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

We consider solutions of the semi-classical Einstein-Klein-Gordon system with a cosmological constant $\Lambda\in\mathbb{R}$, 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 $\omega_2$ that respects all the symmetries of the metric. We assume that the mass $m\ge0$ and scalar curvature coupling $\xi\in\mathbb{R}$ of the field satisfy $m^2+\xi R>0$, 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 $(a,\Lambda,m,\xi)$ 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 $\beta$-KMS state. We argue that all these conclusions remain valid in the reduced order formulation of the semi-classical Einstein equation.