From: baez@math.removethis.ucr.andthis.edu (John Baez)
Subject: This Week's Finds in Mathematical Physics (Week 298)
Organization: University of California, Riverside
Sender: baez@math.removethis.ucr.andthis.edu (John Baez)
Newsgroups: sci.math.research
Also available at http://math.ucr.edu/home/baez/week298.html
May 14, 2010
This Week's Finds in Mathematical Physics (Week 298)
John Baez
Next week I'm going to New York to talk about the stuff I've been
explaining here lately: electrical circuits and category theory.
Then - volcanos permitting - I'll fly to Oxford to attend a course
on quantum computation:
1) Foundational Structures in Quantum Computation and Information,
May 24-28, 2010, Oxford University, organized by Bob Coecke and
Ross Duncan,
http://web.comlab.ox.ac.uk/people/Bob.Coecke/QICS_School.html
I look forward to lots of interesting conversations. A bunch of my
math pals will be attending - folks like Bruce Bartlett, Eugenia Cheng,
Simon Willerton, Jamie Vicary and maybe even my former student Alissa
Crans, who lives here in California, but may swing by. I'll talk to
Thomas Fischbacher about environmental sustainability and computational
field theory, and Dan Ghica about hardware description languages. I
also plan to meet Tim Palmer, a physicist at Oxford who works on climate
and weather prediction, and one of the people I've been interviewing
for the new This Week's Finds. I'm quite excited about that.
(My plan, you see, is to interview people who are applying math and
physics to serious practical problems. It'll be a lot more interesting
than what I've been doing so far, I promise.)
The weekend after the course, there will be a workshop:
2) Quantum Physics and Logic, May 29-30, 2010, Oxford University,
organized Bob Coecke, Prakash Panangaden, and Peter Selinger,
http://web.comlab.ox.ac.uk/people/Bob.Coecke/QPL_10.html
Peter Selinger is the guy who got me interested in categories where
the morphisms are electrical circuits! I'll be giving a talk in this
workshop - you can see the slides here:
3) John Baez, Duality in logic and physics,
http://math.ucr.edu/home/baez/dual/
Alas, I've been so busy getting ready for my talks that I don't feel
like writing about electrical circuits today. I was going to, but I
need a change of pace. So let me say a bit about octonions, higher
gauge theory, string theory and hyperdeterminants.
Integers are very special real numbers. But there are also "integers"
for the complex numbers, the quaternions and octonions. The most
famous are the "Gaussian integers", which are complex numbers like
a + bi
where a and b are integers. They form a square lattice like this:
* * * *
* * * *
* * * *
The second most famous are the "Eisenstein integers", which form a
hexagonal lattice like this:
* * * *
* * *
* * * *
I explained how these lattices are important in string theory back
in "week124" - "week126".
People have also thought about various kinds of far less well-known are
various kinds of quaternionic and octonionic "integers". To learn about
these, there's no better place than the magnificent book by Conway and
Smith:
4) John H. Conway and Derek A. Smith, On Quaternions and Octonions:
Their Geometry, Arithmetic, and Symmetry, A. K. Peters, Ltd., Natick,
Massachusetts, 2003.
For a little taste, try my summary in "week194", and my review here:
5) John Baez, review of On Quaternions and Octonions, Bull. Amer.
Math. Soc. 42 (2005), 229-243. Also available at
http://www.ams.org/journals/bull/2005-42-02/S0273-0979-05-01043-8/
and as a webpage at http://math.ucr.edu/home/baez/octonions/conway_smith/
You'll meet the Lipschitz integers and Hurwitz integers sitting
inside the quaternions - and the Lipschitzian, Hurwitzian and Gravesian
integers sitting inside the octonions. I suspect all these charming
names are due to Conway, who has a real gift for terminology. But
best of all are the Cayley integers, which form a famous 8-dimensional
lattice: the E8 lattice! This gives the densest lattice packing of
spheres in 8 dimensions: each sphere touches 240 others.
But you've heard me rhapsodizing about E8 for years. What's new this
week? Well, recently I was perusing a draft of a paper on various
kinds of quaternionic and octonionic integers, which was kindly sent to
me by Norman Johnson. I'm afraid that paper is still top secret, but
an interesting issue came up.
People know all the finite subgroups of the unit quaternions, otherwise
known as SU(2), the double cover of the 3d rotation group. The most
famous of these are:
the 24-element "binary tetrahedral group"
the 48-element "binary octahedral group"
the 60-element "binary icosahedral group"
These are the double covers of the rotational symmetry groups of the
Platonic solids. Since beautiful structures like this have a way of
connecting diverse subjects, you shouldn't be surprised that these
groups show up all over in math, from the McKay correspondence and
Klein's work on the quintic equation (see "week65" and "week230") to
the theory of modular forms (see "week197"). But the best place to
learn the classification of these finite subgroups is the book by
Conway and Smith.
For some idiotic reason I'd never pondered the analogous question for
the octonions until Norman Johnson brought it up! The unit octonions
form a 7-dimensional sphere. They don't form a group, since
multiplication of octonions isn't associative. But they form a "loop",
which is just like a group but with the associative law dropped. Since
they're a smooth manifold, and the group operations are smooth, they
form a "Lie loop".
In fact the only spheres that are Lie groups are:
the unit real numbers: the 0-sphere, also called Z/2 or O(1)
the unit complex numbers: the 1-sphere, also called SO(2) or U(1)
the unit quaternions: the 3-sphere, also called SU(2) or Sp(1)
And the only other sphere that's a "Lie loop" is the unit octonions!
So we can - and should! - ask: what are the finite subloops of the
unit octonions?
Michael Kinyon, an expert on loops, quickly provided two references
that settle the question:
6) R. T. Curtis, Construction of a family of Moufang loops,
Math. Proc. Cambridge Philos. Soc. 142 (2007), 233-237.
7) P. Boddington and D. Rumynin, On Curtis' theorem about finite
octonionic loops, Proc. Amer. Math. Soc. 135 (2007), 1651-1657.
Available at
http://www.ams.org/journals/proc/2007-135-06/S0002-9939-07-08707-2/
There are no huge surprises here. The most exciting finite subloop has
240 elements: it consists of the Cayley integers of length 1, otherwise
known as the root vectors of E8. The rest of the finite subloops are
either finite subgroups of the unit quaternions or "doubles" of these.
No exotic beasts I hadn't dreamt of. But it's very nice to know the
full story!
I also have some other news on the octonionic front. John Huerta and
I wrote a second paper on division algebras and supersymmetry,
where we explain how to construct the "Lie n-superalgebras" that govern
classical superstring and super-2-brane theories:
8) John Baez and John Huerta, Supersymmetry and division algebras II,
available as arXiv:1003.3436.
As you may know, Lie groups and their Lie algebras are incredibly
important in gauge theory, which describe how particles change when
you move them around. Lately people have been developing "higher
gauge theory", which does the same job for strings and higher-
dimensional membranes. For strings we need Lie 2-groups and their Lie
2-algebras. Lie 2-groups are like Lie groups except that they're
*categories* instead of sets... and similarly for Lie 2-algebras. Go
up one more dimension and you need math based on *2-categories*. So,
for 2-branes, which look like soap bubbles instead of loops of string,
you need Lie 3-groups and their Lie 3-algebras. Etcetera.
So, the race is on to construct, classify and understand Lie n-groups
and Lie n-algebras - and redo all of geometry to take advantage of
these higher structures! For an easy introduction, try:
9) John Baez and John Huerta, An invitation to higher gauge theory,
available as arXiv:1003.4485.
But for supersymmetric theories, geometry based on manifolds isn't
enough: we need supermanifolds. The world is made of bosons and
fermions, and supersymmetry is an attempt to unite them. So, a
supermanifold has ordinary "even" or "bosonic" coordinate functions
that commute with each other, but also "odd" or "fermionic" coordinate
functions that anticommute. For the last few decades people have been
redoing geometry using supermanifolds. As part of this, they've done
a lot of work to construct, classify and understand Lie supergroups
and their Lie superalgebras.
Superstring theory combines supersymmetry and higher-dimensional
membranes in a beautiful way. It's never made any predictions about
the real world, and it may never succeed in doing that. But it's been
a real boon for mathematicians. And here's another example: we can
now enjoy ourselves developing a theory of Lie n-supergroups and their
Lie n-superalgebras!
I might feel guilty indulging in such decadent pleasures, were it not
that I plan to start work on more practical projects. But having
spent years thinking about division algebras and higher gauge theory,
it was irresistible to combine them - especially since my student John
Huerta has a knack for this stuff.
And here's what we discovered:
The real numbers give rise to a Lie 2-superalgebra which describes
the symmetries of classical superstrings in 3d spacetime.
The complex numbers give rise to a Lie 2-superalgebra which describes
the symmetries of classical superstrings in 4d spacetime.
The quaternions give rise to a Lie 2-superalgebra which describes
the symmetries of classical superstrings in 6d spacetime.
The octonions give rise to a Lie 2-superalgebra which describes
the symmetries of classical superstrings in 10d spacetime.
3, 4, 6 and 10 - these are 2 more than the dimensions of the real
numbers, complex numbers, quaternions and octonions. I've discussed
this pattern many times here. But then we discovered something else:
The real numbers give rise to a Lie 3-superalgebra which describes
the symmetries of classical super-2-branes in 4d spacetime.
The complex numbers give rise to a Lie 3-superalgebra which describes
the symmetries of classical super-2-branes in 5d spacetime.
The quaternions give rise to a Lie 3-superalgebra which describes
the symmetries of classical super-2-branes in 7d spacetime.
The octonions give rise to a Lie 3-superalgebra which describes
the symmetries of classical super-2-branes in 11d spacetime.
4, 5, 7 and 11 - these are *3* more than the dimensions of the real
numbers, complex numbers, quaternions and octonions! And the 11d
case is related to "M-theory" - that mysterious dream you've probably
heard people muttering about.
You might ask if the pattern keeps going on, like this:
Do the real numbers give rise to a Lie 4-superalgebra which
describes the symmetries of classical super-3-branes in 5d
spacetime?
Do the complex numbers give rise to a Lie 4-superalgebra which
describes the symmetries of classical super-3-branes in 6d
spacetime?
Do the quaternions give rise to a Lie 4-superalgebra which
describes the symmetries of classical super-3-branes in 8d
spacetime?
Do the octonions give rise to a Lie 4-superalgebra which
describes the symmetries of classical super-3-branes in 12d
spacetime?
But some calculations by Tevian Dray and John Huerta, together with
a lot of physics lore, suggest that the pattern does *not* keep
going on - at least not for the most exciting case, the octonionic
case. You can see what I mean by looking at the "brane scan" in this
classic paper by Duff:
10) Michael J. Duff, Supermembranes: the first fifteen weeks,
Classical and Quantum Gravity 5 (1988), 189-205. Also available at
http://ccdb4fs.kek.jp/cgi-bin/img_index?8708425
The story seems to fizzle out after 11 dimensions. And that's part of
what intrigues me about division algebras and related exceptional
structures in math: the funny fragmentary patterns that don't go on
forever.
Recently Peter Woit criticized Duff for some remarks in this article:
11) Michael Duff, Black holes and qubits, CERN Courier May 5, 2010.
Available at http://cerncourier.com/cws/article/cern/42328
12) Peter Woit, Applying string theory to quantum information theory,
http://www.math.columbia.edu/~woit/wordpress/?p=2952
Duff's article describes how he noticed some of the same math showing
up in superstring calculations of black hole entropy and patterns of
quantum entanglement between qubits. This leads into some nice math
involving octonions and related exceptional structures like the Fano
plane and Cayley's "hyperdeterminants".
Unfortunately, Duff gets a bit carried away. For example, he says
that string theory "predicts" the various ways that three qubits can
be entangled. Someone who didn't know physics might jump to the
conclusion that this is a prediction whose confirmation lends credence
to string theory as a description of the fundamental constituents of
nature. It's not!
I also doubt that "superquantum computing" is likely to be
practical... though I've read interesting things about supersymmetry
in graphene, so I could wind up eating my words.
On the other hand, the math is fascinating. For details, try these
papers:
13) Akimasa Miyake and Miki Wadati, Multipartite entanglement and
hyperdeterminants, Quant. Info. Comp. 2 (2002), 540-555. Also
available as arXiv:quant-ph/0212146.
14) Michael J. Duff and S. Ferrara, E7 and the tripartite entanglement
of seven qubits, Phys. Rev. D 76 025018 (2007). Also available as
arXiv:quant-ph/0609227.
15) Michael J. Duff and S. Ferrara, E6 and the bipartite entanglement
of qutrits, Phys. Rev. D 76 124023 (2007). Also available as
arXiv:0704.0507.
or this very nice recent one:
16) Bianca L. Cerchiai and Bert van Geemen, From qubits to E7,
available as arXiv:1003.4255.
Since I've spent a lot of time talking about the Fano plane, the
octonions, and exceptional groups, let me say just a word or two
about hyperdeterminants.
In the 1840's, after his work on determinants, Arthur Cayley invented
a theory of "hyperdeterminants" for 2 x 2 x 2 arrays of numbers.
They're a bit like determinants of 2 x 2 matrices, but more
complicated. They lay dormant for about a century, but were recently
revived by three bigshots: Gel'fand, Kapranov and Zelevinsky. I never
understood them, but when Woit called them "extraordinarily obscure",
it was like waving a red flag in front of a bull. I charged
forward... and now I sort of understand them, at least a little.
Suppose we have a 2 x 2 matrix of complex numbers. We can think of
this as an element of the Hilbert space
C^2 tensor C^2
so we get a linear functional
C^2 tensor C^2 -> C
sending any vector to its inner product with this element. We can
then regard this linear functional as a function
f: C^2 x C^2 -> C
that's linear in each argument. Is there a nonzero point in C^2 x C^2
where the gradient of f vanishes? The answer is yes if and only if
the determinant of our 2 x 2 matrix is zero!
Next, suppose we have a 2 x 2 x 2 array of complex numbers.
We can think of this as an element of the Hilbert space
C^2 tensor C^2 tensor C^2
so we get a linear functional
C^2 tensor C^2 tensor C^2 -> C
sending any vector to its inner product with this element. We can
then regard this linear functional as a function
f: C^2 x C^2 x C^2 -> C
that's linear in each argument. Is there a nonzero point in
C^2 x C^2 x C^2 where the gradient of f vanishes? The answer is
yes if and only if the hyperdeterminant of our 2 x 2 x 2 array is
zero!
Here's another way to think about it. Suppose you have a quantum
system made of 3 subsystems, each described by a 2d Hilbert space.
The Hilbert space of the whole system is thus
C^2 tensor C^2 tensor C^2
The hyperdeterminant is a function on this space that can be thought
of as measuring how correlated - or to use a bit of jargon,
"entangled" - the 3 subsystems are. In particular, if we look at unit
vectors modulo phase, we get the projective space CP^7. The group
GL(2,C) x GL(2,C) x GL(2,C)
acts on this. There's an open dense orbit, coming from the vectors
whose hyperdeterminant is nonzero. These states are quite entangled,
as you'd expect: entanglement is not an exceptional situation, it's
generic. And then there are various smaller orbits, going down to
the 1-point orbit coming from vectors like
u tensor v tensor w
These describe states with no entanglement at all!
We can make what I'm saying a bit more precise at the expense of a
little more math. The polynomial functions on C^2 tensor C^2 tensor C^2
that are invariant under the action of SL(2,C) x SL(2,C) x SL(2,C)
form an algebra, and this algebra is generated by the hyperdeterminant,
which is a homogeneous cubic polynomial. (Why did I switch from
GL(2,C) to SL(2,C)? Because the hyperdeterminant is not invariant
under the bigger group GL(2,C) x GL(2,C) x GL(2,C). However, it comes
close: it gets multiplied by a scalar in a simple way.)
If we try to generalize this idea to tensor products of more spaces,
lik C^2 tensor C^2 tensor C^2 tensor C^2, we typically get an algebra
that's not generated by a single polynomial. The same happens if we
jack up the dimension, for example replacing C^2 by C^9. So, the
situations where a single polynomial does the job are special.
As an extra lure, let me add that you can write the Lagrangian for a
string in (2+2)-dimensional spacetime using hyperdeterminants! The
formula is very pretty and simple:
17) Michael J. Duff, Hidden symmetries of the Nambu-Goto action, Phys.
Lett. B641 (2006), 335-337. Also available as arXiv:hep-th/0602160.
This being the 21st century, there is even a blog on hyperdeterminants:
18) Hyperdeterminacy,
http://hyperdeterminant.wordpress.com/2008/09/25/hello-world/
The author begins with some introductory posts which are definitely
worth reading, so I've linked to the first of those. He expresses
the worry that "this may turn out to be one of the least read blogs in
the blogosphere". Go visit, leave a comment, and prove him wrong!
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Quote of the Week:
"Quaternions appear to exude an air of nineteenth century decay, as a
rather unsuccessful species in the struggle-for-life of mathematical
ideas. Mathematicians, admittedly, still keep a warm place in their
hearts for the remarkable algebraic properties of quaternions but,
alas, such enthusiasm means little to the harder-headed physical
scientist." - Simon L. Altmann
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Addenda: On August 8th 2010, I received the following mail from
Péter Lévay, who has allowed me to quote it here:
Dear John,
I used to read your fascinating blog. Recently I have come across
your "week298" post (14 May) concerning hyperdeterminants and
reflections on a paper by Duff which appeared in CERN Courier on
the black hole qubit correspondence.
Since I know that you are really fascinated by octonions,
Freudenthal systems, etc., with this mail I intend to draw your
attention to papers of mine connected to the stuff of your
"week298" post. Moreover, I would also like to add some further
hints (which Duff does not mention) why this analogy is worth
working out further.
Originally I used to work on the field of quantum entanglement, and
group theoretical and geometrical methods in quantum physics. My
paper:
19) Péter Lévay, The twistor geometry of three-qubit entanglement,
available as arXiv:quant-ph/0403060.
on the "Cayley" hyperdeterminant, twistors and three qubit
entanglement was the one that motivated originally Duff's work
(as he mentions in the CERN Courier). Later I have written
another paper on the four qubit case (hyperdeterminant):
20) Péter Lévay, On the geometry of four qubit invariants,
available as arXiv:quant-ph/0605151.
After Duff's and Kallosh and Linde's paper I have shown that the
black hole qubit analogy also works for issues concerning DYNAMICS.
I have shown that the attractor mechanism (a process used for moduli
stabilization on the BH horizon) can be rephrased in this picture
as a distillation procedure of GHZ-like states:
21) Péter Lévay, Stringy black holes and the geometry of
entanglement, available as arXiv:hep-th/0603136.
Péter Lévay, A three-qubit interpretation of BPS and non-BPS
STU black holes, available as arXiv:0708.2799.
Péter Lévay and Szilárd Szalay, The attractor mechanism as a
distillation procedure, available as arXiv:1004.2346.
The physical nature of these moduli, charge and warp factor
dependent "states" (or whether is it legitimate to call them
states at all) is, however, still unclear.
Independently of Duff and Ferrara during the summer of 2006 I worked
out the E7 tripartite entanglement of seven qubits picture. My
paper came out later because I have bogged down on some maths
details (I also wanted to see the E7 generators in the 56 dim rep.
acting as qubit gates - without success!) However, the idea that
this curious type of entanglement is related to the structure of
the Fano plane appears first in this paper:
22) Péter Lévay, Strings, black holes, the tripartite entanglement
of seven qubits and the Fano plane, available as
arXiv:hep-th/0610314.
Here I made the conjecture that Freudenthal systems and issues
concerning the magic square could be relevant to further
developments of the black hole qubit analogy, initiating the nice
study of Duff et al, culminating in their valuable Phys. Reports
paper.
In the meanwhile with my student we have shown that Freudenthal
systems are capable of giving hints for solving the classification
problem of entanglement classes for systems consisting of special
types of indistinguishable constituents (fermions and bosons). The
black hole entropy formulas of string theory and supergravity also
gave suggestions how to define sensible (tripartite) entanglement
measures for these systems:
23) Péter Lévay and Péter Vrana, Three fermions with six single
particle states can be entangled in two inequivalent ways,
available as arXiv:0806.4076.
Péter Lévay and Péter Vrana, Special entangled quantum systems
and the Freudenthal construction, available as arXiv:0902.2269.
Apart from these studies after realizing the relevance of finite
geometric ideas with a Slovak co-worker Metod Saniga we have shown
that the structure of the E6 and E7 invariants giving rise to black
hole entropy formulas are related to generalizations of Mermin
Square like configurations, and generalized polygons. Our studies
motivated partly the nice paper of van Geemen and B. L Cerchiai you
mention in your post ("From qubits to E7"). Since the finite
geometric structures relevant to the black hole qubit
correspondence turned out to be just geometric hyperplanes of
incidence geometries for n-qubit systems, we conducted a
mathematical study on the structure of such hyperplanes giving rise
to another incidence geometry: the Veldkamp space. The nice maths
results can be found in:
24) Péter Vrana and Péter Lévay, The Veldkamp space of multiple
qubits, available as arXiv:0906.3655.
Such notions as symplectic structure, quadratic forms,
transvections, and the group Sp(2n,2) connected to n-qubit
systems and black holes appear here. The language of this paper
is similar to the van Geemen paper.
Recently I have shown that certain solutions of the STU model
living in 4D and coming from one sector of the E7(7) invariant
N=8 SUGRA Duff mentions in his paper in connection to three qubits,
is really a model living in 3D coming from a coset of E8(8)
related to FOUR QUBITS. In this paper I have given hints that the
classification problem for four qubits can be translated to the
classification problem of black hole solutions (BPS and non BPS,
extremal and even possibly non extremal).
25) Péter Lévay, STU black holes as four qubit systems, available
as arXiv:1004.3639.
(The idea that four qubit systems show up in STU truncations
first appeared in my Fano-E7 paper. This is related to the
structure of a coset of SO(4,4), with triality making its debut
via permutation of the qubits.)
Based on the results of this paper the challenge to relate
the BH classification problem based on four qubit systems by
fitting together the existing results in the literature was
recently taken up by Duff et al.
I hope that these results add some useful hints to update the
picture on the black hole qubit correspondence.
I think that the main virtue of this field is that fascinating
maths (like octonions, Freudenthal systems, finite geometries
etc.) will finally makes its debut to understanding quantum
entanglement better. On the string theory side such studies
might initiate some new way of looking at existing results in
the field of stringy black hole solutions.
Of course finding the underlying physics (if any) is still out
there!
With best regards,
Peter Levay
Department of Theoretical Physics
Institute of Physics
Budapest University of Technology
HUNGARY
For more discussion, visit the n-Category Cafe at:
http://golem.ph.utexas.edu/category/2010/05/this_weeks_finds_in_mathematic_59.html
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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