Journal Article

An intuitive construction of modular flow

Sorce, Jonathan

2023

Overview

A bstract 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.

• Publisher: Springer Berlin Heidelberg,Springer Nature B.V,SpringerOpen
• Subject: Algebra / Banach spaces / Classical and Quantum Gravitation / Elementary Particles / Fourier analysis / High energy physics / Modular construction / Non-Equilibrium Field Theory / Operators (mathematics) / Pedagogy / Physicists / Physics / Physics and Astronomy / Quantum Field Theories / Quantum Field Theory / Quantum Physics / Quantum theory / Regular Article - Theoretical Physics / Relativity Theory / Space-Time Symmetries / String Theory / Textbooks / Theorems / Thermodynamic equilibrium