An intuitive construction of modular flow

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.

38 pages; 9 figures; 9 pages in appendices; v2 corrects some typos, adds some comments about the GNS construction, and is accepted to JHEP

References in corpus (3)

• Towards a derivation of holographic entanglement entropy
• Entanglement hamiltonians in two-dimensional conformal field theory
• Proof of a Quantum Bousso Bound

Cited by in corpus (7)

• Modular Hamiltonian for de Sitter diamonds
• The Unruh-DeWitt model and its joint interacting Hilbert space
• On the Nonequilibrium Dynamics of Gravitational Algebras
• Relational bulk reconstruction from modular flow
• A short proof of Tomita's theorem
• Gravitational algebras and applications to nonequilibrium physics
• Fiducial observers and the thermal atmosphere in the black hole quantum throat