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arXiv 2007.14311math-phgr-qcmath.MP
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半经典Einstein-Klein-Gordon系统的静态对称解

Static symmetric solutions of the semi-classical Einstein-Klein-Gordon system

Ko Sanders

备注：
31p, v2: added reference and minor corrections
发表信息：
Ann. Henri Poincaré (2021) (online first)
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摘要：

我们考虑具有宇宙学常数$Λ\in\mathbb{R}$的半经典爱因斯坦-克莱因-戈登系统的解，其中时空由爱因斯坦在$\mathbb{R}\times\mathbb{S}^3$上的静态度规给出，该度规具有半径为$a>0$的圆球，并且标量量子场的状态具有两点分布$ω_2$，该分布尊重度规的所有对称性。我们假设场的质量$m\ge0$和标量曲率耦合$ξ\in\mathbb{R}$满足$m^2+ξR>0$，这意味着基态的存在。我们不要求状态是阿达玛或准自由的，但准自由的解决方案的特点是在充分的细节。半经典Einstein-Klein-Gordon系统的解集依赖于参数$(a,Λ,m,ξ)$的选择和标量场的重整化应力张量中的重整化常数。我们证明了解的集合是（i）空集，或者（ii）只包含基态的单例集，或者（iii）具有无穷多个元素的集合。我们的范围内的参数和重整化常数，这些替代品发生。我们还表明，所有的准自由的解决方案是由密度矩阵在基态表示，我们表明，在情况（ii）和（iii）有一个独特的准自由的解决方案，使冯诺依曼熵最小化。当$m=0$时，此唯一状态为$β$ -KMS状态。我们认为，所有这些结论仍然有效的半经典爱因斯坦方程的降阶制定。

英文摘要：

We consider solutions of the semi-classical Einstein-Klein-Gordon system with a cosmological constant $Λ\in\mathbb{R}$, where the spacetime is given by Einstein's static metric on $\mathbb{R}\times\mathbb{S}^3$ with a round sphere of radius $a>0$ and the state of the scalar quantum field has a two-point distribution $ω_2$ that respects all the symmetries of the metric. We assume that the mass $m\ge0$ and scalar curvature coupling $ξ\in\mathbb{R}$ of the field satisfy $m^2+ξR>0$, which entails the existence of a ground state. We do not require states to be Hadamard or quasi-free, but the quasi-free solutions are characterised in full detail. The set of solutions of the semi-classical Einstein-Klein-Gordon system depends on the choice of the parameters $(a,Λ,m,ξ)$ and on the renormalisation constants in the renormalised stress tensor of the scalar field. We show that the set of solutions is either (i) the empty set, or (ii) the singleton set containing only the ground state, or (iii) a set with infinitely many elements. We characterise the ranges of the parameters and renormalisation constants where each of these alternatives occur. We also show that all quasi-free solutions are given by density matrices in the ground state representation and we show that in cases (ii) and (iii) there is a unique quasi-free solution which minimises the von Neumann entropy. When $m=0$ this unique state is a $β$-KMS state. We argue that all these conclusions remain valid in the reduced order formulation of the semi-classical Einstein equation.

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