# Dyson’s Equations for Quantum Gravity in the Hartree–Fock Approximation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dyson’s Equations and the Hartree–Fock Approximation

## 3. The Case of the Nonlinear Sigma Model

## 4. Nonlinear Sigma Model in the Large-$\mathbf{N}$ Limit

## 5. Hartree–Fock Method for the Nonlinear Sigma Model

## 6. Gauge Theories

## 7. The Quantum Gravity Case

## 8. A Sample Application to Cosmology

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wilson, K.G. Feynman-graph Expansion for Critical Exponents. Phys. Rev. Lett.
**1972**, 28, 548–551. [Google Scholar] [CrossRef] - Wilson, K.G.; Fisher, M.E. Critical Exponents in 3.99 Dimensions. Phys. Rev. Lett.
**1972**, 28, 240–243. [Google Scholar] [CrossRef] - Wilson, K.G. Quantum Field-Theory Models in Less than 4 Dimensions. Phys. Rev. D
**1973**, 7, 2911–2926. [Google Scholar] [CrossRef] - Parisi, G. On the Renormalizability of not Renormalizable Theories. Lett. Nuovo Cim.
**1973**, 6, 450–452. [Google Scholar] [CrossRef] - Parisi, G. Theory of Non-Renormalizable Interactions—The large N expansion. Nucl. Phys. B
**1975**, 100, 368–388. [Google Scholar] [CrossRef] - Parisi, G. On Non-Renormalizable Interactions. In New Developments in Quantum Field Theory and Statistical Mechanics; Levy, M., Mitter, P., Eds.; Nato Advanced Study Institutes Series; Plenum Press: New York, NY, USA, 1977. [Google Scholar]
- Parisi, G. Statistical Field Theory; Benjamin Cummings: San Francisco, CA, USA, 1981. [Google Scholar]
- Itzykson, C.; Drouffe, J.M. Statistical Field Theory; Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]
- Cardy, J.L. Scaling and Renormalization in Statistical Physics; Cambridge Lecture Notes in Physics; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Zinn-Justin, J. Quantum Field Theory and Critical Phenomena, 4th ed.; Oxford University Press: Oxford, UK, 2002. [Google Scholar]
- Brezin, E. Introduction to Statistical Field Theory; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Regge, T. General Relativity without Coordinates. Nuovo Cim.
**1961**, 19, 558–571. [Google Scholar] [CrossRef] - Wheeler, J.A. Geometrodynamics and the Issue of the Final State. In Relativity, Groups and Topology; DeWitt, B., DeWitt, C., Eds.; Les Houches Lectures; Gordon and Breach: New York, NY, USA, 1964. [Google Scholar]
- Hamber, H.W. Quantum Gravitation. In Springer Tracts in Modern Physics; Springer: Berlin, Germany; New York, NY, USA, 2009. [Google Scholar]
- Feynman, R.P. Quantum Theory of Gravitation. Acta Phys. Pol.
**1963**, 24, 697–722. [Google Scholar] - Feynman, R.P. Lectures on Gravitation. In Advanced Book Program; Morinigo, F.B., Wagner, W.G., Hatfield, B., Eds.; Caltech Lecture Notes, 1962–1963; Addison-Wesley: Boston, MA, USA, 1995. [Google Scholar]
- Faddeev, L.D.; Popov, V.N. Covariant Quantization of the Gravitational Field. Sov. Phys. Uspekhi
**1974**, 16, 777–788. (First published in Uspekhi Fiz. Nauk 1973, 111, 427–450.). [Google Scholar] [CrossRef] - ’t Hooft, G.; Veltman, M. One-Loop Divergences in the Theory of Gravitation. Ann. Inst. Henri Poincaré
**1974**, 20, 69–94. [Google Scholar] - ’t Hooft, G. Recent Developments in Gravitation; Levy, M., Deser, S., Eds.; Cargése Lecture Notes 1978; Nato Science Series; Springer: New York, NY, USA, 1979. [Google Scholar]
- ’t Hooft, G. Perturbative Quantum Gravity; Zichichi, A., Ed.; Erice Lecture Notes, Subnuclear Physics Series; World Scientific: Singapore, 2002; Volume 40. [Google Scholar]
- Veltman, M. Quantum Theory of Gravitation. In Methods in Field Theory; Les Houches Lecture Notes Session XXVIII; North Holland: Amsterdam, The Netherlands, 1975. [Google Scholar]
- Capper, D.M.; Leibbrandt, G.; Medrano, M.R. Calculation of the Graviton Self-energy Using Dimensional Regularization. Phys. Rev. D
**1973**, 8, 4320–4331. [Google Scholar] [CrossRef] - Deser, S.; van Nieuwenhuizen, P. One loop divergences of quantized Einstein-Maxwell field. Phys. Rev. D
**1974**, 10, 401. [Google Scholar] [CrossRef] [Green Version] - Deser, S.; Tsao, H.S.; van Nieuwenhuizen, P. One Loop Divergences of the Einstein Yang-Mills System. Phys. Rev. D
**1974**, 10, 3337–3342. [Google Scholar] [CrossRef] [Green Version] - Deser, S. Conference on Gauge Theories and Modern Field Theories; Arnowitt, R., Nath, P., Eds.; MIT Press: Cambridge, UK, 1975. [Google Scholar]
- Tsao, H.S. Conformal Anomalies In A General Background Metric. Phys. Lett. B
**1977**, 68, 79–80. [Google Scholar] [CrossRef] - Goroff, M.H.; Sagnotti, A. Quantum Gravity At Two Loops. Phys. Lett. B
**1985**, 160, 81–86. [Google Scholar] [CrossRef] - Goroff, M.H.; Sagnotti, A. The Ultraviolet Behavior Of Einstein Gravity. Nucl. Phys. B
**1986**, 266, 709–736. [Google Scholar] [CrossRef] - van de Ven, A.E.M. Two-Loop Quantum Gravity. Nucl. Phys. B
**1992**, 378, 309–366. [Google Scholar] [CrossRef] - Gastmans, R.; Kallosh, R.; Truffin, C. Quantum Gravity near two dimensions. Nucl. Phys. B
**1978**, 133, 417–434. [Google Scholar] [CrossRef] [Green Version] - Christensen, S.M.; Duff, M.J. Quantum Gravity in two plus epsilon dimensions. Phys. Lett. B
**1978**, 79, 213–216. [Google Scholar] [CrossRef] - Weinberg, S. Ultraviolet Divergences in Quantum Gravity. In General Relativity—An Einstein Centenary Survey; Hawking, S.W., Israel, W., Eds.; Cambridge University Press: Cambridge, UK, 1979. [Google Scholar]
- Bethe, H.A.; Jackiw, R. Intermediate Quantum Mechanics, 3rd ed.; Advanced Book Classics; Benjamin: New York, NY, USA, 1986; Chapter 4. [Google Scholar]
- Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Microscopic Theory of Superconductivity. Phys. Rev.
**1957**, 106, 162. [Google Scholar] [CrossRef] [Green Version] - Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Theory of Superconductivity. Phys. Rev.
**1957**, 108, 1175. [Google Scholar] [CrossRef] [Green Version] - Bogoliubov, N.N. A New Method in the Theory of Superconductivity. Sov. Phys. JETP
**1958**, 34, 58–65. [Google Scholar] [CrossRef] - Abrikosov, A.A.; Gorkov, L.P.; Dzyaloshinski, I.E. Methods of Quantum Field Theory in Statistical Physics; Moscow 1962; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 1963. [Google Scholar]
- Fetter, L.; Walecka, J.D. Quantum Theory of Many Particle Systems; McGraw-Hill: Boston, MA, USA, 1971. [Google Scholar]
- Nambu, Y.; Jona-Lasinio, G. Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I. Phys. Rev.
**1961**, 122, 345–358. [Google Scholar] [CrossRef] [Green Version] - Nambu, Y.; Jona-Lasinio, G. Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II. Phys. Rev.
**1961**, 124, 246–254. [Google Scholar] [CrossRef] - Cornwall, J.M.; Jackiw, R.; Tomboulis, E. Effective Action for Composite Operators. Phys. Rev. D
**1974**, 10, 2428–2445. [Google Scholar] [CrossRef] - Jackiw, R. Lectures Given at the V. J. A. Swieca Brazil Summer School; Gobies, M., Ed.; 1977.
- Jackiw, R. Quantum Meaning of Classical Field Theory. Rev. Mod. Phys.
**1977**, 49, 681–706. [Google Scholar] [CrossRef] - Parisi, G. Statistical Field Theory; The Hartee-Fock Approximation; Benjamin Cummings: San Francisco, CA, USA, 1981; Section 6.1. [Google Scholar]
- Dyson, F.J. The S-Matrix in Quantum Electrodynamics. Phys. Rev.
**1949**, 75, 1736–1755. [Google Scholar] [CrossRef] - Schwinger, J.S. On the Green’s Functions of Quantized Fields. Proc. Natl. Acad. Sci. USA
**1951**, 37, 452–455. [Google Scholar] [CrossRef] [Green Version] - Schwinger, J.S. On the Green’s Functions of Quantized Fields II. Proc. Natl. Acad. Sci. USA
**1951**, 37, 455–459. [Google Scholar] [CrossRef] [Green Version] - Itzykson, C.; Zuber, J.-B. Quantum Field Theory; McGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Alkofer, R.; Smekal, L.V. The Infrared Behavior of QCD Green’s Functions—Confinement, Dynamical Symmetry Breaking, and Hadrons as Relativistic Bound States. Phys. Rep.
**2001**, 353, 281–465. [Google Scholar] [CrossRef] - Roberts, C.D.; Williams, A.G. Dyson-Schwinger Equations and their Application to Hadronic Physics. Prog. Part. Nucl. Phys.
**1994**, 33, 477–575. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Williams, R.M. Discrete Wheeler-DeWitt Equation. Phys. Rev. D
**2011**, 84, 104033. [Google Scholar] [CrossRef] [Green Version] - Gross, D.J.; Neveu, A. Dynamical Symmetry Breaking In Asymptotically Free Field Theories. Phys. Rev. D
**1974**, 10, 3235. [Google Scholar] [CrossRef] - Brezin, E.; Guillou, J.C.L.; Zinn-Justin, J. Critical Exponents from Field Theory. In Phase Transitions and Critical Phenomena; Domb, C., Green, M.S., Eds.; Academic Press: New York, NY, USA, 1976; Volume 6. [Google Scholar]
- Guida, R.; Zinn-Justin, J. Critical Exponents of the N-vector model. J. Phys. A
**1998**, 31, 8103–8121. [Google Scholar] [CrossRef] [Green Version] - Polyakov, A.M. Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields. Phys. Lett. B
**1975**, 59, 79–81. [Google Scholar] [CrossRef] - Brezin, E.; Zinn-Justin, J. Renormalization of the nonlinear sigma model in 2 + epsilon dimensions. Application to the Heisenberg ferromagnets. Phys. Rev. Lett.
**1976**, 36, 691–694. [Google Scholar] [CrossRef] - Hikami, S.; Brezin, E. Three Loop Calculations in the Two-Dimensional Nonlinear Sigma Model. J. Phys. A
**1978**, 11, 1141–1150. [Google Scholar] [CrossRef] - Bernreuther, W.; Wegner, F.J. Four Loop Order Beta Function for Two-dimensional Nonlinear Sigma Models. Phys. Rev. Lett.
**1986**, 57, 1383. [Google Scholar] [CrossRef] - Wegner, F. Four Loop Order Beta Function of Nonlinear Sigma Models in Symmetric Spaces. Nucl. Phys. B
**1989**, 316, 663–678. [Google Scholar] [CrossRef] - Brezin, E.; Hikami, S. Fancy and facts in the (d-2) expansion of the nonlinear sigma models. arXiv
**1996**, arXiv:cond-mat/9612016. [Google Scholar] - Kosterlitz, J.M.; Thouless, D.J. Ordering, Metastability and Phase Transitions in Two-dimensional Systems. J. Phys. C
**1973**, 6, 1181–1203. [Google Scholar] [CrossRef] - Stanley, H.E. Spherical Model as the Limit of Infinite Spin Dimensionality. Phys. Rev.
**1968**, 176, 718–722. [Google Scholar] [CrossRef] - Brezin, E.; Wallace, D.J. Critical Behavior of a Classical Heisenberg Ferromagnet with Many Degrees of Freedom. Phys. Rev. B
**1973**, 7, 1967. [Google Scholar] [CrossRef] - Ma, K.S. Introduction to the Renormalization Group. Rev. Mod. Phys.
**1973**, 45, 589–614. [Google Scholar] [CrossRef] - Abe, R.; Hikami, S. Critical Exponents and Scaling Relations in the 1/N Expansion. Prog. Theor. Phys.
**1973**, 49, 442–452. [Google Scholar] [CrossRef] [Green Version] - Abe, R. Critical Exponent η up to 1/N
^{2}for the Three-Dimensional System with Short-Range Interaction. Progr. Theor. Phys.**1973**, 49, 1877–1888. [Google Scholar] [CrossRef] [Green Version] - Zinn-Justin, J. Vector Models in the Large-N Limit: A few Applications; Taiwan Spring School: Taipei, Taiwan, 1997. [Google Scholar]
- Pfeuty, P.; Toulouse, G. Introduction to the Renormalization Group and to Critical Phenomena; John Wiley and Sons: New York, NY, USA; London, UK; Sydney, Australia; Toronto, ON, Canada, 1976; p. 79. [Google Scholar]
- Nielsen, N.K. Asymptotic Freedom as a Spin Effect. Am. J. Phys.
**1981**, 49, 1171. [Google Scholar] [CrossRef] - Hughes, R.J. Some Comments on Asymptotic Freedom. Phys. Lett. B
**1980**, 97, 246–248. [Google Scholar] [CrossRef] - Hughes, R.J. More Comments on Asymptotic Freedom. Nucl. Phys. B
**1981**, 186, 376–396. [Google Scholar] [CrossRef] - Parisi, G. Hausdorff Dimensions and Gauge Theories. Phys. Lett. B
**1979**, 81, 357–360. [Google Scholar] [CrossRef] - Drouffe, J.M.; Parisi, G.; Sourlas, N. Strong Coupling Phase In Lattice Gauge Theories At Large Dimension. Nucl. Phys. B
**1979**, 161, 397–416. [Google Scholar] [CrossRef] - Aizenman, M. Proof of the Triviality Of Phi**4 in D-Dimensions Field Theory and Some Mean Field Features of Ising Models for D>4. Phys. Rev. Lett.
**1981**, 47, 1–4. [Google Scholar] [CrossRef] - Aizenman, M. The Intersection of Brownian Paths as a Case Study of a Renormalization Group Method For Quantum Field Theory. Commun. Math. Phys.
**1985**, 97, 91. [Google Scholar] [CrossRef] - Fröhlich, J. On the Triviality of Lambda (phi**4) in D-Dimensions Theories and the Approach to the Critical Point in D >= Four-Dimensions. Nucl. Phys. B
**1982**, 200, 281–296. [Google Scholar] [CrossRef] - Feynman, R.P.; Hibbs, A. Quantum Mechanics and Path integrals; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Zinn-Justin, J. Path Integrals in Quantum Mechanics; Oxford University Press: Oxford, UK, 2005. [Google Scholar]
- Kleinert, H. Path Integrals in Quantum Mechanics, Statistics and Polymer Physics; World Scientific: Singapore, 2006. [Google Scholar]
- Campostrini, M.; Giacomo, A.D.; Gunduc, Y. Gluon Condensation in SU(3) Lattice Gauge Theory. Phys. Lett. B
**1989**, 225, 393–397. [Google Scholar] [CrossRef] - Ji, X. Gluon Condensate from Lattice QCD. arXiv
**1995**, arXiv:hep-ph/9506413. [Google Scholar] - Brodsky, S.J.; Shrock, R. Condensates in Quantum Chromodynamics and the Cosmological Constant. Proc. Natl. Acad. Sci. USA
**2009**, 108, 45–50. [Google Scholar] [CrossRef] [Green Version] - Dominguez, C.A.; Hernandez, L.A.; Schilcher, K. Determination of the Gluon Condensate from Data in the Charm-Quark Region. J. High Energy Phys.
**2015**, 7, 110. [Google Scholar] [CrossRef] [Green Version] - McNeile, C.; Bazavov, A.; Davies, C.T.H.; Dowdall, R.J.; Hornbostel, K.; Lepage, G.P.; Trottier, H.D. Direct determination of the strange and light quark condensates from full lattice QCD. Phys. Rev. D
**2011**, 87, 034503. [Google Scholar] [CrossRef] [Green Version] - Mennessier, G.; Narison, S.; Ochs, W. Glueball nature of the sigma/f(0)(600) from pi-pi and gamma-gamma scatterings. Phys. Lett. B
**2008**, 665, 205–211. [Google Scholar] [CrossRef] [Green Version] - Tanabashi, M.; Grp, P.D.; Hagiwara, K.; Hikasa, K.; Nakamura, K.; Sumino, Y.; Takahashi, F.; Tanaka, J.; Agashe, K.; Aielli, G.; et al. [Particle Data Group]. Review of Particle Physics. Phys. Rev. D
**2018**, 98, 030001. Available online: http://pdg.lbl.gov/2018/reviews/rpp2018-rev-qcd.pdf (accessed on 1 April 2020). [CrossRef] [Green Version] - Polyakov, A.M. Gauge Fields and Strings; Oxford University Press: Oxford, UK, 1989. [Google Scholar]
- Kawai, H.; Ninomiya, M. Renormalization Group and Quantum Gravity. Nucl. Phys. B
**1990**, 336, 115–145. [Google Scholar] [CrossRef] - Kawai, H.; Kitazawa, Y.; Ninomiya, M. Operator Product Expansion in two-dimensional Quantum Gravity. Nucl. Phys. B
**1996**, 474, 512–528. [Google Scholar] - Kawai, H.; Kitazawa, Y.; Ninomiya, M. Ultraviolet Stable Fixed Point and Scaling Relations in 2+epsilon-dimensional Quantum Gravity. Nucl. Phys. B
**1993**, 404, 684–716. [Google Scholar] [CrossRef] [Green Version] - Kitazawa, Y.; Ninomiya, M. Scaling Behavior of the Ricci Curvature at Short Distances Near Two Dimensions. Phys. Rev. D
**1997**, 55, 2076–2081. [Google Scholar] [CrossRef] [Green Version] - Aida, T.; Kitazawa, Y. Two-Loop Prediction for Scaling Exponents in (2+epsilon)-Dimensional Quantum Gravity. Nucl. Phys. B
**1997**, 491, 427–460. [Google Scholar] [CrossRef] [Green Version] - Deser, S.; Jackiw, R.; ’t Hooft, G. Three-dimensional Einstein Gravity: Dynamics of Flat Space. Ann. Phys.
**1984**, 152, 220–235. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Williams, R.M. Simplicial quantum gravity in three-dimensions: Analytical and numerical results. Phys. Rev. D
**1993**, 47, 510–532. [Google Scholar] [CrossRef] - Hamber, H.W.; Toriumi, R.; Williams, R.M. Wheeler-DeWitt Equation in 2+1 Dimensions. Phys. Rev. D
**2012**, 86, 084010. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Toriumi, R.; Williams, R.M. Wheeler-DeWitt Equation in 3+1 Dimensions. Phys. Rev. D
**2013**, 88, 084012. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Toriumi, R. On the Exact Solution of 2+1-Dimensional Lattice Quantum Gravity. Manuscript in preparation.
- Hamber, H.W. Scaling Exponents for Lattice Quantum Gravity in Four Dimensions. Phys. Rev. D
**2015**, 92, 064017. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W. Vacuum Condensate Picture of Quantum Gravity. Symmetry
**2019**, 11, 87. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Williams, R.M. Nonperturbative Gravity and the Spin of the Lattice Graviton. Phys. Rev. D
**2004**, 70, 124007. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Williams, R.M. Quantum Gravity in Large Dimensions. Phys. Rev. D
**2006**, 73, 044031. [Google Scholar] [CrossRef] [Green Version] - Englert, F. Linked Cluster Expansion in the Statistical Theory of Ferromagnetism. Phys. Rev.
**1963**, 129, 567–577. [Google Scholar] [CrossRef] - Fisher, M.E.; Gaunt, D.S. Ising Model and Self-Avoiding Walks on Hypercubical Lattices and High-Density Expansions. Phys. Rev.
**1964**, 133, A224–A239. [Google Scholar] [CrossRef] - Abe, R. Critical Exponent of the Ising Model in the High Density Limit. Progr. Theor. Phys.
**1972**, 47, 62–68. [Google Scholar] [CrossRef] [Green Version] - Strominger, A. The Inverse Dimensional Expansion In Quantum Gravity. Phys. Rev. D
**1981**, 24, 3082. [Google Scholar] [CrossRef] - Bjerrum-Bohr, N.E.J. Quantum gravity at a large number of dimensions. Nucl. Phys. B
**2004**, 684, 209–234. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Williams, R.M. Gravitational Wilson Loop and Large Scale Curvature. Phys. Rev. D
**2007**, 76, 084008. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Williams, R.M. Gravitational Wilson Loop in Discrete Gravity. Phys. Rev. D
**2010**, 81, 084048. [Google Scholar] [CrossRef] [Green Version] - Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. [Planck Collaboration]. Planck 2018 results. I. Overview and the cosmological legacy of Planck. arXiv
**2018**, arXiv:1807.06205v1. [Google Scholar] - Hamber, H.W.; Williams, R.M. Nonlocal effective gravitational field equations and the running of Newton’s G. Phys. Rev. D
**2005**, 72, 044026. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Toriumi, R. Scale-Dependent Newton’s Constant G in the Conformal Newtonian Gauge. Phys. Rev. D
**2011**, 84, 103507. [Google Scholar] [CrossRef] [Green Version] - Weinberg, S. Cosmology; Oxford University Press: Oxford, UK, 2008. [Google Scholar]
- Dodelson, S. Modern Cosmology; Academic Press: Amsterdam, The Netherlands, 2003. [Google Scholar]
- Lewis, A.; Challinor, A.; Lasenby, A. Efficient Computation of CMB anisotropies in closed FRW models. arXiv
**1999**, arXiv:astro-ph/9911177. [Google Scholar] - Antony Lewis. CAMB Notes. Available online: https://cosmologist.info/notes/CAMB.pdf (accessed on 1 April 2020).
- Alex, Z.; Levon, P.; Alessandra, S.; Gong-Bo, Z. MGCAMB with massive neutrinos and dynamical dark energy. arXiv
**2019**, arXiv:1901.05956. [Google Scholar] - Lesgourgues, J. The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview. arXiv
**2011**, arXiv:astro-ph.IM. [Google Scholar] - Tessa, B.; Philip, B. Observational signatures of modified gravity on ultra-large scales. arXiv
**2015**, arXiv:1506.00641. [Google Scholar] - Garcia-Quintero, C.; Ishak, M.; Fox, L.; Dossett, J. ISiTGR: Testing deviations from GR at cosmological scales including dynamical dark energy, massive neutrinos, functional or binned parametrizations, and spatial curvature. arXiv
**2019**, arXiv:1908.00290. [Google Scholar] - Garcia-Quintero, C.; Ishak, M. ISiTGR Version 3.1 Released in February 2020 (with Python Wrapper). GitHub Repository. Available online: https://github.com/mishakb/ISiTGR (accessed on 1 April 2020).
- Antony, L.; Sarah, B. Cosmological parameters from CMB and other data: A Monte-Carlo approach. arXiv
**2002**, arXiv:astro-ph/0205436. [Google Scholar] - Bellini, E.; Barreira, A.; Frusciante, N.; Hu, B.; Peirone, S.; Raveri, M.; Zumalacárregui, M.; Avilez-Lopez, A.; Ballardini, M.; Battye, R.A.; et al. Comparison of Einstein-Boltzmann solvers for testing general relativity. Phys. Rev. D
**2018**, 97, 023520. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Yu, L.H.S. Gravitational Fluctuations as an Alternative to Inflation. Universe
**2019**, 5, 31. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Yu, L.H.S. Gravitational Fluctuations as an Alternative to Inflation II. CMB Angular Power Spectrum. Universe
**2019**, 5, 216. [Google Scholar] [CrossRef] [Green Version] - Hamber, H.W.; Yu, L.H.S.; Kankanamge, H.E.P. Gravitational Fluctuations as an Alternative to Inflation III. Numerical Results. Universe
**2020**, 6, 92. [Google Scholar] [CrossRef]

**Figure 1.**Dyson’s equations in QED relating the dressed fermion propagator, the dressed photon propagator and the dressed vertex function. Thin lines represent bare propagators, while thicker lines represent the dressed ones.

**Figure 5.**Pure Yang–Mills vacuum polarization contributions to lowest order in perturbation theory. The dashed line represents the ghost contribution.

**Figure 7.**Dyson’s equations for the dressed Yang–Mills boson propagator. Ghost contributions have been omitted from the diagrams.

**Figure 10.**Dressed graviton propagator ${D}_{\alpha \beta ,\mu \nu}\left(p\right)$, with a graviton proper vacuum polarization insertion ${\mathsf{\Pi}}_{\alpha \beta ,\mu \nu}\left(p\right)$. The thin wavy line denotes the tree-level graviton propagator ${D}_{\alpha \beta ,\mu \nu}^{0}\left(p\right)$.

**Figure 11.**Graviton three-point vertex ${V}_{\alpha \beta ,\mu \nu ,\rho \sigma}^{\left(3\right)}(p,q,r)$.

**Figure 12.**Graviton four-point vertex ${V}_{\alpha \beta ,\gamma \delta ,\mu \nu ,\rho \sigma}^{\left(4\right)}(p,q,r,s)$.

**Figure 13.**Quantum gravity lowest order vacuum polarization ${\mathsf{\Pi}}_{\alpha \beta ,\mu \nu}\left(p\right)$ diagrams. A thin wavy lines denotes a bare graviton propagator, and small sized dots stand for bare graviton vertices. A dashed line appears in the ghost loop. Dots indicate higher order corrections.

**Figure 14.**Dyson’s equations for the dressed graviton propagator ${D}_{\alpha \beta ,\mu \nu}\left(p\right)$. Small dots denote bare graviton vertices, while larger dots stand for dressed vertices. Thin wavy lines denote bare graviton propagators, while thicker lines stand for dressed propagators.

**Figure 15.**Dyson’s equations for the proper graviton vacuum polarization insertion ${\mathsf{\Pi}}_{\alpha \beta ,\mu \nu}\left(p\right)$. Small dots denote bare graviton vertices, while larger dots stand for dressed vertices. A thin wavy lines denotes a bare graviton propagator, while thicker lines stand for dressed propagators.

**Figure 16.**Self-consistent Hartree–Fock approximation for the graviton proper vacuum polarization tensor. Small dots denote bare graviton vertices, and the thick wavy line indicates a dressed graviton propagator, to be determined self-consistently.

**Figure 17.**Next higher order terms in the Hartree–Fock approximation to the graviton proper vacuum polarization tensor. Small dots denote bare graviton vertices, and the thick wavy line indicates a dressed graviton propagator, to be determined later self-consistently.

**Figure 18.**Universal correlation length scaling exponent $\nu $ as a function of spacetime dimension d. Shown are the $2+\u03f5$ expansion result to one (lower curve) and two (upper curve) loops [92], the value in $2+1$ dimensions obtained from the exact solution of the Wheeler–DeWitt equation [95,97], the numerical lattice result in four spacetime dimensions [98], the large d result ${\nu}^{-1}\simeq d-1$ [100], and the Hartree–Fock approximation results discussed in the text.

**Figure 20.**Comparison of the gravitational running coupling $G\left(r\right)$ versus r, obtained from $G\left(q\right)$ in Equations (163) (the Hartree–Fock result) and (164) (the Regge–Wheeler lattice result) by setting $q\sim 1/r$. Note that the approximate Hartree–Fock analytical result of Equation (163) (red line) initially rises more rapidly for small r. The nonperturbative scale $\xi $ used here is related to the gravitational vacuum condensate via Equations (160) and (161).

**Figure 21.**Matter power spectrum $P\left(k\right)$ with lattice versus Hartree–Fock running of Newton’s constant G. The middle solid (orange) curve shows $P\left(k\right)$ implemented with the Hartree–Fock running of Newton’s constant G factor as given in Equation (163). The lower dashed (green) curve shows the original lattice RG running of Newton’s constant G (with the original lattice coefficient of Equation (164) equal to ${c}_{0}=16.0$) for comparison. The original spectrum with no running is also displayed by the top dotted (blue) curve for reference.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hamber, H.W.; Yu, L.H.S.
Dyson’s Equations for Quantum Gravity in the Hartree–Fock Approximation. *Symmetry* **2021**, *13*, 120.
https://doi.org/10.3390/sym13010120

**AMA Style**

Hamber HW, Yu LHS.
Dyson’s Equations for Quantum Gravity in the Hartree–Fock Approximation. *Symmetry*. 2021; 13(1):120.
https://doi.org/10.3390/sym13010120

**Chicago/Turabian Style**

Hamber, Herbert W., and Lu Heng Sunny Yu.
2021. "Dyson’s Equations for Quantum Gravity in the Hartree–Fock Approximation" *Symmetry* 13, no. 1: 120.
https://doi.org/10.3390/sym13010120