Vacuum Condensate Picture
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.
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
One loop diagram quartically divergent in d=4.
Two Graviton Loops : worse
Perturbation theory badly divergent …
Wrong ground state ??
Cosmological Constant
Add to the action λ (Volume of Space-Time), or :
A new length scale
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
Quarks and gluons are confined. Main theoretical
evidence for confinement and chiral symmetry
breaking is possibly from the lattice.
BCS Theory: Fermions close to the Fermi surface
condense into Cooper pairs.
Described by condensate density.
Degenerate Electron Gas
Screening due to Thomas-Fermi mechanism.
Homogeneous Turbulence
Rc= critical Reynolds no.
Spontaneous Symmetry Breaking & dimensional
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).
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.
• 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
“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
Unphysical (branched
polymer-like, d ≈ 2)
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 :
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 :
( Phys Rev D Sept. 2015 )
Recent runs on 2400 node cluster
324 data
324 fit
Distribution of zeros in complex k space
R k
ν = 0.334(4)
324 data
324 fit
More calculations in progress …
Finite Size Scaling (FSS) Analysis
Exponent Comparison (D=4)
Exponent ν determines the “running” of G
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)
• Modified results for (Relativistic) Matter Density
• Non-Vanishing “Slip” Function in conformal Newtonian
• 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
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
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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
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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 ?
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