Many-Particle Systems in Mathematics and Physics

Many-particle systems play a fundamental role in both mathematics and physics.

• In physics, we encounter systems of molecules (e.g., gases or liquids) or systems of elementary particles in quantum field theory.

• In mathematics, for example, we want to study the system of prime numbers.

In the 19th century, physicists developed the methods of statistical mechanics for studying many-particle systems, whereas mathematicians proved the distribution law for prime numbers. It turns out that the two apparently different approaches can be traced back to the same mathematical root, namely, the notion of partition function. In modern quantum field theory, the Feynman functional integral can be viewed as a partition function, as we will discuss later on. The typical procedure proceeds in the following two steps.