Dyson's Equations for Quantum Gravity in the Hartree-Fock Approximation
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 calculations that can be done 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 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. 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 G slowly increasing on cosmological scales.