New Classical Limits of Quantum Theories

TL;DR

This paper explores new classical limits of quantum theories where quantum fluctuations vanish under parameters other than Planck's constant, leading to better approximations and connections to random matrix spectra.

Contribution

It introduces novel classical limits in quantum theories, including large spatial dimension and Hecke operators, linking quantum fluctuations to classical behavior and spectral properties.

Findings

Quantum fluctuations vanish as parameters like 1/N go to zero.

Classical limits improve approximation accuracy to quantum theories.

Hecke operators' spectra tend to random matrix spectra at large weights.

Abstract

Quantum fluctuations of some systems vanish not only in the limit $\hbar\to 0$, but also as some other parameters (such as $1\over N$, the inverse of the number of `colors' of a Yang-Mills theory) vanish. These lead to new classical limits that are often much better approximations to the quantum theory. We describe two examples: the familiar Hartree--Fock-Thomas-Fermi methods of atomic physics as well as the limit of large spatial dimension. Then we present an approach of the Hecke operators on modular forms inspired by these ideas of quantum mechanics. It explains in a simple way why the spectra of these operators tend to the spectrum of random matrices for large weight for the modular forms.