Bosonization of non-relativistic fermions on a circle: Tomonaga's problem revisited

TL;DR

This paper revisits Tomonaga's problem by applying exact bosonization to non-relativistic fermions on a circle, revealing a relativistic boson description at large N and low energies, with implications for quantum field theory and black hole physics.

Contribution

It introduces an exact bosonization approach for finite fermion systems, clarifies the large-N low-energy limit, and connects the results to Yang-Mills theory and black hole nonperturbative effects.

Findings

Bosonized Hamiltonian splits into O(N) and O(1) parts.

Large-N, low-energy limit yields a massless relativistic boson.

Exact bosonization captures finite-N, high-energy effects and nonperturbative corrections.

Abstract

We use the recently developed tools for an exact bosonization of a finite number $N$ of non-relativistic fermions to discuss the classic Tomonaga problem. In the case of noninteracting fermions, the bosonized hamiltonian naturally splits into an O$(N)$ piece and an O$(1)$ piece. We show that in the large-N and low-energy limit, the O$(N)$ piece in the hamiltonian describes a massless relativistic boson, while the O$(1)$ piece gives rise to cubic self-interactions of the boson. At finite $N$ and high energies, the low-energy effective description breaks down and the exact bosonized hamiltonian must be used. We also comment on the connection between the Tomonaga problem and pure Yang-Mills theory on a cylinder. In the dual context of baby universes and multiple black holes in string theory, we point out that the O$(N)$ piece in our bosonized hamiltonian provides a simple understanding of the origin of two different kinds of nonperturbative O$(e^{-N})$ corrections to the black hole partition function.