Vacuum Condensate Picture Quantum Gravitation · Vacuum Condensate Picture of Quantum Gravitation General reference: “Quantum Gravitation” ... Misner Thorne Wheeler, “Gravitation”

“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

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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

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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

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