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A Geometric Theory of Everything Deep Down, the Particles and Forces of the Universe Are a Manifestation of Exquisite Geometry

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A Geometric Theory of Everything Deep Down, the Particles and Forces of the Universe Are a Manifestation of Exquisite Geometry Nature’s zoo of elementary particles is not a random mish- mash; it has striking patterns and interrelationships that can be depicted on a diagram correspond- ing to one of the most intricate geometric objects known to mathematicians, called E8. 54 Scientific American, December 2010 Photograph/Illustration by Artist Name © 2010 Scientific American A. GarrettAuthor Lisi balances Bio Tex this until time an between ‘end nested research style incharacter’ theoretical (Command+3) physics andxxxx surfing. xxxx xx xxxxxAs an itinerantxxxxx xxxx scientist, xxxxxx hexxxxxx is in thexxx xxprocess xxxxxxxxxx of xxxx realizingxxxxxxxxxx a lifelong dream: xxxxxx founding xxxxxxxx the xxxxxxxx. Pacific Science Institute, located on the Hawaiian island of Maui. James Owen Weatherall, having recently completed his doctorate in physics and mathematics at the Stevens Institute of Technology, is now finishing a second Ph.D. in philosophy at the University of California, Irvine. He also manages to find time to work on a book on the history of ideas moving from physics into financial modeling. PHYSICS A Geometric Theory of Everything Deep down, the particles and forces of the universe are a manifestation of exquisite geometry By A. Garrett Lisi and James Owen Weatherall odern physics began with a sweeping unification: in 1687 isaac Newton showed that the existing jumble of disparate theories describing everything from planetary motion to tides to pen- dulums were all aspects of a universal law of gravitation. Unifi- cation has played a central role in physics ever since. In the middle of the 19th century James Clerk Maxwell found that electricity and magnetism were two facets of electromagnetism. One hundred years later electromagnetism was unified with the weak nuclear force Mgoverning radioactivity, in what physicists call the electroweak theory. This quest for unification is driven by practical, philosophical and aesthetic consid- erations. When successful, merging theories clarifies our understanding of the universe and leads us to discover things we might otherwise never have suspected. Much of the activity in experimental particle physics today, at accelerators such as the Large Hadron Collider at CERN near Geneva, involves a search for novel phenomena predicted by the IN BRIEF In 2007 physicist A. Garrett Lisi wrote the most talked radical shift in our conception of reality. Lisi, in contrast, about theoretical physics paper of the year. Outlets argues that the geometric framework of modern quan- from the New Yorker to Outside magazine were drawn tum physics can be extended to incorporate Einstein’s to the story, partly on account of his surfer lifestyle. Lisi theory, leading to a long-sought unification of physics. and others have continued to develop the theory. Even if Lisi turns out to be wrong, the E8 theory he has Most physicists think reconciling Einstein’s general pioneered showcases striking patterns in particle phys- theory of relativity with quantum theory will require a ics that any unified theory will need to explain. Photograph/Illustration by Artist Name Illustrations by Chad Hagen December 2010, ScientificAmerican.com 55 © 2010 Scientific American unified electroweak theory. In addition to predicting new physi- different, internal spaces attached to each point of our space- cal effects, a unified theory provides a more aesthetically satisfy- time, with shapes corresponding to particles’ properties. ing picture of how our universe operates. Many physicists share This idea, introduced in 1918 by mathematician Hermann an intuition that, at the deepest level, all physical phenomena Weyl, is now a well-established principle of physics [see “Fiber match the patterns of some beautiful mathematical structure. Bundles and Quantum Theory,” by Herbert J. Bernstein and An- The current best theory of nongravitational forces—the elec- thony V. Phillips; Scientific American, July 1981]. Distinct from tromagnetic, weak and strong nuclear force—was largely com- the speculated undulating spatial dimensions of string theory, pleted by the 1970s and has become familiar as the Standard these internal space fibers are of fixed shape. Their dynamics Model of particle physics. Mathematically, the theory describes arise from how they are attached to four-dimensional spacetime. these forces and particles as the dynamics of elegant geometric The electric and magnetic fields existing everywhere in our objects called Lie groups and fiber bundles. It is, however, some- space are the result of fibers with the simplest shape: the circle. what of a patchwork; a separate geometric object governs each A circle, called U(1) by physicists, is the simplest example of a force. Over the years physicists have proposed various Grand Lie group (pronounced “Lee,” after 19th-century Norwegian Unified Theories, or GUTs, in which a single geometric object mathematician Sophus Lie). It has a single symmetry: if we ro- would explain all these forces, but no one yet knows which, if tate a circle, it remains the same. A small rotation like this is any, of these theories is true. called a generator of the Lie group. Following a generator, just And an even deeper unification problem faces today’s physi- like drawing with a compass, takes us around a circle. cists. In a fully unified theory, gravity and matter should also The fiber bundle of electromagnetism consists of circles at- combine naturally with the other forces, all as parts of one math- tached to every point of spacetime [see box on opposite page]. ematical structure—a Theory of Everything. Since the 1980s Crucially, each circle can rotate a little relative to its spacetime string theory, the dominant research program in theoretical neighbors. The so-called connection field of a fiber bundle de- particle physics, has been an attempt to describe gravity and the scribes how neighboring fibers are related by these symmetry Standard Model using elaborate constructs of strings and mem- rotations. The electric and magnetic force fields filling space- branes vibrating in many spacetime dimensions. time correspond to the curvature of this fiber bundle—geomet- But string theory is not the only effort. An alternative, loop rically, the electric and magnetic fields are how the circular -fi quantum gravity, uses a more minimal framework, closer to that bers twist over time and space. An electromagnetic wave is the of the Standard Model [see “Atoms of Space and Time,” by Lee undulation of circles over spacetime. One quantum of an elec- Smolin; Scientific American, January 2004]. Building on its in- tromagnetic wave—a photon—is a propagating particle of light. sights, one of us (Lisi) proposed a new unified theory in 2007. Each kind of elementary particle corresponds to a different -fi The basic idea is to extend Grand Unified Theories and include ber over spacetime; the Chia Pet has many different kinds of gravity as part of a consistent geometric framework. In this uni- sprouts. All the electrons of the world result from the twisting of a fied field theory, called E8 theory, all forces and matter are -de single kind of fiber—explaining, among other things, why all elec- scribed as the twisting of a single geometric object. trons are identical. The fibers of electrically charged particles, such All new ideas must endure a trial by fire, and this one is no as electrons, wrap around the circular fibers of electromagne- exception. Many physicists are skeptical—and rightly so. The tism like threads around a screw. How fast a particle’s fiber twists theory remains incomplete. But even in this early stage of devel- around the circle is equal to its electric charge, determining how opment, it unveils some of the beautiful structures in play at the the particle responds to the force of electromagnetism. deepest levels of nature, and it makes predictions for new parti- Because twists must meet around the circle, these charges cles that the Large Hadron Collider might find. Although physi- are integer multiples of some standard unit of electric charge. cists are not yet at the culmination of our centuries-long quest Of the elementary matter particles, called fermions, electrons for unity, E8 theory is an important step on that journey. have electric charge –1 (three twists), up quarks have electric 2 charge + ⁄3 (two opposite twists), down quarks have electric 1 EVERY FIBER OF OUR BEING charge – ⁄3 (one twist), and neutrinos have 0. The antimatter to describe e8 theory, we first need to set out the widely accept- particles, such as positrons and antiquarks, have twists in the ed geometric principles that govern all known forces and parti- opposite direction around the electromagnetic circle, giving cles. Geometry is the study of shape, but in the case of funda- them the opposite electric charges. mental physics, you might wonder: shape of what? Plato thought When particles collide, they may be converted into new elements such as earth and air were associated with little cubes types, but the outgoing particles have exactly the same total and octahedra. Similarly, in modern physics, the geometric ob- charge as the incoming ones did. This crucial fact is a conse- jects associated with elementary particles are perfect, smooth quence of fiber geometry: When any two particles meet, their shapes, existing outside our space yet connected to it. We cannot twists add. In this way, the fiber-bundle picture explains what see these shapes directly, but we see their effects. we know about electromagnetism. The electric charges describe The main geometric idea underlying the Standard Model is the geometric structure of the combined electromagnetic and that every point in our space time has shapes attached to it, called matter fiber bundle, determining what interactions are possible fibers, each corresponding to a different kind of particle. You can between electrically charged particles. envision the universe as a Chia Pet (a terra-cotta figurine covered with sprouts). The surface of the figurine is analogous to space- DIFFERENT CHARGES FOR DIFFERENT FORCES time, and the sprouts are fibers.
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