“Some good/bad news about string theory:

Good: String theory is a consistent theory of quantum gravity. Bad: It’s really a generator of an infinite number of mostly disconnected theories of quantum gravity, each around a different ground state. No background independent truely off-shell formulation of string theory is known (yet).

Good: String theory is unique, i.e. there is only one distinct consistent theory of “fundamental” strings. Bad: It has an infinite number of continously connected ground states plus a google of discrete ones. There appears to be no vacuum selection principle, other than the stability of supersymmetric vacua, which gives the wrong answer.

Good: String theory gives you chiral gauge theories, with big gauge groups, for free and complicated flavor structure at low energies is mapped into the geometry of extra dimensions. Bad: Doesn’t like to give the standard model as the low energy theory. A “typical” string compactification is either much simpler (with more SUSY and bigger gauge groups) or much more complicated (lot’s of extra exotic matter, extra U(1) gauge groups, etc.).

Good: String theory predicts supersymmetry and extra dimensions of space. Bad: It’s happy to hide the both up at the Planck scale.

Good: No length or energy scales are put in by hand; all scales should be determined diynamically. Bad: Appears to be too many (hundreds!) scalar fields (moduli) with too much SUSY to get determined dynamically; may be forced to appeal to cosmic initial conditions (the Landscape).

Good: String theory gives a microphysical description of (at least some) black holes, resolves their singularities. Bad: Doesn’t seen to resolve the singularity of the Big Bang (good for inflation, though).

Good: Lots of powerful dualities including weak ↔ strong coupling dualities and short ↔ long distance dualities. Bad: Can’t tell what are the “fundamental” degrees of freedom. String theory not necessarily a theory of strings.

Good: Unification of all the forces is almost for free, may need an (interesting) extra dimensional assist. Bad: In our most realistic string constructions so far, SU(3)_C, SU(2)_W, and U(1)_Y have essentially nothing to do with each other: related to different features of complicated D-brane setups.

Good: AdS/CFT duality shows that 10-dimensional string theory in a certain background is equivalent to a 4-dimensional gauge theory!! Use this e.g. to show that RHIC QCD physics maps onto quantum gravity/black holes. Bad: Adds more confusion: can’t tell an extra dimension apart from technicolor.

Good: We are starting to use string theory to learn tricks for perturbative QCD, understanding the QCD string, etc. Bad: The QCD community was already doing fine, thank you.

That’s all folks!”

The body of evidence is quite suggestive: “DAMA/LIBRA, CoGeNT, and CRESST have each reported signals which are inconsistent with known backgrounds, and (roughly) consistent with the elastic scattering of ~5-10 GeV dark matter particles; the spectrum of gamma rays from the region surrounding the Galactic Center peaks at a few GeV, consistent with a ~7-10 GeV dark matter particle annihilating largely to leptons, with a cross section on the order of that predicted by relic abundance considerations.”

However, “the case is not yet incontrovertible.”

“String theory has not yet been able to make contact with experiments that would give us strong reasons to accept it as the ‘sure winner’ in the race to construct a theory of quantum gravity. However, though experiment can often function as a decisive arbiter in situations where there are several competing theories, there are many more theoretical virtues that play a role in our evaluation of theories. Taking these extraexperimental factors into account, string theory is very virtuous indeed, it is arguably the most mathematically fertile theory of the past century or so. I would go further and say that no direct experiment is likely to ever come about (other than ones that could be explained by multiple approaches), so we can assume that non-experimental factors will have to be relied upon more strongly in our assessments of future research in fundamental physics.”

Why is there an arrow of time? A common idea is that the arrow of time is cosmologically defined by the macroscopic increase of entropy (the second law of thermodynamics). But this raises the associated question of why the universe starts in a low entropy state. This approach also suggests that the notion of time is inherently connected to the coarse graining of an underlying quantum gravitational configuration space.

Why is there only one time? Geometrically, time is different from space because the geometry of spacetime is locally Minkowski (Lorentzian metric signature (1, 3)), not Euclidean (metric signature (0, 4)). From a geometrical point of view we could equally well imagine a signature (2, 2), with two times, which is more symmetric between space and time. In the context of string theory with its many extra dimensions one can ask why we seem to have extra spatial dimensions, not temporal dimensions.

Is there a connection between the existence of a time, and the quantumness of the universe? The difference between time and space is somehow implicated in the difference between quantum mechanics with its characteristic features of quantum interference and entanglement, and classical statistical physics which lacks these features. This kind of difference appears in nonrelativistic quantum mechanics, in quantum field theory, and even in string theory.

Could the real, Lorentzian structure of conventional spacetime be simply a convenient way of summarizing analytic information about an underlying complexified geometry? Physical quantities seem to be described by analytic functions of space and time in both quantum field theory and string theory. 

How can singularities localized in time be resolved in string theory or some other quantum theory of gravity? A prediction of General Relativity is that spacetime singularities exist, either timelike (i.e. localized in space), lightlike (i.e. localized on a null curve), or spacelike (i.e. localized in time). One of the goals of a quantum theory of gravity such as string theory is to resolve such singularities.

Why is the area of a horizon, a causal construct, related to entropy, a thermodynamic concept, and can this entropy be given a statistical explanation for general horizons? Semiclassical analyses of quantum mechanics in spacetimes containing horizons like black holes and accelerating geometries such as de Sitter space suggest that inertial observers perceive the horizon as having an entropy proportional to area and a temperature proportional to the surface gravity at the horizon. Neither is there any explanation of why entropy becomes associated to a geometrical construct – the area of a horizon.

How precisely is physics beyond a black hole horizon encoded in a unitary description of spacetime? The “information loss paradox” for black holes is due to the non-unitary semiclassical evolution of quantum states in Hawking radiation. The apparent loss of unitarity can be traced ultimately to the causal disconnection of the region behind the horizon. A solution is required since there is simply no room in the full quantum theory for information loss in black holes.

Can time be emergent from the dynamics of a timeless theory? In  the AdS/CFT correspondence, string theory in a (d+1)-dimensional, asymptotically Anti-de Sitter (AdS) spacetime is exactly equivalent to a d-dimensional quantum field theory defined on the timelike boundary of such a universe. Thus, the radial dimension of AdS spacetime (as well as any additional compact dimensions of the bulk string theory) must be regarded as somehow “emergent” from the dynamics of the d-dimensional field theory. The field theory contains a time and the emergent gravitational theory inherits its time directly from the field theory. 

Are time and space concepts that only become effective in “phases” where the primordial degrees of freedom self-organize with appropriate relations of conditional dependence and entanglement? The spacetime and its metric are generally be thought of as a coarse-grained description of some underlying degrees of freedom which may, or may not, be organized with the proximity and continuity relations associated to smooth spacetime. The spacetime can be viewed as an emergent description of relations of conditional dependence of underlying fundamental variables.

If you have enjoyed the questions, please refer to the paper “What we don’t know about  time” for possible lines of research in order to obtain the answers in string theory.

“In 2007 physicist A. Garrett Lisi wrote the most talked about theoretical physics paper of the year. He argues that the geometric framework of modern quantum physics can be extended to incorporate Einstein’s theory, leading to a long-sought unification of physics” based on a geometrical object referred to as the exceptional Lie Group E8. Lisi, the surfer physicist, has a Midas touch in Mathematical Physics. Everybody talks about his great achievements, even if they are criticized by the mainstream. In the December 2010 issue of Scientific American appears an 8-page article entitled “A Geometric Theory of Everything,” written with James Owen Weatherall. Let us extract some paragraphs from the paper.

“The current best theory of nongravitational forces—the electromagnetic, weak and strong nuclear force—was largely completed by the 1970s and has become familiar as the Standard Model of particle physics. Mathematically, the theory describes theseforces and particles as the dynamics of elegant geometric objects called Lie groups and fiber bundles. Over the years physicists have proposed various Grand Unified Theories, or GUTs, in which a single geometric object would explain all these forces, but no one yet knows which, if any, of these theories is true. In Lisi’s theory a single geometric object unifies all forces and matter into a single geometric object.

The main geometric idea underlying the Standard Model is that every point in our spacetime has shapes attached to it, called fibers, each corresponding to a different kind of particle. The entire geometric object is called a fiber bundle. The fibers are in internal spaces corresponding to particles’ properties. This idea was introduced by  Hermann Weyl in 1918 for the unification of gravity and electromagnetism. The electric and magnetic fields existing everywhere in our space are the result of fibers with the simplest shape: the circle, called U(1) by physicists, the simplest example of a Lie group. The fiber bundle of electromagnetism consists of circles attached to every point of spacetime. An electromagnetic wave is the undulation of circles over spacetime. Photons and electrons have different fiber bundles over spacetime. The fibers of electrons wrap around the circular fibers of electromagnetism like threads around a screw. Because twists must meet around the circle, these charges are integer multiples of some standard unit of electric charge.

Physicists apply these same principles to the weak and strong nuclear forces. Each of these forces has its own kind of charge and its own propagating particles. They are described by more complicated fibers, made up not just of a single circle but of sets of intersecting circles, interacting with themselves and with matter according to their twists. The weak force is associated with a three-dimensional Lie group fiber called SU(2). Its shape has three symmetry generators, corresponding to the three weak-force boson particles: W⁺, W^– and W₃. Matter particles, fermions, come in two varieties, related to how their spin aligns with their momentum: left-handed and right-handed. Only the left-handed fermions have weak charges, with the left-handed up quark and neutrino having weak charge +1/2 and the left-handed down quark and electron having weak charge –1/2. For antiparticles, this is reversed. Our universe is not left-right symmetrical, one of many mysteries a unified theory seeks to explain.

Electroweak force unifies the weak force with electromagnetism by combining the SU(2) fiber with a U(1) circle. This circle is not the same as the electromagnetic one; it represents a precursor to electromagnetism known as the hypercharge force, with particles twisting around it according to their hypercharge, labeled Y. The W₃ circles combine with the hypercharge circles to form a two-dimensional torus. The fibers of particles known as Higgs bosons twist around the electroweak Lie group and determine a particular set of circles, breaking the symmetry. The Higgs does not twist around these circles, which then correspond to the massless photon of electromagnetism. Perpendicular to these circles are another set that should correspond to another particle, which the developers of electroweak theory called the Z boson. The fibers of the Higgs bosons twist around the circles of the Z boson, as well as the circles of the W⁺ and W^–, making all three particles massive. Experimental physicists discovered the Z in 1973, vindicating the theory and demonstrating how geometric principles have real-world consequences.

The strong nuclear force that binds quarks into atomic nuclei corresponds geometrically to an even larger Lie group, SU(3). The SU(3) fiber is an eight-dimensional internal space composed of eight sets of circles twisting around one another in an intricate pattern, producing interactions among eight kinds of photonlike particles called gluons on account of how they “glue” nuclei together. This fiber shape can be broken into comprehensible pieces. Embedded within it is a torus formed by two sets of untwisted circles, corresponding to two generators, g₃ and g₈. The remaining six gluon generators twist around this torus and their resulting g₃ and g₈ charges form a hexagon in the weight diagram. The quark fibers twist around this SU(3) Lie group, their strong charges forming a triangle in the weight diagram. These quarks are whimsically labeled with three colors: red, green and blue. A collection of matter fibers forming a complete pattern, such as three quarks in a triangle, is called a representation of the Lie group. The colorful description of the strong interactions is known as the theory of quantum chromodynamics.

Together, quantum chromodynamics and the electroweak model make up the Standard Model of particle physics, with a Lie group formed by combining SU(3), SU(2) and U(1), as well as matter in several representations. The Standard Model is a great success, but it presents several puzzles: Why does nature use this combination of Lie groups? Why do these matter fibers exist? Why do the Higgs bosons exist? Why is the weak mixing angle what it is? How is gravity included? The quarks, electrons and neutrinos that constitute common matter are called the first generation of fermions; they have second- and third-generation doppelgängers with identical charges but much larger masses. Why is that? And what are cosmic dark matter and dark energy? A unified theory should be able to provide answers to these and other questions.

A Grand Unified Theory use a large Lie group with a single fiber encompassing both the electroweak and strong forces. The first attempt at such a theory was proposed in 1973, by Howard Georgi and Sheldon Glashow. They found that the combined Lie group of the Standard Model fits snugly into the Lie group SU(5) as a subgroup. This SU(5) GUT made some distinctive predictions. First, fermions should have exactly the hypercharges that they do. Second, the weak mixing angle should be 38 degrees, in fair agreement with experiments. And finally, in addition to the 12 Standard Model bosons, there are 12 new force particles in SU(5), called X bosons. It was the X bosons that got the theory into trouble. These new particles would allow protons to decay into lighter particles. In impressive experiments, including the observation of 50,000 tons of water in a converted Japanese mine, the predicted proton decay was not seen. Thus, physicists have ruled out this theory.

A related Grand Unified Theory, developed around the same time, is based on the Lie group Spin(10). It produces the same hypercharges and weak mixing angle as SU(5) and also predicts the existence of a new force, very similar to the weak force. This new “weaker” force, mediated by relatives of the weak-force bosons called W’⁺, W’^– and W’₃, interacts with right-handed fermions, restoring leftright symmetry to the universe at short distances. Although this theory predicts an abundance of X bosons—a full 30 of them—it also indicates that proton decay would occur at a lower rate than for the SU(5) theory. So the theory remains viable. The Spin(10) Lie group with its 45 bosons, along with its representations of 16 fermions and their 16 antifermions, are in fact all parts of a single Lie group, a special one known as the exceptional
Lie group E6.

The classification of all the Lie groups found the existence of five exceptional ones that stand out: G2, F4, E6, E7 and E8. The fact that the bosons and fermions of Spin(10) and the Standard Model tightly fit the structure of E6, with its 78 generators, is remarkable. It provokes a radical thought. Up until now, physicists have thought of bosons and fermions as completely different. Bosons are parts of Lie group force fibers, and fermions are different kinds of fibers, twisting around the Lie groups. But what if bosons and fermions are parts of a single fiber? That is what the embedding of the Spin(10) GUT in E6 suggests. The structure of E6 includes both types of particles. In a radical unification of forces and matter, bosons and fermions can be combined as parts of a superconnection field. But E6 does not include the Higgs bosons or gravity.

A Lie group formulation of gravity uses the group Spin(1,3) for rotations in three spaces and one time direction. Now it is just a matter of putting the pieces together. With gravity described by Spin(1,3) and the favored Grand Unified Theory based on Spin(10), it is natural to combine them using a single Lie group, Spin(11,3), yielding a Gravitational Grand Unified Theory—as introduced last year by Roberto Percacci of the International School for Advanced Studies in Trieste and Fabrizio Nesti of the University of Ferrara in Italy. It brings us close to a full Theory of Everything. The Spin(11,3) Lie group allows for blocks of 64 fermions and, amazingly, predicts their spin, electroweak and strong chargesperfectly. It also automatically includes a set of Higgs bosons and the gravitational frame; in fact, they are unified as “frame-Higgs” generators in Spin(11,3). The curvature of the Spin(11,3) fiber bundle correctly describes the dynamics of gravity, the other forces and the Higgs. It even includes a cosmological constant that explains cosmic dark energy. Everything falls into place.

Skeptics objected that the Spin(11,3) theory should be impossible. It appears to violate a theorem in particle physics, the Coleman-Mandula theorem, which forbids combining gravity with the other forces in a single Lie group. But the theorem has an important loophole: it applies only when spacetime exists. In the Spin(11,3) theory (and in E8 theory), gravity is unified with the other forces only before the full Lie group symmetry is broken, and when that is true, spacetime does not yet exist. Our universe begins when the symmetry breaks: the frame-Higgs field becomes nonzero, singling out a specific direction in the unifying Lie group. At this instant, gravity becomes an independent force, and spacetime comes into existence with a bang. Thus, the theorem is always satisfied. The dawn of time was the breaking of perfect symmetry.

Lisi’s theory uses the most beautiful structure in all of mathematics, the largest simple exceptional Lie group, E8. Just as E6 contains the structure of the Spin(10) Grand Unified Theory, with its 16 fermions, the E8 Lie group contains the structure of the Spin(11,3) Gravitational Grand Unified Theory, with its 64 Standard Model fermions, including their spins. In this way, gravity and the other known forces, the Higgs, and one generation of Standard Model fermions are all parts of the unified superconnection field of an E8 fiber bundle. The E8 Lie group, with 248 generators, has a wonderfully intricate structure. In addition to gravity and the Standard Model particles, E8 includes W’, Z’ and X bosons, a rich set of Higgs bosons, novel particles called mirror fermions, and axions—a cosmic dark matter candidate.

Even more intriguing is a symmetry of E8 called triality. Using triality, the 64 generators of one generation of Standard Model fermions can be related to two other blocks of 64 generators. These three blocks might intermix to reproduce the three generations of known fermions. In this way, the physical universe could emerge naturally from a mathematical structure without peer. The theory tells us what Higgs bosons are, how gravity and the other forces emerge from symmetry-breaking, why fermions exist with the spins and charges they have, and why all these particles interact as they do.

Although Lisi’s theory continues to be promising, much work remains to be done. We need to figure out how three generations of fermions unfold, how they mix and interact with the Higgs to get their masses, and exactly how E8 theory works within the context of quantum theory. If E8 theory is correct, it is likely the Large Hadron Collider will detect some of its predicted particles. If, on the other hand, the collider detects new particles that do not fit E8’s pattern, that could be a fatal blow for the theory. In either case, any particles that experimentalists uncover will lead us toward some geometric structure at the heart of nature. And if the structure of the universe at the tiny scales of elementary particles does turn out to be described by E8, with its 248 sets of circles wrapping around one another in an exquisite pattern, twisting and dancing over spacetime in all possible ways, then we will have achieved a complete unification and have the satisfaction of knowing we live in an exceptionally beautiful universe.

Lisi’s papers on ArXiv.

A forthcoming paper of the Fermi-LAT Collaboration will describe method and results yielding to the left plot (the right one is widely known).  A map of the galactic center after 2 years of Fermi operation by the Large Area Telescope (LAT) γ-ray spectrum above 1 GeV. The announcement appears in Michael Gustafsson (Padova University, On behalf of the Fermi Collaboration), “Fermi Gamma-ray Space Telescope: Gamma-ray Observations and their Dark Matter Interpretations,” PPC 2011 @ CERN, June 14, 2011.

Clockwise from top left: Cladonota benitezi; Umbelligerus peruviensis; Nassunia binotata; and a nymph of a Cymbomorpha species.

“Treehoppers are insects that would resemble miniature cicadas were it not for the presence of the helmet. This structure appears to reside on top of the animal’s thorax, and extends dorsally, and in remarkably varied ways, to mimic thorns, animal droppings or aggressive ants. Entomologists joke that some treehoppers use their helmets to send signals to their home planet, so other-worldly is their appearance. Helmets are generally thought to aid in camouflage by disrupting the animal’s shape and outline, or by mimicking thorns, animal droppings or aggressive ants and wasps. Understanding the origin of helmets and other complex traits is among the most enduring puzzles in evolutionary biology. On the one hand, evolution operates within a framework of descent with modification — everything new must come from something old. On the other hand, structures such as the eye, the wing and the turtle’s shell stand out because they lack obvious correspondence to the old. A new paper in Nature by Prud’homme et al. provide evidence that treehoppers have overcome such suppression to produce their helmets. This addresses the puzzle by connecting a complex and highly diverse trait — the helmet of membracid treehoppers — to its origins in both development and evolution.” See Armin P. Moczek, “Evolutionary biology: The origins of novelty,” Nature 473: 34–35, 05 May 2011, which refers to Benjamin Prud’homme et al., “Body plan innovation in treehoppers through the evolution of an extra wing-like appendage,” Nature 473: 83–86, 05 May 2011.

“Helmets have been interpreted as an extension of the pronotum, the dorsal portion of the first segment of the three-segmented thorax shared by all insects. We have long known from fossil evidence that insects arrived at this organization following a period of progressive loss of wings or wing-like appendages from all abdominal segments, as well as from the first thoracic segment. Fossils of an extinct species (Stenodyctya lobata) show that expression of the wing-development program in the first thoracic segment (arrow) was common in early insects. In extant winged insects, wings are borne only on the second and third thoracic segments, with wing development on the first segment being suppressed.

Enter the treehopper Publilia modesta and its helmet. Through careful analysis of this structure’s anatomy, placement and attachment to the thorax, Prud’homme et al. discovered that the helmet may not be a mere extension of the pronotum. Instead, it is attached bilaterally to the thorax by paired articulations reminiscent of joints, much like regular wings. Moreover, when they examined its early developmental stages, the authors found that the helmet forms from paired buds — again, much like wings. The expanding buds subsequently fuse along the midline, creating the continuous helmet. Study of the expression of one gene, nubbin, normally specific to insect wing development, and two genes specific to appendage formation in general, provided additional evidence that helmet development may rely on developmental mechanisms involved in the formation of wings.

“Octonions were largely neglected since their discovery in 1843, but in the past few decades they have assumed a curious importance in string theory. And indeed, if string theory is a correct representation of the universe, they may explain why the universe has the number of dimensions it does.” John C. Baez & and John Huerta, “The Strangest Numbers in String Theory,” Scientific American, May 2011.

“The set of all real numbers forms a line, so we say that the collection of real numbers is one-dimensional. We could also turn this idea on its head: the line is one-dimensional because specifying a point on it requires one real number. The set of all complex numbers of the form a+b i, where i²=–1, and a and b are ordinary real numbers, describe points on the plane and their basic operations —addition, subtraction, multiplication and division— describe geometric manipulations in the plane. Almost everything we can do with real numbers can also be done with complex numbers.

On October 16, 1843, William Rowan Hamilton was walking with his wife along the Royal Canal to a meeting of the Royal Irish Academy in Dublin when he had a sudden revelation: quaternions a + b i + c j + d k, with i²=j²=k²=–1. Quaternions provide an efficient way to represent threedimensional rotations. Hamilton’s college friend, John Graves, discovered on December 26 a new eight-dimensional number system that he called the octaves and that are now called octonions. In 1845 the young genius Arthur Cayley rediscovered the octonions. For this reason, the octonions are also sometimes known as Cayley numbers.”

In the generalization of numbers as tuples, division is the hard part: a number system where we can divide is called a division algebra. Not until 1958 did three mathematicians prove an amazing fact that had been suspected for decades: any division algebra must have dimension one (which is just the real numbers), two (the complex numbers), four (the quaternions) or eight (the octonions).

Hamilton didn’t like the octonions because they break some cherished laws of arithmetic. Real an complex numbers are commutatitve, but quaternions are noncommutative. The octonions are much stranger. Not only are they noncommutative, they also break another familiar law of arithmetic: the associative law (xy)z = x(yz). What the octonions would be good for? They are closely related to the geometry of seven and eight dimensions, and we can describe rotations in those dimensions using the multiplication of octonions.

In the 1970s and 1980 s theoretical physicists developed a strikingly beautiful idea called supersymmetry, a symmetry between matter and the forces of nature. Every matter particle (such as an electron) has a partner particle that carries a force, and vice versa. Supersymmetry also encompasses the idea that the laws of physics would remain unchanged if we exchanged all the matter and force particles. Even though physicists have not yet found any concrete experimental evidence in  support of supersymmetry, the theory is so seductively beautiful and has led to so much enchanting mathematics that many physicists hope and expect that it is real.

In the standard three-dimensional version of quantum mechanics that physicists use every day, spinors describes the wave motion of matter particles and vectors describes that of force particles. Particle interactions require the combination of spinors and vectors by a simulacrum of multiplication. As an alternative, imagine a strange universe with no time, only space. If this universe has dimension one, two, four or eight, both matter and force particles would be waves described by a single type of vectorial object (vectors and spinors coincide), just real numbers, complex numbers, quaternions or octonions, respectively. Supersymmetry emerges  naturally, providing a unified description of matter and forces.

In string theory, every object corresponds to a little string with one dimension in space and another one in time, hence two dimensions have to be added to every point in sapce. Instead of supersymmetry in dimension one, two, four or eight, we get supersymmetry in dimension three, four, six or 10. Coincidentally string theorists have for years been saying that only 10-dimensional versions of the theory are self-consistent: anomalies appear in anything other than 10 dimensions, breaking down string theory. But 10-dimensional string theory is, as we have just seen, the version of the theory that uses octonions. So if string theory is right, the octonions are not a useless curiosity, on the contrary, they provide the deep reason why the universe must have 10 dimensions: in 10 dimensions, matter and force  particles are embodied in the same type of numbers—the octonions.

Recently physicists have started to go beyond strings to consider membranes. In string theory we had to add two dimensions to our standard collection of one, two, four and eight, now we must add three. Supersymmetric membranes naturally emerge in dimensions four, five, seven and 11. Researchers tell us that M-theory (the “M” typically stands for “membrane”) requires 11 dimensions—implying that it should naturally make use of octonions.

Neither string theory nor M-theory have as of yet made no experimentally testable predictions. They are beautiful dreams—but so far only dreams. The universe we live in does not look 10- or 11-dimensional, and we have not seen any symmetry between matter and force particles. Only time will tell if the strange octonions are of fundamental importance in understanding the world we see around us or merely a piece of beautiful mathematics.”

More information about these especulations in Peter Woit, “This Week’s Hype,” Not Even Wrong, April 28th, 2011, where the expository article about octonions by John Baez that appeared in the AMS Bulletin (copy here, a web-site here) is recommended. In the comments, Thomas Larsson recalls that “octonions is the last division algebra, but if you relax your axioms a little the Cayley-Dickson construction gives an infinite tower of increasingly uninteresting algebras: n=1, Reals; n=2, Complex numbers; n=4, Quaternions, not commutative; n=8, Octonions, not associative; n=16: Sedenions, not alternative but power associative, n=32: 32-ions?; …

See also Philip Gibbs, “Octonions in String Theory,” viXra log, April 29, 2011, and Lubos Motl, “John Baez, octonions, and string theory,” The Reference Frame, April 29, 2011.