Many-Particle Systems in Mathematics and Physics

Abstract

Partition functions are the main tool for studying many-particle systems. Folklore 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. (i) Coding: The many-particle system is encoded into one single function called a partition function (e.g., the Boltzmann partition function in statistical physics, Riemann's zeta function or Dirichlet's L-function for describing prime numbers, the Feynman functional integral in quantum field theory). The idea goes back to Euler; in 1737 he proved the identity Y p " 1 − 1 p s « −1 = ∞ X n=1 1 n s = ζ(s) for all real numbers s > 1. The product refers to all prime numbers p. (ii) Decoding: The task is to get crucial information about the many-particle system by studying the properties of the partition function. The idea goes back to Riemann. He recognized that the zeta function ζ extends holomorphically to the punctured complex plane C \ {0}, and that the detailed knowledge on the distribution of the zeros of the zeta function allows far-reaching statements about the asymptotic distribution of the prime numbers. This is related to the famous Riemann hypothesis to be considered on page 298. The complexity of justifying the Riemann hypothesis and the mathematical trouble with the Feynman path integral prove that step (ii) is much more complex than step (i). By analytic continuation, the series ∞ X n=1 1 n s can be assigned the value ζ(s) for all complex numbers s with s = 1. In particular, for the uncritical value s = 0, we can assign the value