“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.