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Modular dynamics in diamonds
Mathematical Physics · Submitted 25 Sep 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In particular, we prove a long-standing conjecture that says that the generators of the mentioned actions differ by a pseudo-differential operator of order zero.
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The abstract, sentence by sentence
We investigate the relation between the actions of Tomita-Takesaki modular operators for local von Neumann algebras in the vacuum for free massive and massless bosons in four dimensional Minkowskian spacetime. In particular, we prove a long-standing conjecture that says that the generators of the mentioned actions differ by a pseudo-differential operator of order zero. To get that, one needs a careful analysis of the interplay of the theories in the bulk and at the boundary of double cones (a.k.a. diamonds). After introducing some technicalities, we prove the crucial result that the vacuum state for massive bosons in the bulk of a double cone restricts to a KMS state at its boundary, and that the restriction of the algebra at the boundary does not depend anymore on the mass. The origin of such result lies in a careful treatment of classical Cauchy and Goursat problems for the Klein-Gordon equation as well as the application of known general mathematical techniques, concerning the interplay of algebraic structures related with the bulk and algebraic structures related with the boundary of the double cone, arising from quantum field theories in curved spacetime. Our procedure gives explicit formulas for the modular group and its generator in terms of integral operators acting on symplectic space of solutions of massive Klein-Gordon Cauchy problem.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the relation between the actions of Tomita-Takesaki modular operators for local von Neumann algebras in the vacuum for free massive and massless bosons in four dimensional Minkowskian spacetime. | contribution 0.69 | 0.82 | 0.04 | unclear 0.58 |
| s1 | In particular, we prove a long-standing conjecture that says that the generators of the mentioned actions differ by a pseudo-differential operator of order zero. | result 0.90 | 0.94 | 0.84 | asserted 1.00 |
| s2 | To get that, one needs a careful analysis of the interplay of the theories in the bulk and at the boundary of double cones (a.k.a. diamonds). | method 0.92 | 0.35 | 0.04 | asserted 0.93 |
| s3 | After introducing some technicalities, we prove the crucial result that the vacuum state for massive bosons in the bulk of a double cone restricts to a KMS state at its boundary, and that the restriction of the algebra at the boundary does not depend anymore on the mass. | result 1.00 | 0.93 | 0.42 | asserted 1.00 |
| s4 | The origin of such result lies in a careful treatment of classical Cauchy and Goursat problems for the Klein-Gordon equation as well as the application of known general mathematical techniques, concerning the interplay of algebraic structures related with the bulk and algebraic structures related with the boundary of the double cone, arising from quantum field theories in curved spacetime. | method 0.96 | 0.60 | 0.06 | asserted 1.00 |
| s5 | Our procedure gives explicit formulas for the modular group and its generator in terms of integral operators acting on symplectic space of solutions of massive Klein-Gordon Cauchy problem. | result 0.52 | 0.95 | 0.13 | asserted 1.00 |
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Original abstract
We investigate the relation between the actions of Tomita-Takesaki modular operators for local von Neumann algebras in the vacuum for free massive and massless bosons in four dimensional Minkowskian spacetime. In particular, we prove a long-standing conjecture that says that the generators of the mentioned actions differ by a pseudo-differential operator of order zero. To get that, one needs a careful analysis of the interplay of the theories in the bulk and at the boundary of double cones (a.k.a. diamonds). After introducing some technicalities, we prove the crucial result that the vacuum state for massive bosons in the bulk of a double cone restricts to a KMS state at its boundary, and that the restriction of the algebra at the boundary does not depend anymore on the mass. The origin of such result lies in a careful treatment of classical Cauchy and Goursat problems for the Klein-Gordon equation as well as the application of known general mathematical techniques, concerning the interplay of algebraic structures related with the bulk and algebraic structures related with the boundary of the double cone, arising from quantum field theories in curved spacetime. Our procedure gives explicit formulas for the modular group and its generator in terms of integral operators acting on symplectic space of solutions of massive Klein-Gordon Cauchy problem.
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1 arXiv paper cite this in 1 sentence
Papers citing it
Cites it as background 1 paper
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Despite their importance, our understanding of modular flows in QFT is limited (previous results include [ref] ).
Modular Flow of Excited States · Introduction · 0.95
What it cites · 17 sentences without a confident label
Cites as background 7 papers
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In this respect, one of the milestones has been the connection of Tomita and Takesaki modular theory[For short lucid introductions we refer to the papers of Guido [ref] , Lledó [ref] and Summers [ref] .
Modular Theory by example · Introduction · 1.00
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In this respect, one of the milestones has been the connection of Tomita and Takesaki modular theory[For short lucid introductions we refer to the papers of Guido [ref] , Lledó [ref] and Summers [ref] .
Modular theory for the von Neumann algebras of Local Quantum Physics · Introduction · 1.00
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In this respect, one of the milestones has been the connection of Tomita and Takesaki modular theory[For short lucid introductions we refer to the papers of Guido [ref] , Lledó [ref] and Summers [ref] .
Tomita-Takesaki Modular Theory · Introduction · 1.00
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