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Prof. Robbert Dijkgraaf on string theory

Duke University
Durham, North Carolina

Monday, Feb. 16, 1998


PHILADELPHIA -- Duke University mathematician David Morrison says he is applying his expertise in algebraic geometry to the exotic field of string theory in a quest for simplicity.

"It is kind of a philosophical question," the James B. Duke professor of mathematics and physics said in an interview. "Do we expect the laws of nature to be based on simple mathematical principles that we can understand and then derive other things from?"

Morrison planned to discuss how string theory modifies the world around us in a talk Monday entitled "The Small-Scale Structure of Spacetime" during the American Association for the Advancement of Science's 1998 annual meeting.

String theory posits that each type of elementary subatomic particle is a kind of vibrating geometric entity that mathematically resembles a string.

One of those vibrational modes corresponds to gravity. Others correspond to the additional forces in nature, including electromagnetism, as manifested in light rays and radio waves. Still other modes include the basic elementary particles out of which matter is built.

These strings would also be infinitesimally tiny, each as small in relation to a subatomic proton as a proton is small in relation to the entire solar system.

The catch in the promising theory, however, was that it requires these entities to vibrate in 10 dimensions rather than the four that humans live in -- three spatial and one time dimension. Somehow, those six additional dimensions must be made to fold into spaces too tiny to be apparent to humans.

In the recent interview, Morrison described how string theory evolved in an attempt to reconcile mathematical incompatibilities between two of physics' most highly successful theories: quantum mechanics and the general theory of relativity.

Quantum mechanics describes the nature and behavior of a multitude of different elementary particles envisioned as the building blocks of our physical universe. And the general theory of relativity describes the nature of gravity and its geometric relationship with space and time.

Despite their sometimes counterintuitive natures, both theories have been consistently borne out by scientific experiments. And since both matter and light are affected by the force of gravity, one theory logically should be consistent with the other.

"We don't understand how they can be made to interact in a consistent fashion," Morrison said of both theories. "The only concrete proposal that we have for doing that -- at the moment -- is string theory, which is why many of us are excited about it."

The problem has been that attempts to reconcile the two theories result in mathematical inconsistencies at the extremely small scales at which quantum mechanics operate. Fluctuations which are an integral part of quantum mechanics require that the geometry of time and space be torn asunder, which violates a fundamental rule of general relativity, he said.

Another motivation for string theory, Morrison said, is that "there is something quite dissatisfying about the current Standard Model of particle physics." That's the quantum mechanics-based set of rules describing how all those elementary particles interact.

The Standard Model uses equations "whose forms are rather complicated and whose origins are not understood," he added. "It doesn't sound like a fundamental theory. It doesn't sit right with a lot of people that such should be the way that fundamental equations of the world work at the smallest level."

String theorists have been seeking to overcome these drawbacks of current theories by modifying both quantum mechanics and Albert Einstein's relativity theory. The latter drew on the earlier geometric insights of 19th century mathematician Georg Reimann to show how gravity causes space and time to curve.

Mathematicians Theodr Kaluza and Oscar Klein showed, beginning in the 1920s, that Reimann's geometry could indeed be extended in a way that allowed for tiny extra dimensions. And they showed that, when this is done, other physical forces are included along with gravity.

Modern string theorists use a variant of that idea to attempt to address the multidimensional problem in string theory. But another obstacle, Morrison said, is that these "superstring" equations can be solved in tens of thousands of different ways.

In 1995, Morrison joined physicists Brian Greene of Cornell University and Andrew Strominger of the University of California at Santa Barbara to deal with both those barriers.

They drew on work over the last two decades by mathematicians, including Eugenio Calabi of the University of Pennsylvania and Shing-Tung Yau of Harvard University, who discovered so-called "Calabi-Yau manifolds." Those manifolds are six-dimensional mathematical shapes, Morrison said, that can meet the "stringent requirements"of string theory.

Mathematicians have found that, by employing algebraic geometry, these Calabi-Yau manifolds "can be described by some rather simple looking types of equations, the simplest kinds of equations we know how to manipulate," he added.

Greene, Morrison and Strominger showed that certain geometrical features of these manifolds would resemble special kinds of submicroscopic "black holes" in our four-dimensional world. But as the manifolds' shapes change, these black holes would have the capacity to tear space-time in a way the relativity theory doesn't allow.

Just before that could happen, though, the investigators also found these black holes could undergo a "phase change" similar to ice freezing into water. After that shift, each type of the special black holes would be transformed into a string, which would appear to us like a corresponding type of elementary particle.

Morrison said this shift in the black hole's phase would be as dramatic as cutting a hole in a drum to change the way it vibrates. Yet the rules of quantum mechanics would permit it to happen, just like they allow electrons to "tunnel" through a forbidden region, or permit the nuclei of some atoms to decay.

"Similarly, a process like moving from one shape to a radically different shape is something that's possible quantum-mechanically," he said. As a bonus, Morrison, Greene and Strominger found their mathematics allows the many different solutions of string theory to be linked together by these phase transitions.

And what about all those extra dimensions that string theory also requires to exist? "It's the folded up structure of that space which gives us interesting clues about the small-scale structure of space time," he said.

Morrison stressed that string theory is still unproven, and could be replaced by something else. "You can have beautiful theories which are wrong -- not wrong as mathematics but as interpretations of the physical world," he said.

But he also noted that Isaac Newton invented calculus to successfully explain his theory of how gravity governs the motions of planets, and Einstein later used more advanced forms of mathematics to reveal something deeper about gravity.

"I think a number of people are still hoping for an analogous formulation for the most fundamental of theories that would explain elementary particles," he said. "And that's one of the motivations, certainly, that drives the formulation of string theory."

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