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An intuitive construction of modular flow

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arxiv 2309.16766 v2 pith:FEQB66OT submitted 2023-09-28 hep-th math-phmath.MPquant-ph

classification hep-thmath-phmath.MPquant-ph
keywords modularflowfundamentaloperatortheoryanalysisconstructioncontribution
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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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Uniqueness of null-local modular flow

    hep-th 2026-07 conditional novelty 7.0 of 10

    For the free massless scalar in a Minkowski causal diamond, the vacuum is the unique state or weight in the vacuum sector whose modular flow is local on the null boundary.

  2. Excitability in quantum field theory

    hep-th 2026-04 unverdicted novelty 7.0 of 10

    For zero-mean Gaussian states in generalized free field theories, one-way local excitability always implies two-way excitability, generalizing the quasiequivalence theorems of Powers, Stormer, van Daele, Araki, and Yamagami.

  3. Geometric modular flows in 2d CFT and beyond

    hep-th 2025-02 conditional novelty 7.0 of 10

    In 2d CFTs, every suitably regular Unruh flow on the Rindler wedge is the modular flow of a state obtained by a conformal unitary acting on the vacuum or thermal state, and local entropy and stress-tensor formulas follow.

  4. Black hole thermodynamics at null infinity. Part 1: Dual Generalized Second Law

    hep-th 2026-01 conditional novelty 5.0 of 10

    At future null infinity, the generalized second law for a Schwarzschild black hole becomes the monotonic decrease of a free energy, or grand potential, constructed from the Bondi mass and angular-mode chemical potentials.

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