Vacuum Condensate Picture of Quantum Gravitation General reference: “Quantum Gravitation” (Springer Tracts in Modern Physics, 2009) “Miami 2015” Conference [ with R.M. Williams and R. Toriumi ] Like QED and QCD, Gravity is a Unique Theory “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 Feynman 1963: Unique Theory of an m=0 s=2 particle. Problems… Some Serious Inherent Problems: • Theory is Highly Non-Linear (“Gravity gravitates”) • Perturbation Theory in G is useless (see below) • Conformal Instability • Physical distances depend on the Metric (OPE) Feynman Diagrams Infinitely many graviton interaction terms in L : Gravitons self-interact, or ‘’Gravitate” : + + + +… One Graviton Loop G G One loop diagram quartically divergent in d=4. Two Graviton Loops : worse G G G G Perturbation theory badly divergent … Wrong ground state ?? Cosmological Constant Add to the action λ (Volume of Space-Time), or : A new length scale or Einstein’s “biggest blunder” (1917, 1922). 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 Life After Perturbation Theory 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 Common thread? Vacuum Condensation True QM ground state ≠ Perturbative ground state Theory of “Non-Renormalizable” Interactions Perturbatively Non-renormalizable Theories are in fact theoretically rather well understood. Common Thread: Non-trivial RG fixed point. A few representative references : 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 … Only known reliable Method : The Lattice • 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 Feynman Path Integral Defined via Lattice 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 ! 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 chromoelectric condensate / Quark condensates QCD is Very Hard. Very Serious Supercomputers. Fermilab LQCD Cluster Lattice Gauge Theory Works Running of α strong : Wilson’s lattice gauge theory provides to this day the only convincing theoretical evidence for : confinement and chiral symmetry breaking in QCD. [Particle Data Group LBL, 2015] Path Integral for Quantum Gravitation DeWitt approach to measure : introduce a Super-Metric G In d=4 this gives a “volume element” : Proper definition of F. Path Integral requires a Lattice (Feynman & Hibbs, 1964). Perturbation theory in 4D about a flat background is useless … badly divergent Only One (Bare) Coupling Pure gravity path integral: In the absence of matter, only one dim.-less coupling: Rescale metric (edge lengths): … similar to g of Y.M. 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. Ω² (x) = conformal factor Gibbons and Hawking PRD 15 1977; Hawking, PRD 18 1978; Gibbons , Hawking and Perry, NPB 1978. Lattice Theory of Gravity T. Regge 1961, J.A. Wheeler 1964 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 “Simplicial Quantum Gravity” Misner Thorne Wheeler, “Gravitation” ch. 42 : Tullio Regge and John A. Wheeler, ca 1971 (Photo courtesy of R. Ruffini) Elementary Building Block = 4-Simplex The metric (a key ingredient in GR) is defined in terms of the edge lengths : The local volume element is obtained from a determinant : … or more directly in terms of the edge lengths : For more details see eg. Ch. 6 in “Quantum Gravitation” (Springer 2009), and refs therein. Curvature - Described by Angles Edge lengths replace the Metric d 2 d 2 d 3 d 4 Curvature determined by edge lengths T. Regge 1961 J.A. Wheeler 1964 Choice of Lattice Structure A not so regular lattice … … and Timothy Nolan, Carl Berg Gallery, Los Angeles Regular geometric objects can be stacked. a more regular one: 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 Lattice Path Integral Lattice path integral follows from edge assignments, Schrader / Hartle / T.D. Lee measure ; Lattice analog of the DeWitt measure Without loss of generality, one can set bare ₀ = 1; Besides the cutoff Λ, the only relevant coupling is κ (or G). 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) … “Minimal area law” follows from loop tiling. … then compare to semi-classical result (from Stokes’ theorem) • suggests ξ related to curvature. • argument can only give a positive cosmological constant. R.M. Williams and H.H., Phys Rev D 76 (2007) ; D 81 (2010) [Peskin and Schroeder, page 783] Numerical Evaluation of Z CM5 at NCSA, 512 processors Lattice Sites are Processed in Parallel Distribute Lattice Sites on 1024 Processor Cores … 256 cores on 32⁴ lattice → ~ 0.8 Tflops 1024 cores on 64⁴ lattice → ~ 3.4 Tflops Curvature Distributions 4⁴ sites 8⁴ sites 16⁴ sites 32⁴ sites 64⁴ sites → → → → → 6,144 simplices 98,304 simplices 1.5 M simplices 25 M simplices 402 M simplices Phases of Quantum Gravity L. Quantum Gravity has two phases … Smooth phase: R ≈ 0 Physical Unphysical (branched polymer-like, d ≈ 2) Unphysical Lattice Continuum Limit Continuum limit requires the existence of an UV fixed point. (Lattice) Continuum Limit Λ → ∞ Use Standard Wilson procedure in cutoff field theory integrated to give : ξ RG invariant correlation length ξ is kept fixed UV cutoff Λ → ∞ (average lattice spacing → 0) Bare G must approach UV fixed point at Gc . The very same relation gives the RG running of G(μ) close to the FP. Determination of Scaling Exponents Use standard Universal Scaling assumption: Find value for ν close to 1/3 : ν≈⅓ ( Phys Rev D Sept. 2015 ) Recent runs on 2400 node cluster 12 10 164 324 data 164 324 fit 6 4 Distribution of zeros in complex k space 2 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 k 30 25 20 k R k 8 ν = 0.334(4) 164 324 data 164 324 fit 15 10 5 More calculations in progress … 0 0.00 0.01 0.02 0.03 0.04 k 0.05 0.06 0.07 Finite Size Scaling (FSS) Analysis Exponent Comparison (D=4) Exponent ν determines the “running” of G ν≈⅓ Phys. Rev. D Sep. 2015 Exponent Comparison (D=3) Comparison for Univ. Exponent 1/ν 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 : Almost identical to 2 + ε expansion result, but with a 4-d exponent ν = 1/3 and a calculable coefficient c0 … “Covariantize” : Running of Newton’s G(□) ν = 1/3 New RG invariant scale of gravity (infrared cutoff) Expect small deviations from GR on largest scales Eg. • Matter density perturbations in comoving gauge • Gravitational “slip” function in Newtonian gauge Newton’s “Constant” no longer Constant Gravitational Field Fluctuations generate anti-screening, and a slow “running” of G(k) virtual graviton cloud infrared cutoff Infrared RG Running of G Phys. Rev. D Sep. 2015 Vacuum Condensate Picture of QG Lattice Quantum Gravity: Curvature condensate Quantum Chromodynamics: Gluon and Fermion condensate Electroweak Theory: Higgs condensate ~ 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) 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 : … 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 Matter Density Perturbations Visualizing Density Perturbations on very large scales … Observed (CfA) … Evolution predicted by GR Simulation (MPI Garching) Density Perturbations with G(□) Standard GR result for density perturbations : GR solution, in matter-dominated era : If Gravity is modified, then k=0 Peebles exponents will change: [with R. Toriumi PRD 2011] (Peebles) Growth Index Parameter γ Growth index parameter γ , as a function of the matter fraction Ω Measured growth parameter γ Alexei Vikhlin et al., Rapetti et al. 2012 Gravitational “Slip” with G(□) In the conformal Newtonian gauge In classical GR : η = ψ/φ – 1 = 0 … Thus a good test for deviations from classical GR. [with R. Toriumi PRD 2013] The End This slide intentionally left blank N-Component Heisenberg model Field theory description of O(N) Heisenberg model unit vector with N components 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) Test of Field Theory Methods 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 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) Second most accurate predictions of QFT, after g-2 This slide intentionally left blank Gravity in 2+ε Dimensions Wilson’s double expansion … Formulate theory in 2+ε dimensions. Wilson 1973 Weinberg 1977 … Kawai, Ninomiya 1995 Kitazawa, Aida 1998 G is dim-less, so theory is now perturbatively renormalizable ( pure gravity : with a non-trivial UV fixed point : { … suggests the existence of two phases ns 0 ) 2+ε Cont’d 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 ? This slide intentionally left blank

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