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This paper provides a rigorous mathematical proof of the Rényi quantum null energy condition (QNEC) for integer parameters $n \geq 2$ within the framework of local Poincaré-invariant quantum field theory. By establishing log-convexity properties for von Neumann algebras with half-sided modular inclusions, the work advances fundamental theoretical bounds on energy conditions essential to quantum gravity and holography.
A general proof of integer R\'enyi QNEC
arXiv:2605.15272v2 Announce Type: replace-cross Abstract: The R\'enyi quantum null energy condition conjectures that the second null shape variation of the sandwiched R\'enyi divergence (SRD) of an excited state relative to the vacuum is non-negative in local Poincar\'e-invariant quantum field theory, giving a one-parameter generalization of the quantum null energy condition (QNEC). We prove R\'enyi QNEC for all integer R\'enyi parameters $n\geq 2$ for von Neumann algebras carrying a half-sided modular inclusion structure. The only assumption on the excited state is finiteness of its SRD relative to the vacuum. Concretely, for any $\sigma$-finite von Neumann algebra with such an inclusion, we prove log-convexity, under the associated null-translation semigroup, of the Kosaki $L^n$ norm of any normal positive functional with finite $L^n$ norm.
Abstract:The Rényi quantum null energy condition conjectures that the second null shape variation of the sandwiched Rényi divergence (SRD) of an excited state relative to the vacuum is non-negative in local Poincaré-invariant quantum field theory, giving a one-parameter generalization of the quantum null energy condition (QNEC). We prove Rényi QNEC for all integer Rényi parameters $n\geq 2$ for von Neumann algebras carrying a half-sided modular inclusion structure. The only assumption on the excited state is finiteness of its SRD relative to the vacuum. Concretely, for any $\sigma$-finite von Neumann algebra with such an inclusion, we prove log-convexity, under the associated null-translation semigroup, of the Kosaki $L^n$ norm of any normal positive functional with finite $L^n$ norm.
57 pages, 1 figure. v2: Significant rewriting to improve exposition; results unchanged
High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Operator Algebras (math.OA); Quantum Physics (quant-ph)
arXiv:2605.15272 [hep-th]
(or arXiv:2605.15272v2 [hep-th] for this version)
https://doi.org/10.48550/arXiv.2605.15272 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 232 words
