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Tomita-Takesaki Modular Theory
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Tomita-Takesaki Modular Theory
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We provide an brief overview of Tomita-Takesaki modular theory and some of its applications to mathematical physics. This is an article commissioned by the Encyclopedia of Mathematical Physics, edited by J.-P. Francoise, G. Naber and T.S. Tsun, to be published by the Elsevier publishing house.
Forward citations
Cited by 10 Pith papers
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Kubo-Martin-Schwinger conditions for non-Hermitian systems
Positivity of the biorthogonal Gibbs functional characterizes quasi-Hermiticity for diagonalisable non-Hermitian operators with real spectra, and the resulting state satisfies the three analytic KMS conditions.
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Kubo-Martin-Schwinger conditions for non-Hermitian systems
For any diagonalisable non-Hermitian H with real spectrum, the biorthogonal Gibbs functional satisfies positivity of ω_bi(A†A) for all A if and only if H is quasi-Hermitian.
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Modular wedge localization, Majorana fields and the Tsirelson limit of the Bell-CHSH inequality
In the 1+1D Majorana QFT the vacuum Bell-CHSH correlator reduces to a modular spectral weight that can be tuned to reach the Tsirelson limit.
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Holography in the linearized quantum gravity regime and modular crossed product
At linearized order, the vacuum-subtracted HRT entropy of a boundary region is the entropy of a coherent graviton state in the modular crossed-product algebra of the dual CFT, assuming a wedge-reconstructing holographic map.
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Modular wedge localization, Majorana fields and the Tsirelson limit of the Bell-CHSH inequality
In the massive Majorana field in 1+1 dimensions, the vacuum Bell-CHSH correlator is reduced to a single spectral weight, with the Tsirelson bound approached as the weight concentrates near modular eigenvalue 1.
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Modular Theory and the Bell-CHSH inequality in relativistic scalar Quantum Field Theory
Modular operators construct wedge-localized one-particle vectors that produce Bell-CHSH violations ~2.3 for free scalar fields, with an outline for approaching Tsirelson's bound via modular-spectrum-aware operators.
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More on Majorana fields, modular localization and near saturation of the Tsirelson bound
Free massless Majorana fields in 1+1D with modular wedge localization nearly saturate the Tsirelson bound for the Bell-CHSH inequality.
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More on Majorana fields, modular localization and near saturation of the Tsirelson bound
Modular wedge localization of free massless Majorana fields in 1+1 dimensions yields Bell-CHSH correlators that approach the Tsirelson bound of 2√2.
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Bosonization, vertex operators and maximal violation of the Bell-CHSH inequality in wedge regions
Vertex operators of a chiral boson realize dichotomic bounded Hermitian operators that saturate the Tsirelson bound of the Bell-CHSH inequality in the vacuum.
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Bohr's complementarity
A review tracing the history of complementarity from Bohr's ideas to its quantification, links with uncertainty and contextuality, and applications in quantum theory.
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