2. “Miami 2015” Conference Vacuum Condensate Picture of Quantum Gravitation General reference: “Quantum Gravitation” (Springer Tracts in Modern Physics, 2009) [ with R.M. Williams and R. Toriumi ]

3. Like QED and QCD, Gravity is a Unique Theory Feynman 1963: Unique Theory of an m=0 s=2 particle. “Quantum Gravity” is the direct combination of: (1) Einstein’s 1916 classical General Relativity (2) Quantum mechanics, in the form of Feynman’s Path Integral

4. Theory is Highly Non-Linear (“Gravity gravitates”) Perturbation Theory in G is useless (see below) Conformal Instability Physical distances depend on the Metric (OPE) Some Serious Inherent Problems: Problems…

5. Feynman Diagrams Infinitely many graviton interaction terms in L : + + + + … Gravitons self-interact, or ‘’Gravitate” :

6. One Graviton Loop One loop diagram quartically divergent in d=4. GG

7. Two Graviton Loops : worse Perturbation theory badly divergent … Wrong ground state ?? G G G G

8. Cosmological Constant Add to the action λ (Volume of Space-Time), or : Einstein’s “biggest blunder” (1917, 1922). A new length scale or

9. Perturbative Non-Renormalizability The usual (diagrammatic) methods of QED and QCD fail because Gravity is not perturbatively renormalizable. This can mean either of two things: Perturbation Theory in G fails (non-analytic) Wrong Theory; need N=8 SuGra or Susy Strings

10.  QCD Quarks and gluons are confined. Main theoretical evidence for confinement and chiral symmetry breaking is possibly from the lattice.  Superconductor BCS Theory: Fermions close to the Fermi surface condense into Cooper pairs.  Superfluid Described by condensate density.  Degenerate Electron Gas Screening due to Thomas-Fermi mechanism.  Homogeneous Turbulence Observables Rc= critical Reynolds no. (Kolmogorov)  Ferromagnets Spontaneous Symmetry Breaking & dimensional transmutation Life After Perturbation Theory Common thread? Vacuum Condensation True QM ground state ≠ Perturbative ground state

11. Theory of “Non-Renormalizable” Interactions Ken Wilson, “Feynman-graph expansion for critical exponents”, PRL 1972 and PRD 1973 Giorgio Parisi, “On the renormalizability of not renormalizable theories”, Lett N Cim 1973 “ Theory of Non-Renormalizable Interactions - The large N expansion” NPB 1975 “ On Non-Renormalizable Interactions”, Lectures at Cargese 1976 Kurt Symanzik, Cargese Lectures 1973, Comm. Math. Phys. 1975 Steven Weinberg, “Ultraviolet Divergences in Quantum Gravity”, Cambridge University Press,1979. + many other … Perturbatively Non-renormalizable Theories are in fact theoretically rather well understood. Common Thread: Non-trivial RG fixed point. A few representative references :

12. Discretization/regularization of the Feynman P.I. Starts from a manifestly covariant formulation No need for gauge fixing (as in Lattice QCD) Dominant paths are nowhere differentiable Allows for non-perturbative calculations Extensively tested in QCD & Spin Systems 30 years experience / high accuracy possible Only known reliable Method : The Lattice

13. Feynman Path Integral Defined via Lattice

14. 40 Years of Statistical Field Theory By now rather well understood methods & concepts include : ● Lattice Field Theory (explicit UV and IR cutoff) ● Lattice Quantum Continuum Limit ● … taken in accordance with the Renormalization Group ● Role of UV and IR fixed points ● Scaling Limit taken at the (perhaps non-trivial) UV fixed point ● Role of Scale Invariance at FP, Universality of Critical Behavior ● Relevant vs. Irrelevant vs. Marginal operators etc. G. Parisi, Statistical Field Theory (Benjamin Cummings1981). J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford U. Press 2002). C. Itzykson and J. M. Drouffe, Statistical Field Theory (Cambridge U. Press 1991). J. L. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge U. Press 1996). E. Brezin, Introduction to Statistical Field Theory (Cambridge U. Press 2010). NO NEED TO RE-INVENT THE WHEEL – For Gravity !

15. Prototype: Wilson’s Lattice QCD In QCD Pert. Th. is next to useless at low energies Nontrivial measure (Haar) Confinement is almost immediate (Area law) Physical Vacuum bears little resemblance to pert. Vacuum Nontrivial Spectrum (glueballs) / Vacuum chromo- electric condensate / Quark condensates

16. QCD is Very Hard. Very Serious Supercomputers. Fermilab LQCD Cluster

17. Lattice Gauge Theory Works [Particle Data Group LBL, 2015] Wilson’s lattice gauge theory provides to this day the only convincing theoretical evidence for : confinement and chiral symmetry breaking in QCD. Running of α strong :

18. Path Integral for Quantum Gravitation DeWitt approach to measure : introduce a Super-Metric G Proper definition of F. Path Integral requires a Lattice (Feynman & Hibbs, 1964). Perturbation theory in 4D about a flat background is useless … badly divergent In d=4 this gives a “volume element” :

19. In the absence of matter, only one dim.-less coupling: … similar to g of Y.M. Only One (Bare) Coupling Rescale metric (edge lengths): Pure gravity path integral:

20. Conformal Instability Euclidean Quantum Gravity - in the Path Integral approach - is affected by a fundamental instability, which cannot be removed. The latter is apparently only overcome in the lattice theory (for G>Gc), because of the entropy (functional measure) contribution. Gibbons and Hawking PRD 15 1977; Hawking, PRD 18 1978; Gibbons, Hawking and Perry, NPB 1978. Ω² (x) = conformal factor

21. Lattice Theory of Gravity  Based on a dynamical lattice  Incorporates continuous local invariance  Puts within the reach of computation problems which in practical terms are beyond the power of analytical methods  Affords in principle any desired level of accuracy by a sufficiently fine subdivision of space-time T. Regge 1961, J.A. Wheeler 1964 Misner Thorne Wheeler, “Gravitation” ch. 42 : “Simplicial Quantum Gravity”

22. Tullio Regge and John A. Wheeler, ca 1971 (Photo courtesy of R. Ruffini)

23. Elementary Building Block = 4-Simplex For more details see eg. Ch. 6 in “Quantum Gravitation” (Springer 2009), and refs therein. The metric (a key ingredient in GR) is defined in terms of the edge lengths : … or more directly in terms of the edge lengths : The local volume element is obtained from a determinant :

24. Curvature - Described by Angles Curvature determined by edge lengths T. Regge 1961 J.A. Wheeler 1964 Edge lengths replace the Metric

25. Choice of Lattice Structure Timothy Nolan, Carl Berg Gallery, Los Angeles Regular geometric objects can be stacked. A not so regular lattice … … and a more regular one:

26. Rotations & Riemann tensor Due to the hinge’s intrinsic orientation, only components of the vector in the plane perpendicular to the hinge are rotated: Exact lattice Bianchi identity (Regge) Regge Action

27. Lattice Path Integral Without loss of generality, one can set bare ₀ = 1 ; Besides the cutoff Λ, the only relevant coupling is κ (or G ). Lattice path integral follows from edge assignments, Schrader / Hartle / T.D. Lee measure ; Lattice analog of the DeWitt measure

28. Gravitational Wilson Loop  Parallel transport of a vector done via lattice rotation matrix For a large closed circuit obtain gravitational Wilson loop; compute at strong coupling (G large) … suggests ξ related to curvature. argument can only give a positive cosmological constant. … then compare to semi-classical result (from Stokes’ theorem) “Minimal area law” follows from loop tiling. R.M. Williams and H.H., Phys Rev D 76 (2007) ; D 81 (2010) [Peskin and Schroeder, page 783]

29. CM5 at NCSA, 512 processors Numerical Evaluation of Z

30. 256 cores on 32 ⁴ lattice → ~ 0.8 Tflops 1024 cores on 64 ⁴ lattice → ~ 3.4 Tflops … Distribute Lattice Sites on 1024 Processor Cores Lattice Sites are Processed in Parallel

31. Curvature Distributions  4 ⁴ sites → 6,144 simplices  8 ⁴ sites → 98,304 simplices  16 ⁴ sites → 1.5 M simplices  32 ⁴ sites → 25 M simplices  64 ⁴ sites → 402 M simplices

32. Phases of Quantum Gravity Smooth phase: R ≈ 0 Unphysical (branched polymer-like, d ≈ 2) Unphysical Physical L. Quantum Gravity has two phases …

33. Lattice Continuum Limit Continuum limit requires the existence of an UV fixed point.

34. (Lattice) Continuum Limit Λ → ∞ Bare G must approach UV fixed point at Gc. UV cutoff Λ → ∞ (average lattice spacing → 0) RG invariant correlation length ξ is kept fixed ξ Use Standard Wilson procedure in cutoff field theory The very same relation gives the RG running of G(μ) close to the FP. integrated to give :

35. Determination of Scaling Exponents Find value for ν close to 1/3 : Use standard Universal Scaling assumption: ν ≈ ⅓ν ≈ ⅓ ( Phys Rev D Sept. 2015 )

36. Recent runs on 2400 node cluster ν = 0.334(4) Distribution of zeros in complex k space More calculations in progress …

37. Finite Size Scaling (FSS) Analysis

38. Exponent Comparison (D=4) ν ≈ ⅓ν ≈ ⅓ Exponent ν determines the “running” of G Phys. Rev. D Sep. 2015

39. Exponent Comparison (D=3)

40. Comparison for Univ. Exponent 1/ν

41. Almost identical to 2 + ε expansion result, but with a 4-d exponent ν = 1/3 and a calculable coefficient c 0 … “Covariantize” : Running Newton’s Constant G In gravity there is a new RG invariant scale ξ : Running of G determined largely by scale ξ and exponent ν :

42. Running of Newton’s G(□) New RG invariant scale of gravity  Expect small deviations from GR on largest scales Eg. Matter density perturbations in comoving gauge Gravitational “slip” function in Newtonian gauge (infrared cutoff) ν = 1/3

43. Newton’s “Constant” no longer Constant Gravitational Field Fluctuations generate anti-screening, and a slow “running” of G(k) virtual graviton cloud infrared cutoff

44. Infrared RG Running of G Phys. Rev. D Sep. 2015

45. Vacuum Condensate Picture of QG  Lattice Quantum Gravity: Curvature condensate  Quantum Chromodynamics: Gluon and Fermion condensate  Electroweak Theory: Higgs condensate

46. ~ Five Main Predictions Running of Newton’s G on very large (cosmological) scales Modified results for (Relativistic) Matter Density Perturbations Non-Vanishing “Slip” Function in conformal Newtonian gauge Non-Trivial Curvature and Matter Density Correlations Modifications of Spectral Indices at very small k No adjustable parameters (as in QCD)

47. Curvature Correlation Functions Need the geodesic distance between any two points : Curvature correlation function : But for ν = ⅓ the result becomes quite simple : If the two parallel transport loops are not infinitesimal :

48. … Related to Matter Density Correlations The classical field equations relate the local curvature to the local matter density For the macroscopic matter density contrast one then obtains From the lattice one computes : Astrophysical measurements of G(r) are roughly consistent with

49. Matter Density Perturbations Visualizing Density Perturbations on very large scales … … Evolution predicted by GR Observed (CfA) Simulation (MPI Garching)

50. Density Perturbations with G(□) Standard GR result for density perturbations : If Gravity is modified, then k=0 Peebles exponents will change: GR solution, in matter-dominated era : [with R. Toriumi PRD 2011]

51. (Peebles) Growth Index Parameter γ Growth index parameter γ, as a function of the matter fraction Ω

52. Alexei Vikhlin et al., Rapetti et al. 2012 Measured growth parameter γ

53. Gravitational “Slip” with G(□) In the conformal Newtonian gauge [with R. Toriumi PRD 2013] In classical GR : η = ψ/φ – 1 = 0 … Thus a good test for deviations from classical GR.

54. The End

55. This slide intentionally left blank

56. N- Component Heisenberg model Field theory description of O(N) Heisenberg model For d > 2 theory is not (perturbatively) renormalizable (J = 1/G² has mass dimensions D-2) Phase Transition (“non-trivial UV fixed point”) makes this an interesting model  new scale (correlation length)  unit vector with N components

57. J.A. Lipa et al, Phys Rev 2003: α = 2 – 3 ν = -0.0127(3) MC, HT, 4-ε exp. to 4 loops, & to 6 loops in d=3: α = 2 – 3 ν ≈ -0.0125(4) O(2) non-linear sigma model describes the phase transition of superfluid Helium Space Shuttle experiment (2003) High precision measurement of specific heat of superfluid Helium He4 (zero momentum energy-energy correlation at UV FP) yields ν Test of Field Theory Methods Second most accurate predictions of QFT, after g-2

58. This slide intentionally left blank

59. Gravity in 2+ε Dimensions Wilson’s double expansion … Formulate theory in 2+ε dimensions. G is dim-less, so theory is now perturbatively renormalizable Wilson 1973 Weinberg 1977 … Kawai, Ninomiya 1995 Kitazawa, Aida 1998 { … suggests the existence of two phases ( pure gravity : ) with a non-trivial UV fixed point :

60. Running of Newton’s G(k) in 2+ε is of the form: Two key quantities : i) the universal exponent ν ii) the new nonperturbative scale  What is left of the above QFT scenario in 4 dimensions ? 2+ε Cont’d

61. This slide intentionally left blank