Unlike scalar and gauge field theories in four dimensions, gravity is not perturbatively renormalizable and as a result perturbation theory is badly divergent. Often the method of choice for investigating nonperturbative effects has been the lattice formulation, and in the case of gravity the Regge–Wheeler lattice path integral lends itself well for that purpose. Nevertheless, lattice methods ultimately rely on extensive numerical calculations, leaving a desire for alternate methods that can be pursued analytically. In this work, we outline the Hartree–Fock approximation to quantum gravity, along lines which are analogous to what is done for scalar fields and gauge theories. The starting point is Dyson’s equations, a closed set of integral equations which relate various physical amplitudes involving graviton propagators, vertex functions, and proper self-energies. Such equations are in general difficult to solve, and as a result they are not very useful in practice, but nevertheless provide a basis for subsequent approximations. This is where the Hartree–Fock approximation comes in, whereby lowest order diagrams get partially dressed by the use of fully interacting Green’s function and self-energies, which then lead to a set of self-consistent integral equations. The resulting nonlinear equations for the graviton self-energy show some remarkable features that clearly distinguish it from the scalar and gauge theory cases. Specifically, for quantum gravity one finds a nontrivial ultraviolet fixed point in Newton’s constant G for spacetime dimensions greater than two, and nontrivial scaling dimensions between $d=2$ and $d=4$, above which one obtains Gaussian exponents. In addition, the Hartree–Fock approximation gives an explicit analytic expression for the renormalization group running of Newton’s constant, suggesting gravitational antiscreening with Newton’s constant slowly increasing on cosmological scales.

中文翻译：

---

Hartree-Fock近似中戴森的量子引力方程

与在四个维度上的标量场和标距场理论不同，重力不是可微扰地重新归一化的，因此，微扰理论的分歧也很大。研究非微扰效应的首选方法通常是晶格公式，在重力情况下，Regge-Wheeler晶格路径积分很适合于此目的。然而，晶格方法最终依赖于大量的数值计算，从而留下了可以通过分析来追求的替代方法的需求。在这项工作中，我们沿着与标量场和规范理论相似的线，概述了对量子引力的Hartree-Fock近似。起点是戴森方程式，这是一组封闭的积分方程式，它涉及引力子传播器，顶点函数，和适当的自我能量。这样的方程式通常很难求解，结果它们在实践中不是很有用，但是仍然为后续的近似提供了基础。这就是Hartree-Fock逼近的用处，通过使用完全相互作用的Green函数和自能量，最低阶图得到了部分修饰，然后得出了一组自洽积分方程。引力子自能量的非线性方程式显示出一些显着特征，这些特征将其与标量和规范理论情况区分开来。具体来说，对于量子引力，人们在牛顿常数中找到了一个非平凡的紫外线固定点 但是尽管如此，它仍可为后续近似提供基础。这就是Hartree-Fock逼近的用处，通过使用完全相互作用的Green函数和自能量，最低阶图得到了部分修饰，然后得出了一组自洽积分方程。引力子自能量的非线性方程式显示出一些显着的特征，这些特征明显地将其与标量和规范理论情况区分开。具体来说，对于量子引力，人们在牛顿常数中找到了一个非平凡的紫外线固定点 但是尽管如此，它仍可为后续近似提供基础。这就是Hartree-Fock逼近的用处，通过使用完全相互作用的Green函数和自能量，最低阶图得到了部分修饰，然后得出了一组自洽积分方程。引力子自能量的非线性方程式显示出一些显着的特征，这些特征明显地将其与标量和规范理论情况区分开。具体来说，对于量子引力，人们在牛顿常数中找到了一个非平凡的紫外线固定点 引力子自能量的非线性方程式显示出一些显着的特征，这些特征明显地将其与标量和规范理论情况区分开。具体来说，对于量子引力，人们在牛顿常数中找到了一个非平凡的紫外线固定点 引力子自能量的非线性方程式显示出一些显着的特征，这些特征明显地将其与标量和规范理论情况区分开。具体来说，对于量子引力，人们在牛顿常数中找到了一个非平凡的紫外线固定点G表示时空维度大于2，且非平凡缩放维度介于$d=2$ 和 $d=4$，在此之上可获得高斯指数。此外，Hartree-Fock逼近为牛顿常数的重整化群运行给出了明确的解析表达式，表明重力反筛查以牛顿常数在宇宙尺度上缓慢增加。