Is God a Mathematician_ - Mario Livio [101]
Since strings are closed loops moving through space, as time progresses, they sweep areas (known as world sheets) in the form of cylinders (as in figure 60). If a string emits other strings, this cylinder forks to form wishbone-shaped structures. When many strings interact, they form an intricate network of fused donutlike shells. While studying these types of complex topological structures, string theorists Hirosi Ooguri and Cumrun Vafa discovered a surprising connection between the number of donut shells, the intrinsic geometric properties of knots, and the Jones polynomial. Even earlier, Ed Witten—one of the key players in string theory—created an unexpected relation between the Jones polynomial and the very foundation of string theory (known as quantum field theory). Witten’s model was later rethought from a purely mathematical perspective by the mathematician Michael Atiyah. So string theory and knot theory live in perfect symbiosis. On one hand, string theory has benefited from results in knot theory; on the other, string theory has actually led to new insights in knot theory.
Figure 60
With a much broader scope, string theory searches for explanations for the most basic constituents of matter, much in the same way that Thomson originally searched for a theory of atoms. Thomson (mistakenly) thought that knots could provide the answer. By a surprising twist, string theorists find that knots can indeed provide at least some answers.
The story of knot theory demonstrates beautifully the unexpected powers of mathematics. As I have mentioned earlier, even the “active” side of the effectiveness of mathematics alone—when scientists generate the mathematics they need to describe observable science—presents some baffling surprises when it comes to accuracy. Let me describe briefly one topic in physics in which both the active and the passive aspects played a role, but which is particularly remarkable because of the obtained accuracy.
A Weighty Accuracy
Newton took the laws of falling bodies discovered by Galileo and other Italian experimentalists, combined them with the laws of planetary motion determined by Kepler, and used this unified scheme to put forth a universal, mathematical law of gravitation. Along the way, Newton had to formulate an entirely new branch of mathematics—calculus—that allowed him to capture concisely and coherently all the properties of his proposed laws of motion and gravitation. The accuracy to which Newton himself could verify his law of gravity, given the experimental and observational results of his day, was no better than about 4 percent. Yet the law proved to be accurate beyond all reasonable expectations. By the 1950s the experimental accuracy was better than one ten-thousandth of a percent. But this is not all. A few recent, speculative theories, aimed at explaining the fact that the expansion of our universe seems to be speeding up, suggested that gravity may change its behavior on very small distance scales. Recall that Newton’s law states that the gravitational attraction decreases as the inverse square of the distance. That is, if you double the distance between two masses, the gravitational force each mass feels becomes four times weaker. The new scenarios predicted deviations from this behavior at distances smaller than one millimeter (the twenty-fifth part of an inch). Eric Adelberger, Daniel Kapner, and their collaborators at the University of Washington, Seattle, conducted