论文标题
半古典爱因斯坦 - 克莱因 - 戈登系统的静态对称解
Static symmetric solutions of the semi-classical Einstein-Klein-Gordon system
论文作者
论文摘要
我们考虑具有宇宙学常数$λ\ in \ mathbb {r} $的半古典爱因斯坦 - 克莱因 - 戈尔登系统的解决方案，其中Einstein在$ \ mathbb {r} \ times \ times \ times \ times \ times \ times \ times \ quont vont $ quorm a fiend $ splay $ nium a d $ splace afthere上给出了时空。两点分布$ω_2$尊重公制的所有对称性。我们假设质量$ m \ ge0 $和标量曲率耦合$ξ\ in \ mathbb {r} $的字段满足$ m^2+ξr> 0 $，这意味着存在地面状态。我们不需要状态是哈达玛或不含准的，但是无准的解决方案的特征是详细的。 半古典爱因斯坦 - 克莱因 - 凯琳 - 戈登系统的解决方案集取决于参数$（a，λ，m，ξ）$的选择以及标量量量的重态应力张量中的重量法代数。我们表明，解决方案的集合是（i）空集，或者（ii）仅包含基态的单例集，或（iii）一个带有无限多个元素的集合。我们表征了这些替代方案中每种替代的参数和重量法常数的范围。我们还表明，所有无准溶液均通过基态表示中的密度矩阵给出，我们表明在（ii）和（iii）的情况下，有一个独特的无准溶液，可以最大程度地减少von neumann熵。当$ M = 0 $时，此唯一状态为$β$ -KMS状态。我们认为，所有这些结论在降低半古典爱因斯坦方程的顺序中仍然有效。
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.
