Integrable Systems

A swinging bridge

Most of us will have crossed a suspension bridge wondering which frequencies of steps will make the bridge swing most, i.e. could produce a so called resonance of the bridge.

The phenomenon of resonances is known from many other aspects of every day live such as for example a violin. Thinking of the strings on a violin, it is clear that the frequencies depend on the material the strings are made of and their tension. The latter is used to tune a violin.

A hiker crossing a swing bridge on the Huxley River in the South Island of New Zealand. Courtesy Alan Liefting.

Smart tools

The systems we have looked at in the previous paragraph belong to the macroscopic world.

But similar questions can also be asked when we go to systems of atoms which are described by quantum rather than classical mechanics. In this case, the systems may be come as string of atoms, spin chains or quantum wires. As in the world of classical mechanics, one may wonder about the resonance frequencies of atomic strings. Starting in particular with the highly influential work of the physicist Hans Bethe, scientists have indeed developed very smart tools to answer such questions. Bethe’s paper was entitled “Zur Theorie der Metalle” (On the Theory of Metals) and it opened an important new chapter in condensed matter theory.

The term integrable systems refers to “quantum strings” for which we can determine resonance frequencies using the techniques that generations of physicists and mathematicians developed after Bethe had initiated this development. But now that we have an idea what integrable systems are about we should wonder what they have to do with gauge theories. Even though one of the most famous equations in the theory of integrable systems was discovered by C.N. Yang and R.J. Baxter, the same C.N. Yang that proposed Yang-Mills gauge theory, the two subjects seemed to have no connection for many decades, until about 10 years ago when GATIS scientists uncovered the surprising link: string theory.