random matrix theory

NewScientist | SUPPOSE we had a theory that could explain everything. Not just atoms and quarks but aspects of our everyday lives too. Sound impossible? Perhaps not.

It's all part of the recent explosion of work in an area of physics known as random matrix theory. Originally developed more than 50 years ago to describe the energy levels of atomic nuclei, the theory is turning up in everything from inflation rates to the behaviour of solids. So much so that many researchers believe that it points to some kind of deep pattern in nature that we don't yet understand. "It really does feel like the ideas of random matrix theory are somehow buried deep in the heart of nature," says electrical engineer Raj Nadakuditi of the University of Michigan, Ann Arbor.

All of this, oddly enough, emerged from an effort to turn physicists' ignorance into an advantage. In 1956, when we knew very little about the internal workings of large, complex atomic nuclei, such as uranium, the Hungarian physicist Eugene Wigner suggested simply guessing.

Quantum theory tells us that atomic nuclei have many discrete energy levels, like unevenly spaced rungs on a ladder. To calculate the spacing between each of the rungs, you would need to know the myriad possible ways the nucleus can hop from one to another, and the probabilities for those events to happen. Wigner didn't know, so instead he picked numbers at random for the probabilities and arranged them in a square array called a matrix.

The matrix was a neat way to express the many connections between the different rungs. It also allowed Wigner to exploit the powerful mathematics of matrices in order to make predictions about the energy levels.

Bizarrely, he found this simple approach enabled him to work out the likelihood that any one level would have others nearby, in the absence of any real knowledge. Wigner's results, worked out in a few lines of algebra, were far more useful than anyone could have expected, and experiments over the next few years showed a remarkably close fit to his predictions. Why they work, though, remains a mystery even today.

What is most remarkable, though, is how Wigner's idea has been used since then. It can be applied to a host of problems involving many interlinked variables whose connections can be represented as a random matrix.

The first discovery of a link between Wigner's idea and something completely unrelated to nuclear physics came about after a chance meeting in the early 1970s between British physicist Freeman Dyson and American mathematician Hugh Montgomery.

Montgomery had been exploring one of the most famous functions in mathematics, the Riemann zeta function, which holds the key to finding prime numbers. These are numbers, like 2, 3, 5 and 7, that are only divisible by themselves and 1. They hold a special place in mathematics because every integer greater than 1 can be built from them. Fist tap Dale.