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Nuclearity and Thermal States in Conformal Field Theory
Mathematical Physics · Submitted 30 Mar 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the semigroup generated by the conformal Hamiltonian L_0, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property.
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The abstract, sentence by sentence
We introduce a new type of spectral density condition, that we call L^2-nuclearity. One formulation concerns lowest weight unitary representations of SL(2,R) and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the semigroup generated by the conformal Hamiltonian L_0, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L_0 is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a beta-KMS state for the translation dynamics on the net of C*-algebras for every inverse temperature beta>0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L^2-nuclearity is satisfied for the scalar, massless Klein-Gordon field.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We introduce a new type of spectral density condition, that we call L^2-nuclearity. | contribution 1.00 | 0.95 | 0.09 | asserted 0.99 |
| s1 | One formulation concerns lowest weight unitary representations of SL(2,R) and turns out to be equivalent to the existence of characters. | result 0.49 | 0.63 | 0.08 | asserted 0.99 |
| s2 | A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. | contribution 0.87 | 0.28 | 0.04 | asserted 0.93 |
| s3 | We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the semigroup generated by the conformal Hamiltonian L_0, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. | result 0.82 | 0.95 | 0.56 | asserted 1.00 |
| s4 | As a corollary, if L_0 is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a beta-KMS state for the translation dynamics on the net of C*-algebras for every inverse temperature beta>0. | result 0.91 | 0.83 | 0.16 | asserted 1.00 |
| s5 | We include further discussions on higher dimensional spacetimes. | contribution 0.83 | 0.72 | 0.02 | unclear 0.59 |
| s6 | In particular, we verify that L^2-nuclearity is satisfied for the scalar, massless Klein-Gordon field. | result 0.97 | 0.93 | 0.08 | asserted 1.00 |
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Original abstract
We introduce a new type of spectral density condition, that we call L^2-nuclearity. One formulation concerns lowest weight unitary representations of SL(2,R) and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the semigroup generated by the conformal Hamiltonian L_0, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L_0 is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a beta-KMS state for the translation dynamics on the net of C*-algebras for every inverse temperature beta>0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L^2-nuclearity is satisfied for the scalar, massless Klein-Gordon field.
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How the literature uses it
9 arXiv papers cite this in 20 sentences
Papers citing it · 16 sentences without a confident label
Cites it as support 1 paper
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The split property follows from the trace class property of $e^-Ḻ_0$, for all $>̱0$ in the vacuum representation [ref] ; it holds automatically for a conformal net [ref] .
Rotational KMS states and type I conformal nets · Completely rational case · 0.96
Cites it as background 3 papers
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For chiral nets, the conformal nuclearity condition was studied in [ref] , together with other notions of nuclearity.
Towards entanglement entropy with UV cutoff in conformal nets · Nuclearity conditions and the split property · 0.97
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We refer to the original works [ref] for more details on this topic.
Conformal covariance and the split property · Introduction · 0.83
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In the literature, there exist several “nuclearity” conditions [ref] , reminiscent of the trace class condition Tr$(e^-βH)<∞$ for Gibbs states in quantum mechanics, which are related to the split property and thermodynamical properties.
Algebraic constructive quantum field theory: Integrable models and deformation techniques · Relative commutants of wedge algebras · 0.90
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