John Baez is fascinated by the existence of four number systems, namely R,C,H,O, whose dimensions are 1,2,4,8. He's thinking about this mysterious stuff even when he tries to

learn M-theory (at the Distler-owned Category Café).Let me admit that it looks like he is being distracted by marginal and irrelevant stuff all the time. When Richard Feynman was trying to teach the electromagnetic induction to his artist friend (who has successfully taught Feynman how to make paintings), the friend screamed "it's just like f*cking" when a piece of metal was attracted inside the solenoid.

In a similar way, it seems that John Baez pays too much attention to marginal topics and wild speculations, including those that are not really the bulk of Mike Duff's review (even though Duff's review of SUGRA surely has more mysterious stuff in it than others). The classification to R,C,H,O is nice but one shouldn't forget that these four algebras have different basic properties.

Only the first one, R, is the "minimal building block" of the other algebras. Only the second one, C, allows us to solve every algebraic equation and find the right number of the solutions. The last two, H and O, are not commutative, and the very last one, O, is not even associative. That makes a difference. The classification to R,C,H,O shouldn't be thought of as being analogous to the ADE classification of the simply laced Lie groups. In the ADE case, we genuinely classify objects of the same kind. In the RCHO case, we gradually "complexify" the type of the things we're "classifying".

*Off-topic: Czech EU presidency has some issues with the Iranian presidential elections.*

The idea that many complex mathematical objects, such as p-branes, should be classified according to a simple R,C,H,O classification looks utterly childish to me. The octonions are the most complicated and discernible among the four division algebras and it is questionable whether they really appear in the places of physics where they're often claimed to be relevant.

When we talk about the octonions as a special division algebra, it's only special if we accept its multiplication rules. Without those rules, the 8-dimensional space is not too different from other 8-dimensional spaces. But with the rules, one immediately sees a G_2 automorphism group. Whenever one "genuinely" encounters the octonions, the correct symmetry - the automorphism group - must play a role, too. The opinion that E_8 is all about the octonions because of some "octo-octonionic planes" look both vague and morally wrong to me.

Let me ask: why doesn't John Baez actually try to learn what Duff is writing, instead of being constantly attracted to some possibly linked "big discoveries" that almost manifestly don't exist?

**Spinors in different dimensions, classical superstring and SYM theories**

There are many issues that Baez tries to present as very, almost religiously mysterious but that look completely mundane to me. One of them are possible representations of spinors in various dimensions.

Imagine that you want to create a minimal supersymmetric gauge theory. A massless gauge field in D dimensions has D-2 transverse, physical polarizations. A supersymmetric theory will add a spinor to it. It should have the same number of components. The dimensions of a spinorial representation depends on the spacetime dimension as well as the "reality/chirality" types of the representation.

In D-dimensional Minkowski spaces, the number of real components of the minimal spinor is proportional to 2^{D/2}, with additional factors of "2" if the representation is complex and "1/2" if it is chiral etc. If you want the number of bosonic degrees of freedom (D-2) of the gauge field to be matched by the spinorial components of the gauginos, the choices are:

- odd dimension, non-chiral spinors: D=3
- even dimension, complex spinors: D=4
- even dimension, pseudoreal spinors: D=6
- even dimension, real spinor: D=10

Note that in 3,4,6,10 dimensions, the gauge field has 1,2,4,8 polarizations. It can be matched to a power of two from the spinorial dimensions, with four solutions depending on your choice of the reality/chirality conditions. It's not mysterious at all.

For these dimensions, Green-Schwarz-like superstring theory (with vectorial bosonic and spinorial fermionic worldsheet fields) may exist classically - recall that an open superstring reduces to a supersymmetrized gauge field at low energies. However, quantum mechanically, the conformal anomaly must cancel and it cancels in D=10 only.

The appearance of four possibilities is less mysterious here than the reason why the list {R,C,H,O} has four entries: the four options simply represent the possible spinor types in D dimensions, including the odd-D case (and by the way, no spinor representations are really "octonionic").

Gravity and 11 dimensions

For these dimensions, Green-Schwarz-like superstring theory (with vectorial bosonic and spinorial fermionic worldsheet fields) may exist classically - recall that an open superstring reduces to a supersymmetrized gauge field at low energies. However, quantum mechanically, the conformal anomaly must cancel and it cancels in D=10 only.

The appearance of four possibilities is less mysterious here than the reason why the list {R,C,H,O} has four entries: the four options simply represent the possible spinor types in D dimensions, including the odd-D case (and by the way, no spinor representations are really "octonionic").

Gravity and 11 dimensions

The theory in 11 dimensions, M-theory, is different than the four choices classified above because its massless field content is not that of a gauge field but instead, it is a supergravity multiplet. If you require as many as 32 real supercharges, gravitons preserve 16 of them. They can be combined into 8 complexified raising/lowering operators, each of which is able to change the polarization of the spin by 1/2. Eight of them are able to climb by 8 times 1/2 which is by 4 - so the spin must be allowed to go from -2 to +2. Gravitons are the particles that have spin 2.

With so many supercharges, you therefore inevitably end up with a supergraviton multiplet. It has 128 bosonic and 128 fermionic components. The fermionic components coincide with a spin 3/2 gravitino in 11 dimensions while the 128 bosonic components can be split to (9 x 10 / 2 x 1 - 1) = 44 components of the graviton and (9 x 8 x 7 / 3 x 2 x 1) = 84 components of the three-form, an antisymmetric rank-three tensor that generalizes the electromagnetic field.

This decomposition can be derived by rather basic group theory. The field content of a supergraviton multiplet in 11 dimensions has no direct link to the octonions. Because John Baez is obsessed with the wrong idea that the biggest object in any classification of interesting mathematical structures must be linked to the octonions, he can't understand the proper derivation of the field content in 11 dimensions. And that's too bad.

In his new text, John Baez is also irritated by the fact that the 11-dimensional spinors have 32 components. Well, they really do. Any formulation of anything that obscures the 32 components of the spinors in 11 dimensions must inevitably obscure the Lorentz symmetry, too. The reduced number of fermionic components is a result of several physical steps. There may exist a description with 16 components that teaches us an important lesson but it's certainly not guaranteed that there is one.

I think it is simply a misconception to insist that the degrees of freedom should be eliminated from the scratch. Quite on the contrary, many important insights in physics - and mathematical physics! - only materialize when a sufficient amount of work is done. And there's nothing wrong with it. In some sense, we might say that the longer the path from the assumptions to the conclusions, the deeper a result in theoretical physics we deal with.

Another reason why John Baez only wants to work with 16 components of the spinor is that in his optics, such a modified 11-dimensional spacetime looks just like the exceptional Jordan algebra of 3x3 Hermitean octonionic matrices, the algebra whose automorphism group is an E_6. I have no idea why he thinks that this is the right mathematical structure for this case. There's no canonical E_6 symmetry in the infinite flat-space 11-dimensional M-theory, so there can't be any exceptional Jordan algebra, either. For me personally, this simple (failed) consistency checks of the group theory kills the idea and reclassifies it as pure numerology.

You know, your humble correspondent is also attracted by many things linked to the exceptional groups and their importance for the dualities, supergravity, and physics in general. However, when I look at at John Baez's reasoning, I can't fail to see that what he is trying to do is pseudoscience simply because he never eliminates hypotheses that don't work. In his methodology, the appearance of the adjective "exceptional" in any sentence (or an ad hoc link involving physics and exceptional groups) beats any rational argument.

I don't believe that physics in 11 dimensions can be described by a proper research of the octonions. The hypothesis rejected in previous sentence is kind of vague and it is hard to operationally define the criterion deciding whether it's correct or not. Nevertheless, I think that despite its vagueness, it's just morally wrong. The group theory doesn't work.

Many of these vague speculations about the character of various theories - speculations based on superficial similarities with the division algebras etc. - have been explicitly ruled out. These hypotheses were based on "prophesies" of the people who believed that they could immediately see the truth, without serious detailed calculations and arguments - except that they couldn't.

Most of the remaining hypotheses of this kind have led to no interesting results. The breakthroughs in theoretical physics were simply based on different ideas. There exists one more way to describe what I find wrong with the very framework of John Baez's reasoning about mathematical physics and the value of different ideas. He really dislikes insights and arguments that are based on physics: only arguments based on elementary parts of mathematics or geometry, those that people knew many decades ago, are "desirable" or "recommended" by him.

This whole structure of reasoning is profoundly counterproductive and "untrue" in some very deep sense. The most important insights about mathematics, physics, and their relationships are those that try to use some structures that were only known on one side - e.g. physical mechanisms - to learn something new on the other side - i.e. in mathematics. And it's always important to learn new things, instead of blindly believing that e.g. the four division algebras one learned when he was 12 must include all important facts about the natural science.

They certainly don't. And that's the memo.

**Positive role of exceptional groups in SUGRA**

As a bonus, let me add a few comments about the actual exceptional groups that appear in M-theory or supergravity. Supergravity theories with 32 supercharges have exceptional noncompact symmetry groups - such as E_{7(7)} for M-theory on a seven-torus. Their discrete groups are the U-duality groups respected by string/M-theory, after one appreciates the (moduli-dependent) quantization rules for the charges etc. A better understanding of the origin of these groups is clearly desirable.

Also, the brane charges in M-theory may be classified with the help of another copy of an E_8 group, as argued by Diaconescu, Moore, Witten. Their argumentation has been extremely solid and modest: in this sense, these Gentlemen are the antipodal points of John Baez. They only viewed the E_8 construction as a mathematical tool to study one mathematical problem: it is the appropriate tool because of the "mostly vanishing" homotopy groups of the E_8 group manifold. No speculation that had not been established was included. They didn't argue that this E_8 gauge field in the bulk had to be important for all of physics, although it is a possibility that an emotional physicist must immediately think about.

But this ambitious generalization may be a wrong idea. Maybe, there's just nothing else we can learn about physics in 11 dimensions by thinking about the E_8 gauge field in the bulk.

Finally, I am amazed by the mysterious duality and still spend some time with it. My idea is that an underlying "worldsheet for worldsheet" theory with a del Pezzo target space may be used to generate its scattering amplitudes.

In this picture, all the power laws for the tensions/masses of all supersymmetric p-dimensional objects in toroidal compactifications of M-theory are encoded in cohomologies of del Pezzo surfaces - whose E_k structure of the intersection numbers "mimicks" the U-duality group of M-theory on tori. Again, the question whether all physics of M-theory can be rephrased in some del Pezzo-friendly way, or whether the observations by Iqbal, Neitzke, and Vafa are pretty much the only thing where this "dictionary" may show its muscles (and this partial success could have been guaranteed by the E_k structure of the del Pezzo surfaces), remains to be seen.

At the same moment, I must say that I have already worked with lots of more specific tantalizing ideas of this kind that I consider "essentially ruled out" today.

Once upon a time, a poor university wanted to create a new cheap department. They wanted to establish a theoretical physics department because theoretical physicists only needed pens and paper. However, they ultimately founded a philosophy departments because theoretical physicists often need a trash bin, too. Sadly, I am not sure whether people such as John need it, too.

Moreover, there is an atmosphere discouraging scientists from providing negative evidence against these seemingly "ambitious" projects. Whenever a rational perspective is adopted, the rational people are seen as hurting the belief of these big believers and potential prophets. In principle, there can exist prophets who "guess" the right answers to all important questions at the very beginning.

The only thing I am sure about is that neither Lee Smolin nor Garrett Lisi (or John Baez, who is closer to the goal) can make it to the list of these supernatural beings.

And the very main selective mechanism of science is to avoid wrong ideas and focus on the promising ones, as judged by rational arguments. Whether an idea looks "religiously mysterious" or not just doesn't matter in science. Some people think that it does and they're often very hypocritical about the criteria, denouncing one idea (usually string theory) because it looks too "religious" to them while uncritically promoting another, even more mysterious theory for the very same reason.

That's very bad.

## snail feedback (1) :

For the case of super YM, your counting argument of components is surely a necessary condition to make the algebra close. However, if you read GSW you'll find, that at a more technical level a certain Fierz identity (with three antisymmetrized gammas) has to hold. Of course it does and you could leave it at this.

You can however note that in some light cone split of the 9+1 dimensions (in +,- and i directions) you only have an SO(8) manifest. But you can use a clever choice of basis where the gammas are given by the octonionic structure constants (a similiar construction holds in the allowed dimensions). In that basis it can be checked that the Fierz identity is noting but the requirement of alternativity of a division algebra.

So you can argue that the division algebras are behind the spinor magic that makes the susy algebra close.

Post a Comment