标题： 引力熵界的形式化与证明 显示英文标题

标题： Formulation and Proof of the Gravitational Entropy Bound

摘要： 我们提供了一种引力熵界的形式化和证明。我们使用了一个最近给出的框架，该框架将量子理论中的可测量量表示为在理论相空间中路径的加权和。 如果将此框架应用于时空上的场论，该时空由超曲面$\Sigma,$叶状结构划分，选择一个无边界的余维2曲面$B$包含在$\Sigma$中，指定相空间中的一个子流形。 我们在此表明，如果场论是微分同胚不变的，这个子流形自然受限于服从熵界。我们通过考虑相空间中路径的量子力学求和，并利用求和的交换性与微分同胚不变性之间的相互作用来证明这一限制。 我们详细陈述并推导了熵界的表述，它涉及与$B.$相关的子流形上的泛函$K$。我们给出了$K$显式构造，以拉格朗日量为基础。 引力熵界则陈述如下：对于任意实数$\frac{\lambda}{\hbar},$，考虑使得$K$的值不超过$\lambda$的状态集合，并令$V$表示该集合的相空间体积。 于是有$\ln (V) \le \frac{\lambda}{\hbar}.$特别地，我们证明了对于具有宇宙常数和任意极小耦合物质的任何维度的爱因斯坦-希尔伯特拉格朗日量，有$K = \frac{A}{4G}.$其中，$A$表示在特定状态下的$B$的面积。 显示英文摘要

摘要： 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 $\Sigma,$ the choice of a codimension-2 surface $B$ without boundary contained in $\Sigma$ 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 $\frac{\lambda}{\hbar},$ consider the set of states where $K$ takes a value not bigger than $\lambda$ and let $V$ denote the phase space volume of this set. One has then $\ln (V) \le \frac{\lambda}{\hbar}.$ 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.