“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 ]
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
• 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…
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.
G G
Two Graviton Loops : worse
Perturbation theory badly divergent …
Wrong ground state ??
G
G
G
G
Cosmological Constant
Add to the action λ (Volume of Space-Time), or :
Einstein’s “biggest blunder” (1917, 1922).
A new length scale or
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
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
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 :
• 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
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 chromo-electric condensate / Quark condensates
QCD is Very Hard.
Very Serious Supercomputers.
Fermilab LQCD Cluster
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 :
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” :
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:
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
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”
Tullio Regge and John A. Wheeler, ca 1971
(Photo courtesy of R. Ruffini)
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 :
Curvature - Described by Angles
2d
Curvature determined by edge lengths 3d
2d
4d
T. Regge 1961
J.A. Wheeler 1964
Edge lengths replace the Metric
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:
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
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
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]
CM5 at NCSA, 512 processors
Numerical Evaluation of Z
256 cores on 32⁴ lattice → ~ 0.8 Tflops
1024 cores on 64⁴ lattice → ~ 3.4 Tflops
…
Distribute Lattice Sites on 1024 Processor Cores
Lattice Sites are Processed in Parallel
Curvature Distributions
4⁴ sites → 6,144 simplices
8⁴ sites → 98,304 simplices
16⁴ sites → 1.5 M simplices
32⁴ sites → 25 M simplices
64⁴ sites → 402 M simplices
Phases of Quantum Gravity
Smooth phase: R ≈ 0
Unphysical (branched
polymer-like, d ≈ 2)
Unphysical
Physical
L. Quantum Gravity has two phases …
Lattice Continuum Limit
Continuum limit requires the existence of an UV fixed point.
(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 :
Determination of Scaling Exponents
Find value for ν close to 1/3 :
Use standard Universal
Scaling assumption:
ν ≈ ⅓ ( Phys Rev D Sept. 2015 )
Recent runs on 2400 node cluster
ν = 0.334(4)
Distribution of zeros in complex k space
More calculations in progress …
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0
2
4
6
8
10
12
k
Rk
164
324
fit
164
324
data
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070
5
10
15
20
25
30
k
k
164
324
fit
164
324
data
Finite Size Scaling (FSS) Analysis
Exponent Comparison (D=4)
ν ≈ ⅓
Exponent ν determines the “running” of G
Phys. Rev. D Sep. 2015
Exponent Comparison (D=3)
Comparison for Univ. Exponent 1/ν
Almost identical to 2 + ε expansion result, but with a 4-d exponent ν = 1/3
and a calculable coefficient c0 … “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 :
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
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 …
… Evolution predicted by GR
Observed (CfA)
Simulation (MPI Garching)
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]
(Peebles) Growth Index Parameter γ
Growth index parameter γ , as a function of the matter fraction Ω
Alexei Vikhlin et al., Rapetti et al. 2012
Measured growth parameter γ
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.
The End
This slide intentionally left blank
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
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
This slide intentionally left blank
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
0sn( pure gravity : )
with a non-trivial UV fixed point :
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
This slide intentionally left blank
