2 Physicists Simplify Study of Four-Dimensional Space : Science: Discovery stuns mathematicians. Finding will help in understanding basic properties of matter.

Ever since Albert Einstein showed that the three-dimensional space we live in actually curves into an unseen fourth dimension--much as the seemingly flat Earth plunges into invisibility over the horizon--mathematicians have been trying to understand the shape of our universe. It could curve, like the Earth, into a familiar sphere. Or it could have a hole, like a donut. Or even two holes, like a two-handled cup.

Now two physicists, Ed Witten of the Institute for Advanced Studies at Princeton and Nathan Seiberg of Rutgers, have made studying these strange spaces tantalizingly simple.

Their discovery has electrified mathematicians who have spent decades trying to find ways to describe four-dimensional space. The finding will almost certainly help physicists understand the most fundamental properties of matter.

Essentially, Witten and Seiberg figured out a way to solve problems in the complex world of four-dimensional space that is as simple as it would be in two dimensions. In other words, they took equations that were essentially unsolvable and transformed them into the ordinary equations of everyday calculus.

The discovery has packed lecture rooms at the prestigious Princeton institute where Einstein once worked and scientists around the world, informed over computer networks, already have begun working out the implications.

In one sense, the physicists’ work is a major embarrassment to mathematicians, who’ve struggled with the problems of four-dimensional space for decades with limited success. “People in the mathematical community were very skeptical about this at first,” said Sylvain Cappell of NYU’s Courant Institute of Mathematics, “but now they’ve been blown away. It’s an awesome thing that a physicist came in and did this.”

According to Cappell, the discovery has the potential of simplifying an entire field of mathematics into a single 10-page paper.

For physicists, the work offers insights into the smallest building blocks of matter, an area of study that has been stuck since the 1970s due to its overwhelming mathematical complexity.

In the innermost heart of the atom, particles called quarks bind together to comprise all matter. Quarks, it turns out, have very perplexing properties. Close together inside a proton, they rattle around freely, like marbles in a bag. But try to pull them apart, and the force between them gets infinitely strong. The harder you pull, the more fiercely they stick together.

Physicists have methods for describing this behavior. “But the equations are impossible to compute,” Seiberg said . “People use massive computers and it still doesn’t work out.”

Now together, Witten and Seiberg have found a way to tackle these complex four-dimensional problems by methods essentially as simple as a two-dimensional problem. The physicists are pleased because they have a model, “a practice case,” Witten calls it, where the problems of quark behavior look workable. Because it’s simpler than the real world, it can be solved. “We hope it’s the same as the real world,” said Seiberg, “but only time will tell.”

In the meantime, mathematicians are both pleased and appalled. “It’s embarrassing because we didn’t notice it first,” Cappell said.

Why so much interest in problems of four-dimensional space? One obvious answer, says mathematician Ron Stern of UC Irvine, is that “it’s the space we live in.” More interesting to the abstract world of mathematics, it’s the most quirky space of all. Quirkier, even, than eight-dimensional space. Or 10,000-dimensional space. “It’s ironic,” Stern said, “because we understand one- and two-dimensional manifolds (surfaces). We understand five-dimensions and higher. But we don’t understand three and four, and those are the ones we live in.”

The problem that Witten and Seiberg has solved has to do with equations that were not “commutative” in four-dimensional space. In other words, A times B did not equal B times A. If you rotated something in one direction, and then rotated the same thing in another direction, you would not get the same result as if you performed the rotations in reverse. This made things enormously complicated.

In order to understand the universe, physicists like laws of nature that are not constantly changing. That is one reason why symmetrical equations are so appealing. It was a similar symmetry, for example, that allowed Einstein to see that energy and matter are just different facets of the same essential stuff.

The buzz around Princeton these days has spread from mathematicians from coast to coast. “What we’re trying to understand is the kind of universe we live in,” said Stern. “It’s a sad thing for a mathematician, but in the last 15 years, the physicists have made all the good insights.”

As for the physicists, they’ve made one step toward understanding the essential quirkiness of quarks. “We’ve made one step,” Seiberg said. “But it’s never clear how difficult the next step will be.”