Nuclei, Primes and the Random Matrix Connection

Frank W K Firk,Steven J Miller

doi:10.3390/sym1010064

Abstract

In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics – random matrix theory – provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys.

Highlights

Number Theory PreliminariesThe primes1 are the building blocks of number theory: every integer can be written uniquely as a product of prime powers [12].2 One of the most important questions we can ask about primes is one of the most basic: how many primes are there at most x? In other words, how many building blocks are there up to a given point?Euclid proved over 2000 years ago that there are infinitely many primes; so, if we let π(x) denote the number of primes at most x, we know limx→∞ π(x) = ∞
In the early 1970’s a remarkable connection was unexpectedly discovered between two very different fields, nuclear physics and number theory, when it was noticed that random matrix theory accurately modeled many problems in each
We assume no familiarity with either subject; for the most part, basic linear algebra and probability theory suffice

Summary

In the early 1970’s a remarkable connection was unexpectedly discovered between two very different fields, nuclear physics and number theory, when it was noticed that random matrix theory accurately modeled many problems in each. As there are many mathematical surveys of the subject, as well as some popular accounts [4,5] of how the connection between the fields was noticed, our goal is to explain the broad brushstrokes of the theory without getting bogged down in the technical details For those interested in a more mathematical survey, we recommend [6,7,8,9,10] (see Section 1.8 of [11]). For the benefit of the reader, we have included in the footnotes definitions and explanations of much of the assumed background material to help keep the paper accessible

Introduction

Dirichlet L-functions

Cuspidal newforms

Elliptic curves

38. Statistical Theories of Spectra