Packaging Spectra (as in Partition Functions and L/$\zeta$-functions) to Reveal Symmetries (Reciprocity) in Nature and in Numbers

TL;DR

This paper explores how packaging functions like partition and zeta functions reveal hidden symmetries in physical systems and number theory, connecting diverse mathematical and physical phenomena.

Contribution

It introduces a unified perspective on packaging functions, highlighting their role in uncovering symmetries across physics and mathematics, inspired by the Langlands Program.

Findings

Packaging functions exhibit symmetries not apparent from the original spectra.

Understanding packaging functions can lead to symmetry discovery independent of the underlying data.

Connections are drawn between physical systems, number theory, and the Langlands Program.

Abstract

In statistical mechanics one packages the possible energies of a system into a partition function. In number theory, and elsewhere in mathematics, one packages the spectrum of a phenomenon, say the prime numbers, into a $\zeta$-function or more generally into an L-function. These packaging functions have symmetries and properties not at all apparent from the energies or the primes themselves, often exhibiting scaling symmetries for example. One might be able to understand those symmetries and compute the packaging function independently of the actual packaging. And so one finds a way of putting together objects into a package, and ways of discerning symmetries of that package independent of the actual mode of packaging. This is a recurrent theme of the Langlands Program as well. Packaging is also found in Weyl's asymptotics and "hearing the shape of a drum" (Kac), the Schwinger Greens function in quantum electrodynamics packaging the Feynman sum of histories, and more generally in the Selberg trace formula.