December 17, 2000

This Week's Finds in Mathematical Physics (Week 162)

John Baez

Since the winter solstice is coming soon, I'll start with some gift suggestions... for the physicist who has everything.

1) The Universe Map, National Geographic Society, 2000, NSG #602011.

I've only seen a picture of this 20 x 31 inch map, but I know I want one! In a series of different 3d views, it shows the solar system, nearby stars, the Milky Way, the Local Group and the observable universe as a whole. I'll put it outside my office so my students can figure out just where they stand in the grand scheme of things.

2) Wil Tirion and Roger W. Sinnot, Sky Atlas 2000.0, 2nd edition, Cambridge U. Press, 1999.

This is a favorite sky atlas among amateur astronomers. It comes in lots of versions, but Kevin Kelly of Whole Earth says that the most useful is the "deluxe version, spiralbound".

3) Lee Smolin, Three Roads to Quantum Gravity, Weidenfeld and Nicholson, 2000.

This is a nontechnical guide to quantum gravity and the different approaches people have taken to this problem: string theory, loop quantum gravity, and the more radical lines of thought pursued by people whom Smolin calls "the true heroes of quantum gravity", like Alain Connes, David Finkelstein, Chris Isham, Roger Penrose and Raphael Sorkin. I haven't gotten ahold of this book, so I can't describe it in detail yet, but it should be lots of fun.

That's enough gift suggestions. Now I want to talk about Jordan algebras and how they show up in projective geometry, quantum logic, special relativity and so on. I'll start by reminding you of some stuff from "week106" and "week145". Then I'll charge ahead and show you how a Jordan algebra built from the octonions is related to 10-dimensional Minkowski spacetime....

Projective geometry is a venerable subject that has its origins in the study of perspective by Renaissance painters. As seen by the eye, any pair of parallel lines - e.g., train tracks - appear to meet at a "point at infinity". Furthermore, when you change your viewpoint, distances and angles appear to change, but points remain points and lines remain lines. This suggests a modification of Euclidean plane geometry based on a set of points, a set of lines, and relation whereby a point "lies on" a line, satisfying the following axioms:

A) For any two distinct points, there is a unique line on which they both lie.

B) For any two distinct lines, there is a unique point which lies on both of them.

C) There exist four points, no three of which lie on the same line.

D) There exist four lines, no three of which have the same point lying on them.

Any structure satisfying these axioms is called a "projective plane". But projective geometry is also interesting in higher dimensions. One can define a "projective space" by the following axioms:

A) For any two distinct points p and q, there is a unique line pq on which they both lie.

B) For any line, there are at least three points lying on this line.

C) If a,b,c,d are distinct points and there is a point lying on both ab and cd, then there is a point lying on both ac and bd.

Given a projective space and a set S of points in this space, we define the "span" of S to be the set of all points lying on lines ab where a,b are distinct points in S. The "dimension" of a projective space is defined to be one less than the smallest number of points that span the whole space. As you would hope, a 2-dimensional projective space is the same thing as a projective plane! It's a fun exercise to show this straight from the above axioms. If you give up, read this book:

4) Lynn E. Garner, An Outline of Projective Geometry, North Holland, New York, 1981.

How can we get our hands on some projective spaces? Well, if K is any field, there is an n-dimensional projective space called KPn where the points are lines through the origin in Kn+1, the lines are planes through the origin in Kn+1, and the relation of "lying on" is inclusion. The example relevant to perspective is the real projective plane, RP2. But it's good to follow Polya's advice:

"Be wise - generalize!"

and study KPn for any field and any n. In fact, we can define KPn even when K is a mere "skew field": a ring such that every nonzero element has a left and right multiplicative inverse. We just need to be a bit careful about defining lines and planes through the origin in Kn+1. To do this, we just take a line through the origin to be any set

L = {ax: a in K}

where x is nonzero element of Kn+1, and take a plane through the origin to be any set

P = {ax + by: a,b in K}

where x,y are elements of Kn+1 such that ax + by = 0 implies a and b are zero.

Around now, you might be wondering whether every projective n-space is of the form KPn for some skew field K. If so, you must have forgotten "week145", where I gave the answer: yes, but only if n > 2. Projective planes are more subtle! A projective plane comes from a skew field if and only if it satisfies an extra axiom, the "axiom of Desargues". I described this axiom in "week145" so I won't do it again here. The main point is that a projective plane coming from a skew field has some extra geometrical properties that a "non-Desarguesian" projective plane will not.

Projective geometry was very fashionable in the 1800s, with such worthies as Poncelet, Brianchon, Steiner and von Staudt making important contributions. Later it was overshadowed by other forms of geometry. However, work on the subject continued, and in 1933 Ruth Moufang constructed a remarkable example of a non-Desarguesian projective plane using the octonions:

5) Ruth Moufang, Alternativkoerper und der Satz vom vollstaendigen Vierseit, Abhandlungen Math. Sem. Hamburg 9, (1933), 207-222.

It turns out that this projective plane deserves the name OP2, where O stands for the octonions.

The 1930s also saw the rise of another reason for interest in projective geometry: quantum mechanics! Quantum theory is distressingly different from the classical Newtonian physics we have learnt to love. In classical mechanics, observables are described by real-valued functions. In quantum mechanics, they are often described by hermitian n x n complex matrices. In both cases, observables are closed under addition and multiplication by real scalars. However, in quantum mechanics, observables do not form an associative algebra. Still, one can raise an observable to any power, and from squaring one can define a commutative product:

x o y = [(x+y)2 - x2 - y2]/2 = (xy + yx)/2

This product is not associative, but it satisfies the weaker identity

x o (y o x2) = (x o y) o x2

In 1932, Pascual Jordan attempted to understand this situation better by isolating the bare minimum axioms that an "algebra of observables" should satisfy:

6) Pascual Jordan, Ueber eine Klasse nichtassociativer hyperkomplexer Algebren, Nachr. Ges. Wiss. Goettingen (1932), 569-575.

He invented the definition of what is now called a "formally real Jordan algebra": a commutative (but not necessarily associative) unital algebra over the real numbers such that:

x o (y o x2) = (x o y) o x2

and also:

a2 + b2 + c2 + ... = 0 implies a = b = c = ... = 0.

The last condition gives our algebra a partial ordering: if we say that x is "less than or equal to" y when the element y - x is a sum of squares, this condition says that if x is less than or equal to y and y is less than or equal to x, then x = y. If we drop this last condition, we get the definition of what is now called a "Jordan algebra".

In 1934, one year after Moufang published her paper on OP2, Jordan published a paper with von Neumann and Wigner classifying all formally real Jordan algebras:

7) Pascual Jordan, John von Neumann, Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29-64.

Their classification is nice and succinct. An "ideal" in the Jordan algebra A is a subspace B such that if b is in B, a o b lies in B for all a in A. A Jordan algebra A is "simple" if its only ideals are {0} and A itself. Every formally real Jordan algebra is a direct sum of simple ones. The simple formally real Jordan algebras consist of 4 infinite families and one exception:

This classification raises some obvious questions. Why does nature prefer the Jordan algebras hn(C) over all the rest? Or does it? Could the other Jordan algebras - even the exceptional one - have some role to play in quantum physics? Despite much research, these questions remain unanswered to this day.

The paper by Jordan, von Neumann and Wigner appears to have been uninfluenced by Moufang's discovery of OP2, but in fact the two are related! A "projection" in a formally real Jordan algebra is defined to be an element p with p2 = p. In the usual case of hn(C), these correspond to hermitian matrices with eigenvalues 0 and 1, so they are used to describe observables that assume only two values - e.g., "true" and "false".

This suggests treating projections in a formally real Jordan algebra as propositions in a kind of "quantum logic". The partial order helps us do this: given projections p and q, we say that p "implies" q if p is less than or equal to q. We can then go ahead and define "and", "or" and "not" in this context, and most of the familiar rules of Boolean logic continue to hold. However, we no longer have the distributive laws:

p and (q or r) = (p and q) or (p and r)

p or (q and r) = (p or r) and (q or r)

The failure of these distributive laws is the hallmark of quantum logic.

Now, the relation between Jordan algebras and quantum logic is already interesting in itself:

8) G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.

... but the real fun starts when we note that projections in the Jordan algebra of n x n self-adjoint complex matrices correspond to subspaces of Cn. This sets up a relationship to projective geometry, since the projections onto 1-dimensional subspaces correspond to points in CPn, while the projections onto 2-dimensional subspaces correspond to lines. Even better, we can work out the dimension of a subspace V from the corresponding projection p: Cn → V using only the partial order on projections: V has dimension d iff the longest chain of distinct projections

p0 < p1 < ... < pi = p

has length i = d. In fact, we can use this to define the "dimension" of any projection in any formally real Jordan algebra. We can then try to construct a projective space whose points are the 1-dimensional projections and whose lines are the 2-dimensional projections, with the relation of "lying on" given by the partial order in our Jordan algebra.

If we try this starting with the Jordan algebra of n x n self-adjoint matrices with real, complex or quaternionic entries, we succeed when n is 2 or more - and we obtain the projective spaces RPn, CPn and HPn, respectively. If we try this starting with the spin factor Rn + R we succeed when n is 2 or more - and we obtain a series of 1-dimensional projective spaces related to Minkowskian geometry, which I'll talk about in a minute. Finally, in 1949 Jordan discovered that if we try this construction starting with the exceptional Jordan algebra, we get the projective plane discovered by Ruth Moufang - OP2!

9) Pascual Jordan, Ueber eine nicht-desarguessche ebene projektive Geometrie, Abhandlungen Math. Sem. Hamburg 16 (1949), 74-76.

Physicists have tried for a long time to find some use for the quantum logic corresponding to the exceptional Jordan algebra. So far they have not succeeded. Jordan hoped this stuff would be related to nuclear physics. Feza Gursey and Murat Gunaydin hoped it was related to quarks, since 3 x 3 hermitian octonionic matrices should describe observables in some 3-state quantum system:

10) Murat Gunaydin and Feza Gursey, An octonionic representation of the Poincare group, Lett. Nuovo Cim. 6 (1973), 401-406.

11) Murat Gunaydin and Feza Gursey, Quark structure and octonions, Jour. Math. Phys. 14 (1973), 1615-1667.

12) Murat Gunaydin and Feza Gursey, Quark statistics and octonions, Phys. Rev. D9 (1974), 3387-3391.

13) Murat Gunaydin, Octonionic Hilbert spaces, the Poincare group and SU(3), Jour. Math. Phys. 17 (1976), 1875-1883.

14) M. Gunaydin, C. Piron and H. Ruegg, Moufang plane and octonionic quantum mechanics, Comm. Math. Phys. 61 (1978), 69-85.

Alas, these ideas never quite worked out, so most physicists discarded the exceptional Jordan algebra as a lost cause.

However, the exceptional Jordan algebra is secretly related to string theory, so there's a sense in which it's still lurking in the collective subconscious. Now, you probably want me to explain this, but I'm not ready to. So I won't say what 3 x 3 hermitian octonionic matrices have to do with string theory. If you want to know that, read these:

15) E. Corrigan and T. J. Hollowood, The exceptional Jordan algebra and the superstring, Commun. Math. Phys., 122 (1989), 393. Also available at

16) E. Corrigan and T. J. Hollowood, A string construction of a commutative nonassociative algebra related to the exceptional Jordan algebra, Phys. Lett. B203 (1988), 47.

17) G. Sierra, An application of the theories of Jordan algebras and Freudenthal triple systems to particles and strings, Class. Quant. Grav. 4 (1987), 227.

Instead, I'll just say what 2 x 2 hermitian octonionic matrices have to do with 10-dimensional Minkowski spacetime. Since superstrings live in 10 dimensions, that's at least a start.

First, we need to think about spin factors.

In case you forgot, spin factors were the fourth infinite family of simple formally real Jordan algebras on my list up there. I gave a lowbrow definition of these guys, but now let's try a highbrow one. Given an n-dimensional real inner product space V, the "spin factor" J(V) is the Jordan algebra generated by V with the relations

v o w = <v,w>

This should remind you of the definition of a Clifford algebra, and indeed, they're related - they have the same representations! This sets up a connection to spinors, which is why these Jordan algebras are called "spin factors".

But anyway: if you think about it a while, you'll see that J(V) is isomorphic to the direct sum V + R equipped with the product

(v,a) o (w,b) = (aw + bv, <v,w> + ab)

which is basically the lowbrow definition of a spin factor.

Though Jordan algebras were invented to study quantum mechanics, the spin factors are also deeply related to special relativity: we can think of J(V) = V + R as "Minkowski spacetime", with V as space and R as time. The reason is that J(V) is naturally equipped with a dot product:

(v,a) . (w,b) = <v,w> - ab

which is just the usual Minkowski metric in slight disguise. This makes it tempting to borrow an idea from special relativity and define the "lightcone" to consist of all nonzero x in J(V) with

x . x = 0

A 1-dimensional subspace of J(V) spanned by an element of the lightcone is called a "light ray", and the space of all light rays is called the "heavenly sphere" S(V). We can identify the heavenly sphere with the sphere of unit vectors in V, since every light ray is spanned by an element of J(V) of the form (v,1) where v is a unit vector in V.

What's the physical meaning of the heavenly sphere? Well, if you were a resident of the spacetime J(V) and gazed up at the sky at night, the stars would seem to lie on this sphere. If you took off in a spaceship and whizzed along at close to the speed of light, all the constellations would look distorted, but all angles would be preserved, since the Lorentz group acts as conformal transformations of the heavenly sphere.

Now, when V is at least 2-dimensional, we can build a projective space from J(V) using the construction I described for any simple formally real Jordan algebra. If we do this, what do we get?

Well, you can easily check that aside from the elements 0 and 1, all projections in J(V) are of the form p = (1/2)(v,1) where v is a unit vector in V. These projections will be the points of our projective space, but as we've seen, they also correspond to points of the heavenly sphere. So our projective space is really just the heavenly sphere! This is cool, because it means points on the heavenly sphere can also be thought of as propositions in a certain sort of quantum logic.

Now, what does this have to do with the exceptional Jordan algebra? Well, we have to sneak up carefully on this wild beast, so first let's think about a smaller Jordan algebra: the 2 x 2 hermitian octonionic matrices. In fact, we can kill four birds with one stone, and think about 2 x 2 hermitian matrices with entries in any n-dimensional normed division algebra, say K. There are not that many normed division algebras, so I really just mean:

the real numbers, R, if n = 1,

the complex numbers, C, if n = 2,

the quaternions, H, if n = 4,

the octonions, O, if n = 8.

The space h2(K) of hermitian 2 x 2 matrices with entries in K is a Jordan algebra with the product

x o y = (xy + yx)/2

Moreover, this Jordan algebra is secretly a spin factor! There is an isomorphism

f: h2(K) → J(K + R) = K + R + R

which sends the hermitian matrix

( a+b   k  )
(  k*  a-b )
to the element (k,b,a) in K + R + R.

Furthermore, the determinant of matrices in h2(K) is just the Minkowski metric in disguise, since the determinant of

( a+b   k  ) 
(  k*  a-b )
a2 - b2 - <k,k>

These facts have a number of nice consequences. First of all, since the Jordan algebras J(K + R) and h2(K) are isomorphic, so are their associated projective spaces. We have seen that the former space is the heavenly sphere S(K + R); unsurprisingly, the latter is the projective line KP1. It follows that these are the same! This shows that:

h2(R) is 3d Minkowski spacetime, and RP1 is the heavenly sphere S1;

h2(C) is 4d Minkowski spacetime, and CP1 is the heavenly sphere S2;

h2(H) is 6d Minkowski spacetime, and HP1 is the heavenly sphere S4;

h2(O) is 10d Minkowski spacetime, and OP1 is the heavenly sphere S8.

Secondly, it follows that the determinant-preserving linear transformations of h2(K) form a group isomorphic to O(n+1,1). How can we find some transformations of this sort? For K = R, it's easy: when g lies in SL(2,R) and x is in h2(R), we have gxg* in h2(R) again, and

det(gxg*) = det(x)

This gives a homomorphism from SL(2,R) to O(2,1). It's easy to see that this makes SL(2,R) into a double cover of the Lorentz group SO0(2,1). The exact same construction works for K = C, so SL(2,C) is a double cover of the Lorentz group SO0(3,1) - which you probably knew already, if you made it this far!

For the other two normed division algebras the above calculation involving determinants breaks down, and it even becomes tricky to define the group SL(2,K), so we'll start by working at the Lie algebra level. We say a 2 x 2 matrix with entries in the normed division algebra K is "traceless" if the sum of its diagonal entries is zero. Any such traceless matrix acts as a real-linear operator on K2. When K is commutative and associative, the space of operators coming from 2 x 2 traceless matrices with entries in K is closed under commutators, but otherwise it is not, so we'll define sl(2,K) to be the Lie algebra of operators on K2 generated by operators of this form. This Lie algebra in turn generates a Lie group of real-linear operators on K2, which we call SL(2,K).

Now, sl(2,K) has an obvious representation on K2, called the "fundamental representation". If we tensor this representation with its dual we get a representation of sl(2,K) on the space of 2 x 2 matrices with entries in K, which is given by

a: x |-> ax + xa*

whenever a is actually a 2 x 2 traceless matrix with entries in K. Since ax + xa* is hermitian whenever x is, this representation restricts to a representation of sl(2,K) on h2(K). This in turn gives a rep of the group SL(2,K). A little calculation at the Lie algebra level shows that this action of SL(2,K) on h2(K) preserves the determinant, so we have a homomorphism

SL(2,K) → SO0(n+1,1)

This is two-to-one and onto, so it follows pretty easily that:

SL(2,R) is the double cover of the Lorentz group SO0(2,1);

SL(2,C) is the double cover of the Lorentz group SO0(3,1);

SL(2,H) is the double cover of the Lorentz group SO0(5,1);

SL(2,O) is the double cover of the Lorentz group SO0(9,1).

and thus:

SL(2,R) acts as conformal transformations of the sphere S1 = RP1;

SL(2,C) acts as conformal transformations of the sphere S2 = CP1;

SL(2,H) acts as conformal transformations of the sphere S4 = HP1;

SL(2,O) acts as conformal transformations of the sphere S8 = OP1.

In the complex case, these conformal transformations are often called "Moebius transformations". For more on the octonionic case, try this:

15) Corinne A. Manogue and Tevian Dray, Octonionic Moebius transformations, Mod. Phys. Lett. A14 (1999) 1243-1256, available as math-ph/9905024.

To round off the story, it helps to bring in spinors:

16) Anthony Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, Jour. Phys. A17 (1984), 939-955.

The fundamental rep of SL(2,K) on K2 is secretly one of the spinor reps of the double cover of the Lorentz group SO0(n+1,1). Moreover, we can get points on the heavenly sphere from these spinors! This has been nicely explained by Penrose in the complex case, but it works the same way for the other normed division algebras. It goes like this:


|ψ> = (x,y) 

is a unit spinor, i.e. an element of K2 with norm one. Then
|ψ> <ψ| =    ( xx*   xy* )
                 ( yx*   yy* )
is a projection in h2(K) which is not 0 or 1 - or in other words, a point on the heavenly sphere. If we identify the heavenly sphere with KP1, this point corresponds to the line through the origin in K2 containing the spinor |ψ>.

To go further, I would want to say more about why this connection between quantum logic, Lorentzian geometry, and spinors is interesting, and what you can do with it. And then I would want to take everything we've seen about OP1 and h2(O) and see how it fits inside the bigger, more interesting story of OP2 and h3(O). But alas, I'm running out of steam here, so I'll just give you a little reading list about the octonionic projective plane and the exceptional Jordan algebra:

20) Hans Freudenthal, Zur ebenen Oktavengeometrie, Indag. Math. 15 (1953), 195-200.

Hans Freudenthal, Beziehungen der e7 und e8 zur Oktavenebene:

I, II, Indag. Math. 16 (1954), 218-230, 363-368.

III, IV, Indag. Math. 17 (1955), 151-157, 277-285.

V - IX, Indag. Math. 21 (1959), 165-201, 447-474.

X, XI, Indag. Math. 25 (1963) 453-471, 472-487.

Hans Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1 (1964), 145-190.

Hans Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie, Geom. Dedicata 19 (1985), 7-63.

21) Jacques Tits, Le plan projectif des octaves et les groupes de Lie exceptionnels, Bull. Acad. Roy. Belg. Sci. 39 (1953), 309-329.

Jacques Tits, Le plan projectif des octaves et les groupes exceptionnels E6 et E7, Bull. Acad. Roy. Belg. Sci. 40 (1954), 29-40.

22) Tonny A. Springer, The projective octave plane, I-II, Proc. Koninkl. Akad. Wetenschap. A63 (1960), 74-101.

Tonny A. Springer, On the geometric algebra of the octave planes, I-III, Proc. Koninkl. Akad. Wetenschap. A65 (1962), 413-451.

23) J. R. Faulkner and J. C. Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 (1977), 1-35.

Finally, for a really good overview of Jordan algebras and related things like "Jordan pairs" and "Jordan triple systems", try this:

24) Kevin McCrimmon, Jordan algebras and their applications, AMS Bulletin 84 (1978), 612-627.

© 2000 John Baez