Also available at http://math.ucr.edu/home/baez/week279.html
September 5, 2009
This Week's Finds in Mathematical Physics (Week 279)
John Baez
I just finished a paper with John Huerta, a math grad student who
really likes particle physics:
1) John Baez and John Huerta, Division algebras and supersymmetry,
available as arXiv:0909.0551.
You can think of this paper as our sequel to:
2) John Baez, OP^1 and Lorentzian geometry,
http://math.ucr.edu/home/baez/octonions/node11.html
A "normed division algebra" is an algebraic gadget where you can add,
multiply, subtract, and divide, satisfying all the usual laws *except*
the commutative and associative laws for multiplication, and
where every element has an "absolute value" or "norm" satisfying
the usual rules, including most notably:
|xy| = |x| |y|
The most popular example is the real numbers. The second most popular
example is the complex numbers. Then comes the quaternions, which are
noncommutative... and then, trailing in a distant fourth place, comes
the octonions, which are noncommutative and nonassociative.
Our paper aims to give a clear and self-contained treatment of the
amazing relation between normed division algebras and supersymmetric
Yang-Mills theory. Let me explain the basic idea! I'll cut some
corners, but you can see all the details in our paper.
Suppose K is a normed division algebra of dimension n. There
are just four choices:
K = R, the real numbers. n = 1
K = C, the complex numbers. n = 2
K = H, the quaternions. n = 4
K = O, the octonions. n = 8
We get all of these by taking the real numbers and throwing in
square roots of minus 1. So, any guy in K has a "real part"
and an "imaginary part" - and we can "conjugate" it by switching
the sign of its imaginary part.
This means we can talk about hermitian matrices with entries
in K: that is, matrices that stay the same when you transpose
them and then conjugate each entry. Let's use h_2(K) to mean
the set of hermitian 2 x 2 matrices with entries in K.
Then a nice thing happens: h_2(K) is the same as (n+2)-dimensional
Minkowski spacetime! To see this, note that any guy in h_2(K) has
this form:
A = t+x y
y* t-x
where t and x are real elements of K, and y is an arbitrary element.
Since K has dimension n, h_2(K) has dimension n+2. And check out
its determinant:
det(A) = t^2 - x^2 - yy*
Note that yy* = y*y = |y|^2, just as in the complex numbers. So,
det(A) is a Minkowski metric with one positive or "timelike" direction,
namely t, and n+1 negative or "spacelike" directions, namely x and the
n components of y.
So:
h_2(R) is 3-dimensional Minkowski spacetime.
h_2(C) is 4-dimensional Minkowski spacetime.
h_2(H) is 6-dimensional Minkowski spacetime.
h_2(O) is 10-dimensional Minkowski spacetime.
And - lo and behold! - these are just the dimensions where classical
superstring theory and super-Yang-Mills theory work best!
More precisely, these are the dimensions where you can write down
the Lagrangian for the "Green-Schwarz superstring" and "pure
super-Yang-Mills theory". There are fancier tricks that give
superstring theories and super-Yang-Mills theories in other
dimensions, but these are mainly offshoots of the four cases listed
here.
So now we have to ask: why do these supersymmetric theories feel so
happy when spacetime is secretly h_2(K)?
Well, supersymmetry is a kind of symmetry that mixes bosons and
fermions. In the simple cases I'm talking about, this means mixing
vectors and spinors. Since vectors are the same as points in
Minkowski spacetime - once we pick an origin - vectors in dimensions
3, 4, 6, or 10 are nicely described by elements of h_2(K).
And it turns out that supersymmetry works well in these dimensions
because we can also describe *spinors* using K. A spinor consists
of 2 guys in K: in other words, an element of K^2.
Indeed, if you've studied physics, you may know that in 4d Minkowski
spacetime, where we apparently live, we use C^2 to describe spinors.
I talked quite a bit about this example and also the example of
3d Minkowski spacetime back in "week196". So go there if you want
more details. For now what matters is this:
R^2 is the space of spinors in 3-dimensional Minkowski spacetime.
C^2 is the space of spinors in 4-dimensional Minkowski spacetime.
H^2 is the space of spinors in 6-dimensional Minkowski spacetime.
O^2 is the space of spinors in 10-dimensional Minkowski spacetime.
This is a bit oversimplified, because physicists use various kinds
of spinors, and I'm not saying which kinds show up here. But I
explained all these kinds back in "week93", and I don't want to
distract you with that here. I'll say more about it later.
Now, from what we've seen so far, there's an obvious way to take a
vector and a spinor and get a new spinor. This is what matrices were
born for! Just take your matrix in h_2(K), multiply your spinor in
K^2 by that matrix, and you get a new spinor in K^2.
In fact, we see this process at work whenever an electron absorbs
a photon. Quite literally, we *see* it - because that's how our eyes
work! A photon is described by a vector, an electron is described
by a spinor, and part of the math involved when an electron absorbs
a photon is this business of matrix multiplication.
Physicists would draw this operation using a Feynman diagram where a
wiggly line (the vector) and a straight one (the spinor) come in and
a straight one goes out:
/
/
/
~~~~~~~~~~
\
\
\
Mathematicians would write it as the operation that takes A in h_2(K)
and psi in K^2 and multiplies them to get A psi in h_2(K).
Now, one cool thing about Feynman diagrams is that you can turn
them around and read them in different ways, and they still make
sense. So as soon as we have a process where a spinor absorbs a vector,
we also get a process where two spinors collide and form a vector:
\ /
\ /
\ /
\ /
S
S
S
S
S
Now what is this process, mathematically speaking? Well, it's some
operation that takes two spinors, say psi in K^2 and phi in K^2, and
creates a vector in h_2(K) that I'll call psi . phi. If you want the
explicit formula for this operation, read our paper.
So, we've got an operation that takes a vector and a spinor and
creates a spinor, and an operation that takes two spinors and creates
a vector. Actually, these operations exist for *any* dimension of
spacetime! In general we need to describe them using the language of
Clifford algebras. Only in dimensions 3, 4, 6 and 10 can we describe
them using the language of normed division algebras, as I've done
here.
But it's this special language that gives the prettiest explanation of
a certain astounding fact. Supersymmetry for the Green-Schwarz
superstring and pure super-Yang-Mills theory relies on a special
identity which is true only in dimensions 3, 4, 6 and 10:
(psi . psi) psi = 0
This is an example of a "Fierz identity". These equations show up
whenever you work with spinors, and they should probably be called
"fierce identities", because they can be pretty scary. In particular,
it's a bit scary how some of them - like this one - hold only in
certain special dimensions.
But this particular one has a beautiful proof in terms of normed
division algebras! It follows from a special property shared by these
algebras: they're all "alternative". In other words, the "associator"
[x,y,z] = (xy)z - x(yz)
changes sign whenever we switch any of the two variables. The
associator is just zero for R, C, and H, since these algebras are
associative. So the only really interesting case is the octonions,
which are not associative, but still alternative. And this is
the case that matters for superstring theory in 10 dimensions!
Anyway, what our paper does is describe the basic operations involving
spinors and vectors using the normed division algebras, then use
this description to prove the identity
(psi . psi) psi = 0,
and then explain how this identity is crucial in supersymmetric
Yang-Mills theory. None of this is particularly new! What's new, we
hope, is that we explain everything in one place, in a way that people
who don't know about division algebras or supersymmetry can follow.
Some of the proofs use a little Clifford algebra technology, but most
of them amount to simple calculations.
Now let me tell you a tiny bit about the history of this subject, with
references. I would love to hear more details from people who were
around at the time. As far as I can tell, this is the first paper that
explained super-Yang-Mills theory and why it only works in dimensions
3, 4, 6 and 10:
3) Lars Brink, John Schwarz and Joel Scherk, Supersymmetric Yang-Mills
theory, Nucl. Phys. B121 (1977), 77-92.
Well, actually 2, 4, 6, and 10, but never mind! This is the paper
that did it for superstring theory:
4) Michael Green and John Schwarz, Covariant description of
superstrings, Phys. Lett. B136 (1984), 367-370.
The bible of string theory contains proofs of these facts, based on
the properties of Clifford algebras in various dimensions:
5) Michael B. Green, John H. Schwarz and Edward Witten, Superstring
Theory, Volume 1, Cambridge U. Press, Cambridge, 1987. Appendix 4.A,
Super Yang-Mills Theory, pages 244-247. Section 5.1.2: The supersymmetric
string action, pp. 253-255.
Back in 1983, Kugo and Townsend showed how spinors in dimension 3, 4,
6, and 10 get special properties from the normed division algebras.
They formulated a supersymmetric model in 6 dimensions using the
quaternions, and speculated about a similar formalism in 10 dimensions
using the octonions:
6) Taichiro Kugo and Paul Townsend, Supersymmetry and the division algebras,
Nucl. Phys. B221 (1983), 357-380. Also available at
http://ccdb4fs.kek.jp/cgi-bin/img_index?198301032
Later, Evans showed that supersymmetry of nonabelian Yang-Mills fields
coupled to massless spinors in (n+2)-dimensional spacetime implies the
existence of a normed division algebra of dimension n:
7) J. M. Evans, Supersymmetric Yang-Mills theories and division
algebras, Nucl. Phys. B298 (1988), 92-108. Also available at
http://ccdb4fs.kek.jp/cgi-bin/img_index?8801412
Shortly after Kugo and Townsend's work, Tony Sudbery used division
algebras to construct vectors, spinors and Lorentz groups in Minkowski
spacetimes of dimensions 3, 4, 6, and 10:
8) Anthony Sudbery, Division algebras, (pseudo)orthogonal groups
and spinors, Jour. Phys. A17 (1984), 939-955.
He then refined his construction with the help of a grad student:
9) Kwok-Wai Chung and Anthony Sudbery, Octonions and the Lorentz and
conformal groups of ten-dimensional space-time, Phys. Lett. B198
(1987), 161-164.
Tony Sudbery and Corinne Manogue then used these ideas to give an
octonionic proof that (psi . psi) psi = 0.
10) Corinne Manogue and Anthony Sudbery, General solutions of
covariant superstring equations of motion, Phys. Rev. D12 (1989),
4073-4077.
Together with her husband and Jason Janesky, Manogue later simplified
the proof, and our argument is based on theirs:
11) Tevian Dray, Jason Janesky and Corinne Manogue, Octonionic
hermitian matrices with non-real eigenvalues, Adv. Appl. Clifford
Algebras 10 (2000), 193-216. Appendix B, The 3-Psi's rule.
Also available as arXiv:math/0006069.
They give a little more history of this wondrous identity. They say
that Manogue's student Schray called it "the 3-Psi's rule" in his
hard-to-obtain thesis:
12) Jörg Schray, Octonions and supersymmetry, Ph.D. thesis, Department
of Physics, Oregon State University, 1994.
and also here:
13) Jörg Schray, The general classical solution of the superparticle,
Class. Quant. Grav. 13 (1996), 27-38. Also available as arXiv:hep-th/9407045.
They also write:
It is well-known that the 3-Psi's rule holds for Majorana spinors in
3 dimensions, Majorana or Weyl spinors in 4 dimensions, Weyl spinors
in 6 dimensions and Majorana-Weyl spinors in 10 dimensions. Thus,
the Green-Schwarz superstring exists only in those cases. As was
shown by Fairlie and Manogue, the 3-Psi's rule in all these cases
is equivalent to an identity on the gamma-matrices, which holds
automatically for the natural representation of the gamma-matrices
in terms of the 4 division algebras R, C, H and O, corresponding
precisely to the above 4 types of spinors. Manogue and Sudbery
then showed how to rewrite these spinor expressions in terms of 2x2
matrices over the appropriate division algebra, thus eliminating
the gamma-matrices completely.
Now I feel like explaining all this Majorana/Weyl business a bit
better - leaving many details to "week93".
First I should admit, for the nonexperts, that I've committed a few
sins of oversimplification for the sake of a nice clean story line.
For starters, remember how I said that the absorption of a photon by
an electron:
/
/
/
~~~~~~~~~~
\
\
\
corresponds to the operation where we take a guy in h_2(K) and a guy
in K^2 and multiply them to get a guy in h_2(K)?
In saying this, I was ignoring everything about energy and momentum,
and focusing on the "spin" aspect of this absorption process. It's
only the "spin", or intrinsic angular momentum, of a photon that's
described by an element of h_2(K) - with K = C, since we live in
4-dimensional spacetime. And it's only the spin of the electron
that's described by an element of K^2.
But it's even worse than that. In 4-dimensional spacetime, spinors
come in left- and right-handed forms. For example, the neutrinos we
most easily see - not that easily, actually! - are left-handed
spinors, while antineutrinos are right-handed. Electrons come in both
left and right-handed forms, so we actually describe them using C^2 +
C^2 = C^4. We call C^4 the space of "Dirac spinors", and we call the
two pieces left- and right-handed "Weyl spinors".
Similar but subtly different things happen in other dimensions. As
far as our division algebras story goes, the crucial fact is that
besides the "obvious" way for an element of h_2(K) to act on K^2,
there is a less obvious way that involves the "traced-reversed"
form of a 2 x 2 hermitian matrix:
A~ = A - tr(A)
where the trace tr(A) is the sum of the diagonal entries. We get
one kind of spinors, say
S+ = K^2
upon which the vectors
V = h_2(K)
act in the obvious way, and another kind of spinors, say
S- = K^2
in which vectors act in a nonobvious way. As vector spaces S+ and
S- are the same - but they differ in how vectors act on them, and we
should think of this action as interchanging the two kinds of spinor.
Here are the formulas:
V tensor S+ -> S-
A tensor psi |-> A psi
V tensor S- -> S+
A tensor psi |-> A~ psi
These actions fit together to yield a Clifford algebra action on
the direct sum of S+ and S-, since
A A~ = A~ A = -det(A)
and the determinant is related to the metric on Minkowski spacetime,
so these are the Clifford algebra relations in deep disguise.
What all this really amounts to depends a lot on which of the four
division algebras we're looking at! Sometimes S+ and S- are secretly
isomorphic, sometimes not. They always start out being real vector spaces,
since as vector spaces they're just K^2, and the only uniform way to think
of all four normed division algebras is as *real* vector spaces. But
sometimes S+ and S- admit Lorentz-invariant complex structures, so we
can think of them as complex vector spaces!
(By "Lorentz-invariant" I really mean invariant under the action of
the *double cover* of the Lorentz group. For brevity, let's just call
this the Lorentz group.)
In fact, each of the four cases has its own unique personality, with
the 4d case being the weirdest - you might call it a "split personality".
Let me just summarize the facts, without much explanation. This is one
of those things where I write stuff down so I can forget it and look it
up later:
If K = R, we're in 3d Minkowski spacetime. Then S+ and S- are
isomorphic as real representations of the Lorentz group - so it's not
important to distinguish them. The secret reason for this is that R
is commutative. Since S+ = S- does not have an invariant complex
structure, we call elements of this space "Majorana" spinors, which is
the name for real spinors that don't have a particular handedness.
If K = C, we're in 4d Minkowski spacetime. Then S+ and S- are
isomorphic as real representations of the Lorentz group - so it's not
important to distinguish them. The secret reason for this is that C
is commutative. If we treat S+ = S- as a real vector space, we call
elements of this space "Majorana" spinors, which is the name for real
spinors that don't have a particular handedness.
But in fact this real vector space has two invariant complex structures,
and the resulting complex representations are *not* isomorphic! If we
think of S+ and S- as two nonisomorphic complex representations we
call their elements left- and right-handed "Weyl" spinors,
respectively - since that's the name for complex spinors that do have
a particular handedness.
If K = H, we're in 6d Minkowski spacetime. Then S+ and S- are not
isomorphic as real representations of the Lorentz group - so it's
important to distinguish them. The secret reason for this is that H
is not commutative. Furthermore, S+ and S- admit invariant complex
structures. If we think of S+ and S- as complex representations we
call their elements left- and right-handed "Weyl" spinors, respectively
- since that's the name for complex spinors that do have a particular
handedness.
If K = O we're in 10d Minkowski spacetime. Then S+ and S- are not
isomorphic as real representations of the Lorentz group - so it's
important to distinguish them. The secret reason for this is that O
is not commutative. Furthermore, S+ and S- do not admit invariant
complex structures. So, we must think of S+ and S- as real
representations, and we call their elements left- and right-handed "
Majorana-Weyl" spinors, respectively - since that's the name for
real spinors that do have a particular handedness.
Wow, I bet that was thrilling!
Now that I'm done with this paper, my life has undergone a phase
change. I've been finishing a lot of old papers for the last 2 years.
This is the last of that batch - and the least old. It comes as an
incredible relief. Working on old projects is tiring, especially when
you have new things you're dying to think about. I feel like I've
been way behind myself, running to catch up... but this week I finally
caught up and ran past myself! It's a strange sensation.
Adding to this strange sensation, I just got word that I'm free to
take leave from UCR and visit Singapore for a year, starting in July
2010. I'll be working at the Centre for Quantum Technologies. That
should be a great adventure.
So, I'm feeling peppy, and I'm dying to tell you about all sorts of
new cool stuff: Stirling numbers and the Poisson operad, stacks and
noncommutative geometry, Adams operations and Galois theory, toric
varieties, octonions and rolling balls, the windmill powered by light,
and the symplectic geometry of electrical circuits. Each of these
deserves a whole Week, but we'll see.
For now, here are a few cool things I *won't* tell you much about,
because I don't understand them well enough. First, as pointed out to
me by Mike Stay:
14) Generalized continued fractions and equal temperament,
Doctroidal Dissertations, April 13th, 2009,
http://doctroid.wordpress.com/2009/04/13/
This starts with the old problem of trying to find a number x such
that x^n = 2 and x^k is almost 3/2. In music jargon, this is called
"finding an equal tempered scale that has a good fifth". In math
jargon, it amounts to finding a good rational approximation to
log(3/2)/log(2) ~ 0.584962501
The theory of continued fractions gives us candidates:
0/1 = 0.000000
1/1 = 1.000000
1/2 = 0.500000
3/5 = 0.600000
7/12 = 0.583333
24/41 = 0.585366
31/53 = 0.584906
and the first really good one is 7/12. Maybe that's why we divide
the scale into 12 equal parts, with the the 7th one up playing a
crucial role!
But what if you want a scale that has a good fifth and something else
too, like a good third? That's where "generalized" continued fractions
come in! I won't tell you what those are.
I also won't tell you about the new revolution linking logic to
weak infinity-groupoids. For that you'll have to read these:
15) Martin Hofmann and Thomas Streicher, The groupoid interpretation
of type theory, in Sambin, Giovanni, et al, Twenty-five years of
constructive type theory, Clarendon Press, Oxford, 1998, pp. 83-111.
Also available at
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.606
16) Steve Awodey and Michael A. Warren, Homotopy theoretic models of
identity types, available as arXiv:0709.0248.
17) Steve Awodey, Pieter Hofstra, Michael A. Warren,
Martin-Löf Complexes, available as arXiv:0906.4521.
18) Benno van den Berg and Richard Garner, Types are weak
omega-groupoids, available at
http://www.dpmms.cam.ac.uk/~rhgg2/Typesom/Typesom.pdf
----------------------------------------------------------------------
Quote of the Week:
A creation of importance can only be produced when its author isolates
himself, it is a child of solitude. - Johann Wolfgang von Goethe
-----------------------------------------------------------------------
Addendum: Francesco Toppan writes:
In relation with your recent, interesting, arxiv:0909.0551 paper I
would like to signal that division algebras also appear in the
N-extended supersymmetric quantum mechanics (in one dimension) for
N=1,2,4,8. This is hardly surprising, of course (arXiv:hep-th/0109073
NPB Pr. Sup.). Perhaps slightly more surprising is the fact that the
octonionic structure constants enter, as coupling constants, N=8
invariant actions, like e.g. the (1,8,7) model of arXiv:hep-th/0511274
(also in JHEP). In this example the 7 auxiliary fields can be
associated with the 7 imaginary octonions, preserving the "octonionic
covariance". I should add that the representations of N-extended 1D
superalgebra are mathematically very interesting and quite
intricate. In the last few years basically two groups, my group and
the group of S. Gates and his collaborators, worked out with
complementary viewpoints and results several features of these
representations: total number of fields, grading of the fields, graph
interpretation, connectivity of the graphs, etc. etc. Basic
references can be found in hep-th/0010135 (in JMP), hep-th/0610180
(PoS), or typing my name (and Gates'name) in the arXiv. Perhaps you
could find these interesting to have a look at.
For more discussion visit the n-Category Cafe at:
http://golem.ph.utexas.edu/category/2009/09/this_weeks_finds_in_mathematic_40.html
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