An intuitive construction of modular flow

TL;DR

This paper introduces an intuitive, physics-based construction of modular flow in quantum field theory, deriving the modular operator from thermal physics principles and providing a clearer, rigorous proof of its fundamental properties.

Contribution

It offers a new pedagogical approach starting from thermal physics to construct modular flow, avoiding complex Fourier analysis and clarifying its symmetry properties.

Findings

Derived modular operator from KMS condition

Provided a new rigorous proof of modular flow symmetry

Simplified understanding of modular flow for physicists

Abstract

The theory of modular flow has proved extremely useful for applying thermodynamic reasoning to out-of-equilibrium states in quantum field theory. However, the standard proofs of the fundamental theorems of modular flow use machinery from Fourier analysis on Banach spaces, and as such are not especially transparent to an audience of physicists. In this article, I present a construction of modular flow that differs from existing treatments. The main pedagogical contribution is that I start with thermal physics via the KMS condition, and derive the modular operator as the only operator that could generate a thermal time-evolution map, rather than starting with the modular operator as the fundamental object of the theory. The main technical contribution is a new proof of the fundamental theorem stating that modular flow is a symmetry. The new proof circumvents the delicate issues of Fourier analysis that appear in previous treatments, but is still mathematically rigorous.