(9), F4 and E8? Can
someone guess what this relation should be? Maybe E8 is built
from so(9) and F4 somehow.
I also wonder if there's a Killing superalgebra interpretation
of the Lie algebra constructions
e6 = so(10) + S_{10} + u(1)
and
e7 = so(12) + S_{12}^+ + su(2)
These would need to be trickier, with the u(1) showing up from
the fact that S_{10} is a complex representation, and the su(2)
showing up from the fact that S_{12}^+ is a quaternionic
representation. The algebra is explained here:
18) John Baez, The octonions, section 4.3: the magic square,
available at http://math.ucr.edu/home/baez/octonions/node16.html
A geometrical interepretation would be nice!
Finally - my former student Aaron Lauda has been working with
Khovanov on categorifying quantum groups, and their work is starting
to really take off. I'm just beginning to read his new papers, but
I can't resist bringing them to your attention:
18) Aaron Lauda, A categorification of quantum sl(2), available as
arXiv:0803.3848.
Aaron Lauda, Categorified quantum sl(2) and equivariant cohomology
of iterated flag varieties, available as arXiv:0803.3652.
He's got a *2-category* that decategorifies to give the quantized
universal enveloping algebra of sl(2)! And similarly for all the
irreps of this algebra!
There's more to come, too...
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
March 29, 2008
This Week's Finds in Mathematical Physics (Week 262)
John Baez
I'm done with teaching until fall, and now I'll be travelling
a lot. I just got back from Singapore. It's an incredibly
diverse place. I actually had to buy a book to understand
all the foods! I'm now acquainted with the charms of appam,
kaya toast, and babi buah keluak. But I didn't get around to
trying a chendol, a bandung, or a Milo dinosaur, even though
they're all available in every hawker center.
Today I'll talk about quantum technology in Singapore, atom
chips, nitrogen-vacancy pairs in diamonds, graphene transistors,
a new construction of E8, and a categorification of sl(2).
But first - the astronomy pictures of the week!
First another planetary nebula - the "Southern Ring Nebula":
1) Hubble Heritage Project, Planetary Nebula NGC 3132,
http://heritage.stsci.edu/1998/39/index.html
This bubble of hot gas is .4 light years in diameter. You
can see *two* stars near its center. The faint one is the
white dwarf remnant of the star that actually threw off the
gas forming this nebula. The gas is expanding outwards at
about 20 kilometers per second. The intense ultraviolet
radiation from the white dwarf is ionizing this gas and making
it glow.
The Southern Ring Nebula is 2000 light years from us. Much
closer to home, here's a new shot of the frosty dunes of Mars:
2) HiRISE (High Resolution Imaging Science Experiment),
Defrosting polar sand dunes, http://hirise.lpl.arizona.edu/PSP_007043_2650
These horn-shaped dunes are called "barchans"; you can read
more about them at "week228". The frost is carbon dioxide,
evaporating as the springtime sun warms the north polar
region. Here's another photo, taken in February:
3) HiRISE (High Resolution Imaging Science Experiment),
Defrosting northern dunes, http://hirise.lpl.arizona.edu/PSP_007193_2640
The dark stuff pouring down the steep slopes reminds me of
water, but they say it's dust!
Meanwhile, down here on Earth, I had some good conversations with
mathematicians and physicists at the National University of Singapore
(NUS), and also with Artur Ekert and Valerio Scarani, who work here:
4) Centre for Quantum Technologies, http://www.quantumlah.org/
I like the name "quantumlah". "Lah" is perhaps the most famous
word in Singlish: you put it at the end of a sentence for
emphasis, to convey "acceptance, understanding, lightness,
jest, and a medley of other positive feelings". Unfortunately
I didn't get to hear much Singlish during my visit.
The Centre for Quantum Technologies is hosted by NUS but
is somewhat independent. It reminds me a bit of the Institute
for Quantum Computing - see "week235" - but it's smaller, and
still getting started. They hope to take advantage of the
nearby semiconductor fabrication plants, or "fabs", to build
stuff.
They've got theorists and experimentalists. Being overly
theoretical myself, I asked: what are the most interesting
real-life working devices we're likely to see soon? Ekert
mentioned "quantum repeaters" - gadgets that boost the power
of a beam of entangled photons while still maintaining
quantum coherence, as needed for long-distance quantum
cryptography. He also mentioned "atom chips", which use tiny
wires embedded in a silicon chip to trap and manipulate cold
atoms on the chip's surface:
5) Atomchip Group, http://www.atomchip.org/
6) Atom Optics Group, Laboratoire Charles Fabry, Atom-chip
experiment,
http://atomoptic.iota.u-psud.fr/research/chip/chip.html
There's also a nanotech group at NUS:
7) Nanoscience and Nanotechnology Initiative, National
University of Singapore, http://www.nusnni.nus.edu.sg/
who are doing cool stuff with "graphene" - hexagonal sheets
of carbon atoms, like individual layers of a graphite crystal.
Graphene is closely related to buckyballs (see "week79") and
polycyclic aromatic hydrocarbons (see "week258").
Some researchers believe that graphene transistors could
operate in the terahertz range, about 1000 times faster than
conventional silicon ones. The reason is that electrons move
much faster through graphene. Unfortunately the difference
in conductivity between the "on" and "off" states is less for
graphene. This makes it harder to work with. People think
they can solve this problem, though:
8) Kevin Bullis, Graphene transistors, Technology Review,
January 28, 2008, http://www.technologyreview.com/Nanotech/20119/
Duncan Graham-Rowe, Better graphene transistors, Technology
Review, March 17, 2008, http://www.technologyreview.com/Nanotech/20424/
Ekert also told me about another idea for carbon-based computers:
"nitrogen-vacancy centers". These are very elegant entities.
To understand them, it helps to know a bit about diamonds.
You really just need to know that diamonds are crystals made
of carbon. But I can't resist saying more, because the geometry
of these crystals is fascinating.
A diamond is made of carbon atoms arranged in tetrahedra, which
then form a cubical structure, like this:
9) Steve Sque, Structure of diamond,
http://Newton.ex.ac.uk/research/qsystems/people/sque/diamond/structure/
Here you see 4 tetrahedra of carbon atoms inside a cube.
Note that there's one carbon at each corner of the cube, and
also one in the middle of each face. If that was all, we'd
have a "face-centered cubic". But there are also 4 more
carbons inside the cube - one at the center of each tetrahedron!
If you look really carefully, you can see that the full
pattern consists of two interpenetrating face-centered
cubic lattices, one offset relative to the other along the
cube's main diagonal!
While the math of the diamond crystal is perfectly beautiful,
nature doesn't always get it quite right. Sometimes a carbon
atom will be missing. In fact, sometimes a cosmic ray will
knock a carbon out of the lattice! You can also do it yourself
with a beam of neutrons or electrons. The resulting hole is
called a "vacancy". If you heat a diamond to about 900
kelvin, these vacancies start to move around like particles.
Diamonds also have impurities. The most common is nitrogen,
which can form up 1% of a diamond. Nitrogen atoms can take
the place of carbon atoms in the crystal. Sometimes these
nitrogen atoms are isolated, sometimes they come in pairs.
When a lone nitrogen encounters a vacancy, they stick together!
We then have a "nitrogen-vacancy center". It's also common for
4 nitrogens to surround a vacancy. Many other combinations are
also possible - and when we get enough of these nitrogen-vacancy
combinations around, they form larger structures called
"platelets".
10) R. Jones and J. P. Goss, Theory of aggregation of nitrogen
in diamond, in Properties, Growth and Application of Diamond,
eds. Maria Helena Nazare and A. J. Neves, EMIS Datareviews
Series, 2001, 127-130.
A nice thing about nitrogen-vacancy centers is that they act
like spin-1 particles. In fact, these spins interact very
little with their environment, thanks to the remarkable properties
of diamond. So, they might be a good way to store quantum
information: they can last 50 microseconds before losing
coherence, even at room temperature. If we could couple them
to each other in interesting ways, maybe we could do some
"spintronics", or even quantum computation:
11) Sankar das Sarma, Spintronics, American Scientist
89 (2001), 516-523. Also available at
http://www.physics.umd.edu/cmtc/earlier_papers/AmSci.pdf
Lone nitrogens are even more robust carriers of quantum
information: their time to decoherence can be as much as a
millisecond! The reason is that, unlike nitrogen-vacancy
centers, lone nitrogens have "dark spins" - their spin
doesn't interact much with light. But this can also makes
them harder to manipulate. So, it may be easier to use
nitrogen-vacancy centers. People are busy studying the options:
12) R. J. Epstein, F. M. Mendoza, Y. K. Kato and D. D.
Awschalom, Anisotropic interactions of a single spin and
dark-spin spectroscopy in diamond, Nature Physics 1 (2005),
94-98. Also available as arXiv:cond-mat/0507706.
13) Ph. Tamarat et al, The excited state structure of the
nitrogen-vacancy center in diamond, available as
arXiv:cond-mat/0610357.
14) R. Hanson, O. Gywat and D. D. Awschalom, Room-temperature
manipulation and decoherence of a single spin in diamond,
Phys. Rev. B74 (2006) 161203. Also available as
arXiv:quant-ph/0608233
But regardless of whether anyone can coax them into quantum
computation, I like diamonds. Not to own - just to contemplate!
I told you about the diamond rain on Neptune back in "week162".
And in "week193", I explained how diamonds are the closest thing
to the E8 lattice you're likely to see in this 3-dimensional world.
The reason is that in any dimension you can define a checkerboard
lattice called Dn, consisting of all n-tuples of integers that
sum to an even integer. Then you can define a set called Dn+ by
taking two copies of the Dn lattice: the original and another
shifted by the vector (1/2,...,1/2). D8+ is the E8 lattice,
but D3 is the face-centered cubic, and D3+ is the pattern formed
by carbons in a diamond!
In case you're wondering: in math, a "lattice" is technically
a discrete subgroup of R^n that's isomorphic to Z^n. Dn+ is
only a lattice when n is even. So, the carbons in a diamond
don't form a lattice in the strict mathematical sense. On the
other hand, the D3 lattice is secretly the same as the A3
lattice, familiar from stacking oranges. It's one of the
densest ways to pack spheres, with a density of
pi/3 sqrt(2) ~ .74
The D3+ pattern, on the other hand, has a density of just
pi sqrt(3)/16 ~ .34
This is why ice becomes denser when it melts: like diamond,
it's arranged in a D3+ pattern.
(Do diamonds become denser when they melt? Or do they always
turn into graphite when they get hot enough, regardless of
the pressure? Inquiring minds want to know. These days
inquiring minds use search engines to answer questions like
this... but right now I'd rather talk about E8.)
As you probably noticed, Garrett Lisi stirred up quite a media
sensation with his attempt to pack all known forces and particles
into a theory based on the exceptional Lie group E8:
15) Garrett Lisi, An exceptionally simple theory of everything,
available as arXiv:0711.0770
Part of his idea was to use Kostant's triality-based description
of E8 to explain the three generations of leptons - see "week253"
for more. Unfortunately this part of the idea doesn't work, for
purely group-theoretical reasons:
16) Jacques Distler, A little group theory,
http://golem.ph.utexas.edu/~distler/blog/archives/001505.html
A little more group theory,
http://golem.ph.utexas.edu/~distler/blog/archives/001532.html
There would also be vast problems trying get all the dimensionless
constants in the Standard Model to pop out of such a scheme - or
to stick them in somehow.
Meanwhile, Kostant has been doing new things with E8. He's mainly
been using the complex form of E8, while Lisi needs a noncompact
real form to get gravity into the game. So, the connection between
their work is somewhat limited. Nonetheless, Kostant enjoys the
idea of a theory of everything based on E8.
He recently gave a talk here at UCR:
17) Bertram Kostant, On some mathematics in Garrett Lisi's
"E8 theory of everything", February 12, 2008, UCR. Video and
lecture notes at http://math.ucr.edu/home/baez/kostant/
He did some amazing things, like chop the 248-dimensional Lie
algebra of E8 into 31 Cartan subalgebras in a nice way, thus
categorifying the factorization
248 = 8 x 31
To do this, he used a copy of the 32-element group (Z/2)^5
sitting in E8, and the 31 nontrivial characters of this group.
Even more remarkably, this copy of (Z/2)^5 sits inside a copy
of SL(2,F_{32}) inside E8, and the centralizer of a certain
element of SL(2,F_{32}) is a product of two copies of the gauge
group of the Standard Model! What this means - if anything -
remains a mystery.
Indeed, pretty much everything about E8 seems mysterious to me,
since nobody has exhibited it as the symmetry group of anything
more comprehensible than E8 itself. This paper sheds some
new light this puzzle:
17) Jose Miguel Figueroa-O'Farrill, A geometric construction
of the exceptional Lie algebras F4 and E8, available as
arXiv:0706.2829.
The idea here is to build the Lie algebra of E8 using Killing
spinors on the unit sphere in 16 dimensions.
Okay - what's a Killing spinor?
Well, first I need to remind you about Killing vectors. Given
a Riemannian manifold, a "Killing vector" is a vector field that
generates a flow that preserves the metric! A transformation
that preserves the metric is called an "isometry", and these
form a Lie group. Killing vector fields form a Lie algebra
if we use the ordinary Lie bracket of vector fields, and this
is the Lie algebra of the group of isometries.
Now, if our manifold has a spin structure, a "Killing spinor" is
a spinor field psi such that
D_v psi = k v psi
for some constant k for every vector field v. Here D_v psi
is the covariant derivative of psi in the v direction, while
v psi is defined using the action of vectors on spinors.
Only the sign of the constant k really matters, since rescaling
the metric rescales this constant.
It's a cute equation, but what's the point of it? Part
of the point is this: the action of vectors on spinors
V tensor S -> S
has a kind of adjoint
S tensor S -> V
This let's us take a pair of spinor fields and form a vector
field. This is what people mean when they say spinors are
like the "square root" of vectors. And, if we do this to
two *Killing* spinors, we get a *Killing* vector! You can
prove this using that cute equation - and that's the main point
of that equation, as far as I'm concerned.
Under good conditions, this fact let's us define a "Killing
superalgebra" which has the Lie algebra of Killing vectors
as its even part, and the Killing spinors as its odd part.
In this superalgebra, the bracket of two Killing vectors
is just their ordinary Lie bracket. The bracket of a Killing
vector and a Killing spinor is defined using a fairly obvious
notion of the "Lie derivative of a spinor field". And, the
bracket of two Killing spinors is defined using the map
S tensor S -> V
which, as explained, gives a Killing vector.
Now, you might think our "Killing superalgebra" should be a
Lie superalgebra. But in some dimensions, the map
S tensor S -> V
is skew-symmetric. Then our Killing superalgebra has a chance
at being a plain old Lie algebra! We still need to check
the Jacobi identity. And this only works in certain special cases:
If you take S^7 with its usual round metric, the isometry group is
SO(8), so the Lie algebra of Killing vectors is so(8). There's
an 8-dimensional space of Killing spinors, and the action of so(8)
on this gives the real left-handed spinor representation S_8^+.
The Jacobi identity holds, and you get a Lie algebra
so(8) + S_8^+
But then, thanks to triality, you knock ourself on the head and
say "I could have had a V_8!" After all, up to an outer
automorphism of so(8), the spinor representation S_8^+ is the
same as the 8-dimensional vector representation V_8. So, your
Lie algebra is just the same as
so(8) + V_8
with its obvious Lie algebra structure. This is just so(9).
So, it's nothing exceptional, though you arrived at it by a
devious route.
If you take S^8 with its usual round metric, the Lie algebra of
Killing vector fields is so(9). Now there's a 16-dimensional
space of Killing spinor fields, and the action of so(9) on this
gives the real (non-chiral) spinor representation S_9. The Jacobi
identity holds, and you get a Lie algebra structure on
so(9) + S_9
This gives the exceptional Lie algebra f4!
Finally, if you take S^{15} with its usual round metric, the Lie
algebra of Killing vector fields is so(16). Now there's a
128-dimensional space of Killing spinor fields, and the action of
so(16) on this gives the left-handed real spinor representation
S_{16}^+. The Jacobi identity holds, and you get a Lie algebra
so(16) + S_{16}^+
This gives the exceptional Lie algebra e8!
In short, what Figueroa-O'Farrill has done is found a nice
geometrical interpretation for some previously known algebraic
constructions of f4 and e8. Unfortunately, he still needs to
verify the Jacobi identity in the same brute-force way. It
would be nice to find a slicker proof. But his new interpretation
is suggestive: it raises a lot of new questions. He lists some
of these at the end of the paper, and mentions a really big one
at the beginning. Namely: the spheres S^7, S^8 and S^{15} all
show up in the Hopf fibration associated to the octonionic projective
line:
S^7 -> S^{15} -> S^8
Does this give a nice relation between so(9), F4 and E8? Can
someone guess what this relation should be? Maybe E8 is built
from so(9) and F4 somehow.
I also wonder if there's a Killing superalgebra interpretation
of the Lie algebra constructions
e6 = so(10) + S_{10} + u(1)
and
e7 = so(12) + S_{12}^+ + su(2)
These would need to be trickier, with the u(1) showing up from
the fact that S_{10} is a complex representation, and the su(2)
showing up from the fact that S_{12}^+ is a quaternionic
representation. The algebra is explained here:
18) John Baez, The octonions, section 4.3: the magic square,
available at http://math.ucr.edu/home/baez/octonions/node16.html
A geometrical interepretation would be nice!
Finally - my former student Aaron Lauda has been working with
Khovanov on categorifying quantum groups, and their work is starting
to really take off. I'm just beginning to read his new papers, but
I can't resist bringing them to your attention:
18) Aaron Lauda, A categorification of quantum sl(2), available as
arXiv:0803.3848.
Aaron Lauda, Categorified quantum sl(2) and equivariant cohomology
of iterated flag varieties, available as arXiv:0803.3652.
He's got a *2-category* that decategorifies to give the quantized
universal enveloping algebra of sl(2)! And similarly for all the
irreps of this algebra!
There's more to come, too...
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