{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:43:51Z","timestamp":1760233431984,"version":"build-2065373602"},"reference-count":125,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T00:00:00Z","timestamp":1610409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2013Wheeler 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\u2013Fock approximation to quantum gravity, along lines which are analogous to what is done for scalar fields and gauge theories. The starting point is Dyson\u2019s 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\u2013Fock approximation comes in, whereby lowest order diagrams get partially dressed by the use of fully interacting Green\u2019s 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\u2019s 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\u2013Fock approximation gives an explicit analytic expression for the renormalization group running of Newton\u2019s constant, suggesting gravitational antiscreening with Newton\u2019s constant slowly increasing on cosmological scales.<\/jats:p>","DOI":"10.3390\/sym13010120","type":"journal-article","created":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T20:11:31Z","timestamp":1610482291000},"page":"120","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Dyson\u2019s Equations for Quantum Gravity in the Hartree\u2013Fock Approximation"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0316-9584","authenticated-orcid":false,"given":"Herbert W.","family":"Hamber","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lu Heng Sunny","family":"Yu","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1103\/PhysRevLett.28.548","article-title":"Feynman-graph Expansion for Critical Exponents","volume":"28","author":"Wilson","year":"1972","journal-title":"Phys. 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