Formulation and proof of the gravitational entropy bound

IF 5.4 1区 物理与天体物理 Q1 Physics and Astronomy
Artem Averin
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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.

引力熵界的公式和证明
我们提供了引力熵界的一个公式和证明。我们使用一个最近给出的框架,该框架将量子理论的可测量量表示为理论相空间中路径的加权和。如果将该框架应用于由超曲面Σ分叶的时空场理论,则选择包含在Σ中的无边界的余维2曲面B指定了相空间中的子流形。我们在这里证明,如果场论是微分不变的,那么这个子流形自然地受限于服从熵界。我们通过考虑相空间中路径的量子力学和,并利用和的交换性与微分同态不变性的相互作用,证明了这一限制的产生。我们详细说明并推导了熵界的公式,它涉及到与b相关的子流形上的泛函K。我们给出了K在拉格朗日量中的显式构造。引力熵界则表示:对于任何实数\( \frac{\lambda }{\hslash } \),考虑K取值不大于λ的状态集,并设V表示该集合的相空间体积。ln(V)≤\( \frac{\lambda }{\hslash } \)。特别地,我们证明了在具有宇宙常数和任意最小耦合物质的任何维度上的爱因斯坦-希尔伯特拉格朗日,有K = \( \frac{A}{4G} \)。其中,A表示B在特定状态下的面积。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of High Energy Physics
Journal of High Energy Physics 物理-物理:粒子与场物理
CiteScore
10.30
自引率
46.30%
发文量
2107
审稿时长
1.5 months
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. Consequently, the Advisory and Editorial Boards, composed of distinguished, active scientists in the field, jointly establish with the Scientific Director the journal''s scientific policy and ensure the scientific quality of accepted articles. JHEP presently encompasses the following areas of theoretical and experimental physics: Collider Physics Underground and Large Array Physics Quantum Field Theory Gauge Field Theories Symmetries String and Brane Theory General Relativity and Gravitation Supersymmetry Mathematical Methods of Physics Mostly Solvable Models Astroparticles Statistical Field Theories Mostly Weak Interactions Mostly Strong Interactions Quantum Field Theory (phenomenology) Strings and Branes Phenomenological Aspects of Supersymmetry Mostly Strong Interactions (phenomenology).
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