Fabric Of The Universe?

Is This The Fabric Of The Universe?

By Roger Highfield | 17 November 2007

A Heroic Mathematical Enterprise That Could Lay Bare The Fundamentals Of The Cosmos, And Lead To The Theory Of Everything.

Mathematicians have successfully scaled their equivalent of Mount Everest. Today they unveil the answer to a problem that, if written out in tiny print, would cover an area the size of Manhattan.

A two dimensional representation of E8, courtesy of Peter McMullen and John Stembridge.     CLICK to enlarge.

At the most basic level, the calculation is an arcane investigation of symmetry— in this case of an object that is 57 dimensional, rather than the usual three dimensional ones that we are familiar with. Although this object was first discovered in the 19th century. there is evidence that it could contain the structure of the cosmos.

A 19th-century Norwegian mathematician, Sophus Lie (rhymes with tree), wrote down what are now known as Lie groups, sets of continuous transformations— meaning changes that could be a little or a lot— that leave an object unchanged in appearance.

For example, rotate a sphere any distance around any axis, and the sphere looks exactly the same.

Later mathematicians found five exceptions to the four classes of Lie groups that Lie knew about. The most complicated of the "exceptional simple Lie groups" is E8. It describes the symmetries of a 57-dimensional object that can in essence be rotated in 248 ways without changing its appearance.

To understand using E8 in all its possibilities requires calculation of 200 billion numbers. That is what Dr. Adams’s team did, a rare collaboration for mathematicians who usually work alone or in small groups and rarely turn to supercomputers.

Robert L. Bryant, a mathematician at Duke who was not involved in the project, gave a biological analogy. Scientists can learn a lot about an animal from its DNA, but to understand it fully "you have to grow the organism and then study it," Dr. Bryant said. "In a certain sense, that is what the E8 team did. They used massive computation to fully develop the group E8 and its representations so that they could list its important features."

Mathematicians are known for their solitary style of working, but the combined assault on what is described as "one of the largest and most complicated structures in mathematics" required the effort of 18 mathematicians from America and Europe for an intensive four-year collaboration.

The feat may baffle most people but could have unforeseen implications in mathematics and physics, which won’t be evident for years to come, said the American Institute of Mathematics.

"The group of symmetries of this strange geometry called E8 is one of the most intriguing structures that Nature has left for the mathematician to play with," commented Prof Marcus du Sautoy of Oxford University, currently in Auckland. "Most of the time mathematical objects fit into nice patterns that we can order and classify. But this one just sits there like a huge Everest. "What makes this group of symmetries so exciting is that Nature also seems to have embedded it at the heart of many bits of physics. One interpretation of why we have such a quirky list of fundamental particles is because they all result from different facets of the strange symmetries of E8. I find it rather extraordinary that of all the symmetries that mathematician’s have discovered, it is this exotic exceptional object that Nature has used to build the fabric of the universe. The symmetries are so intricate and complex that today’s announcement of the complete mapping of E8 is a significant moment in our exploration of symmetry."

For the feat, the team used a mix of theoretical mathematics and intricate computer programming to successfully map E8, (pronounced "E eight") which is an example of a Lie (pronounced "Lee") group. Lie groups were invented by the 19th century Norwegian mathematician Sophus Lie to study symmetry. E8 has rank 8 (the maximum number of mutually commutative degrees of freedom), and dimension 248 (as a manifold).

Underlying any symmetrical object, such as a sphere, is a Lie group. Balls, cylinders or cones are familiar examples of symmetric three-dimensional objects. Today’s feat rests on the drive by mathematicians to study symmetries in higher dimensions.

"E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood," said Prof Jeffrey Adams, Project Leader, at the University of Maryland. "This groundbreaking achievement is significant both as an advance in basic knowledge, as well as a major advance in the use of large scale computing to solve complicated mathematical problems." "This is an exciting breakthrough," said Prof Peter Sarnak at Princeton University. "Understanding and classifying the representations of E8 and Lie groups has been critical to understanding phenomena in many different areas of mathematics and science including algebra, geometry, number theory, physics and chemistry. This project will be invaluable for future mathematicians and scientists."

The ways that E8 manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E8. These representations are extremely complicated, but mathematicians describe them in terms of basic building blocks. The new result is a complete list of these building blocks for the representations of E8, and a precise description of the relations between them, all encoded in a matrix, or grid, with 453,060 rows and columns. There are 205,263,363,600 entries in all, each a mathematical expression called a polynomial. If each entry was written in a one inch square, then the entire matrix would measure more than seven miles on each side.

The result of the E8 calculation, which contains all the information about E8 and its representations, is 60 gigabytes in size. This is enough to store 45 days of continuous music in MP3-format. If written out on paper, the answer would cover an area the size of Manhattan. The computation required sophisticated new mathematical techniques and computing power not available even a few years ago.

"This is an impressive achievement," said Hermann Nicolai, Director of the Albert Einstein Institute in Potsdam, Germany. "While mathematicians have known for a long time about the beauty and the uniqueness of E8, we physicists have come to appreciate its exceptional role only more recently— yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner," he said, referring to efforts to combine the theory of the very big (general relativity) with the very small (quantum mechanics). "Thus, understanding the inner workings of E8 is not only a great advance for pure mathematics, but may also help physicists in their quest for a unified theory."

Garrett Lisi, 39, completed his doctorate in theoretical physics in 1999 at the University of California, San Diego, but has no university affiliation. He spends most of the year surfing in Hawaii, where he has also been a hiking guide and bridge builder (when he slept in a jungle yurt).

Now Lisi says "I think our universe is this beautiful shape."

Lisi's breakthrough came when he noticed that some of the equations describing E8's structure matched his own. "My brain exploded with the implications and the beauty of the thing," he tells New Scientist. "I thought: 'Holy crap, that's it!'"

Lisi's inspiration lies in the most elegant and intricate shape known to mathematics, called E8— a complex, eight-dimensional mathematical pattern with 248 points first found in 1887, but only fully understood by mathematicians this year after workings, that, if written out in tiny print, would cover an area the size of Manhattan.

What Lisi had realised was that he could find a way to place the various elementary particles and forces on E8's 248 points. What remained was 20 gaps which he filled with notional particles, for example those that some physicists predict to be associated with gravity.

Physicists have long puzzled over why elementary particles appear to belong to families, but this arises naturally from the geometry of E8, he says. So far, all the interactions predicted by the complex geometrical relationships inside E8 match with observations in the real world. "How cool is that?" he says. His proposal is all the more remarkable because, by the arcane standards of particle physics, it does not require highly complex mathematics.

Even better, it does not require more than one dimension of time and three of space, when some rival theories need ten or even more spatial dimensions and other bizarre concepts. And it may even be possible to test his theory, which predicts a host of new particles, perhaps even using the new Large Hadron Collider atom smasher that will go into action near Geneva next year.

The crucial test of Lisi's work will come only when he has made testable predictions. Lisi is now calculating the masses that the 20 new particles should have, in the hope that they may be spotted when the Large Hadron Collider starts up. "The theory is very young, and still in development," he told the Telegraph. "Right now, I'd assign a low (but not tiny) likelyhood to this prediction. "For comparison, I think the chances are higher that LHC will see some of these particles than that the LHC will see superparticles, extra dimensions, or micro black holes as predicted by string theory. I hope to get more (and different) predictions, with more confidence, out of this E8 Theory over the next year, before the LHC comes online."

Although the work of Garrett Lisi still has a way to go to convince the establishment, let alone match the achievements of Albert Einstein, the two do have one thing in common: Einstein also began his great adventure in theoretical physics while outside the mainstream scientific establishment. Working as a patent officer in 1905, he formulated the Special, and later (1916) the General, Theories of Relativity, the standard physical models of the macrocosm to this day. However, in later years he sought but failed to achieve the Holy Grail: an overarching explanation to unite all the particles and forces of the cosmos.

Now Lisi, currently in Nevada, has come up with a proposal to do this. Lee Smolin at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada, describes Lisi's work as "Fabulous; it is one of the most compelling unification models I've seen in many, many years," he says.

"Although he cultivates a bit of a surfer-guy image its clear he has put enormous effort and time into working the complexities of this structure out over several years," Prof Smolin tells The Telegraph. "Some incredibly beautiful stuff falls out of Lisi's theory," adds David Ritz Finkelstein at the Georgia Institute of Technology, Atlanta. "This must be more than coincidence and he really is touching on something profound." The new theory is reported in the New Scientist and has been laid out in an online paper entitled "An Exceptionally Simple Theory of Everything" by Lisi. He has high hopes that his new theory could provide what he says is a "radical new explanation" for the three decade old Standard Model, which weaves together three of the four fundamental forces of nature: the electromagnetic force; the strong force, which binds quarks together in atomic nuclei; and the weak force, which controls radioactive decay. The reason for the excitement is that Lisi's model also takes account of gravity, a force that has only successfully been included by a rival and highly fashionable idea called string theory, one that proposes particles are made up of minute strings, which is highly complex and elegant but has lacked predictions by which to do experiments to see if it works. But some are taking a cooler view. Prof Marcus du Sautoy, of Oxford University and author of Finding Moonshine, told the Telegraph: "The proposal in this paper looks a long shot and there seem to be a lot things still to fill in." And a colleague Eric Weinstein in America added: "Lisi seems like a hell of a guy. I'd love to meet him. But my friend Lee Smolin is betting on a very very long shot."

E8 is also the Lie group underlying some superstring theories that physicists are pursuing in an effort to tie gravity and the other fundamental forces of the universe into one theory.

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