论文标题
宇宙学模型中半经典爱因斯坦方程的解决方案的存在和独特性
Existence and uniqueness of solutions of the semiclassical Einstein equation in cosmological models
论文作者
论文摘要
我们证明了由量子大量标量场驱动的平坦宇宙学空间中半经典爱因斯坦方程的溶液的存在和唯一性，并与标量曲率任意耦合。在半经典近似中，通过将爱因斯坦张量等同于合适状态的应力 - 能量张量的预期值来考虑到曲率的逆转。我们在有限时间施加了比例因子的初始条件，我们表明可以选择与这些初始条件兼容的量子物质的常规状态。在应力 - 能量张量的预期值中存在的指标系数的衍生物的贡献和最高衍生物的术语以非本地形式出现。这一事实禁止对半经典方程式进行直接分析，尤其是近似溶液的标准递归方法无法收敛。在本文中，我们表明，在宇宙学中半经典的爱因斯坦方程部分整合后，非本地最高衍生物通过应用线性无界操作员的应用，而不是取决于所选择状态的细节，而在应力 - 能量张量的预期值中出现。我们证明，可以找到该操作员的反转公式，此外，逆向恰好比直接操作员更正常，并且具有智障产品的形式，因此尊重因果关系。因此，应用于追溯的爱因斯坦方程的发现反转公式具有固定点方程的形式。然后，使用Banach固定点定理获得了半经典爱因斯坦方程的局部存在和唯一性的证明。
We prove existence and uniqueness of solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a quantum massive scalar field with arbitrary coupling to the scalar curvature. In the semiclassical approximation, the backreaction of matter to curvature is taken into account by equating the Einstein tensor to the expectation values of the stress-energy tensor in a suitable state. We impose initial conditions for the scale factor at finite time and we show that a regular state for the quantum matter compatible with these initial conditions can be chosen. Contributions with derivative of the coefficient of the metric higher than the second are present in the expectation values of the stress-energy tensor and the term with the highest derivative appears in a non-local form. This fact forbids a direct analysis of the semiclassical equation, and in particular, standard recursive approaches to approximate the solution fail to converge. In this paper we show that, after partial integration of the semiclassical Einstein equation in cosmology, the non-local highest derivative appears in the expectation values of the stress-energy tensor through the application of a linear unbounded operator which does not depend on the details of the chosen state. We prove that an inversion formula for this operator can be found, furthermore, the inverse happens to be more regular than the direct operator and it has the form of a retarded product, hence causality is respected. The found inversion formula applied to the traced Einstein equation has thus the form of a fixed point equation. The proof of local existence and uniqueness of the solution of the semiclassical Einstein equation is then obtained applying the Banach fixed point theorem.
