{"title":"Formulation and proof of the gravitational entropy bound","authors":"Artem Averin","doi":"10.1007/JHEP05(2025)193","DOIUrl":null,"url":null,"abstract":"<p>We provide a formulation and proof of the gravitational entropy bound. We use a recently given framework which expresses the measurable quantities of a quantum theory as a weighted sum over paths in the theory’s phase space. If this framework is applied to a field theory on a spacetime foliated by a hypersurface Σ, the choice of a codimension-2 surface <i>B</i> without boundary contained in Σ specifies a submanifold in the phase space. We show here that this submanifold is naturally restricted to obey an entropy bound if the field theory is diffeomorphism-invariant. We prove this restriction to arise by considering the quantum-mechanical sum of paths in phase space and exploiting the interplay of the commutativity of the sum with diffeomorphism-invariance. The formulation of the entropy bound, which we state and derive in detail, involves a functional <i>K</i> on the submanifold associated to <i>B</i>. We give an explicit construction of <i>K</i> in terms of the Lagrangian. The gravitational entropy bound then states: for any real <span>\\( \\frac{\\lambda }{\\hslash } \\)</span>, consider the set of states where <i>K</i> takes a value not bigger than <i>λ</i> and let <i>V</i> denote the phase space volume of this set. One has then ln(<i>V</i>) ≤ <span>\\( \\frac{\\lambda }{\\hslash } \\)</span>. Especially, we show for the Einstein-Hilbert Lagrangian in any dimension with cosmological constant and arbitrary minimally coupled matter, one has <i>K</i> = <span>\\( \\frac{A}{4G} \\)</span>. Hereby, <i>A</i> denotes the area of <i>B</i> in a particular state.</p>","PeriodicalId":635,"journal":{"name":"Journal of High Energy Physics","volume":"2025 5","pages":""},"PeriodicalIF":5.4000,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/JHEP05(2025)193.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of High Energy Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/JHEP05(2025)193","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Physics and Astronomy","Score":null,"Total":0}
引用次数: 0
Abstract
We provide a formulation and proof of the gravitational entropy bound. We use a recently given framework which expresses the measurable quantities of a quantum theory as a weighted sum over paths in the theory’s phase space. If this framework is applied to a field theory on a spacetime foliated by a hypersurface Σ, the choice of a codimension-2 surface B without boundary contained in Σ specifies a submanifold in the phase space. We show here that this submanifold is naturally restricted to obey an entropy bound if the field theory is diffeomorphism-invariant. We prove this restriction to arise by considering the quantum-mechanical sum of paths in phase space and exploiting the interplay of the commutativity of the sum with diffeomorphism-invariance. The formulation of the entropy bound, which we state and derive in detail, involves a functional K on the submanifold associated to B. We give an explicit construction of K in terms of the Lagrangian. The gravitational entropy bound then states: for any real \( \frac{\lambda }{\hslash } \), consider the set of states where K takes a value not bigger than λ and let V denote the phase space volume of this set. One has then ln(V) ≤ \( \frac{\lambda }{\hslash } \). Especially, we show for the Einstein-Hilbert Lagrangian in any dimension with cosmological constant and arbitrary minimally coupled matter, one has K = \( \frac{A}{4G} \). Hereby, A denotes the area of B in a particular state.
期刊介绍:
The aim of the Journal of High Energy Physics (JHEP) is to ensure fast and efficient online publication tools to the scientific community, while keeping that community in charge of every aspect of the peer-review and publication process in order to ensure the highest quality standards in the journal.
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