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Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

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Page 1: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

���������������

�� �������� ������ ��������

�������������� !

Quantum Gravity on a

Lattice

A Picture Book Description

Page 2: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Outline

" �������������� �������� ��� ��������

" # �$����������$��� ��������� %� ��� ��&'�

" ($��$�)�� �*� ���+, & ��- ��-

" .��������������� %��������� �-/�0 1�2�&��

" ����������� ���������������� ��� ��������

" 0 �� 3 ����� 3����1$�� %�&����$�� ��

" 2������ ��&-4�.�-������-

" 2����&-�����&���� � �1������� � ��-3� �1�& ��- ��-

" � ��������������������� ���������� ����������������

" 5$�� �1����6�7���8-�)

" ����3� ���������3��3��� ������� � -� 3�� ��&��9$� ��-

" 0 ����:3�-���1 3����-�� 3� -����� 3;�-��$� ��-

Page 3: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Perturbative Quantum Gravity

������������� ������������������ �����

��������� ����������

Bad high energy behavior

Page 4: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Non-Renormalizability in Four Dimensions

��� �� ���������� ���������������

���������� � ������� ����

I

� 4-d perturbation theory in (ordinary) gravity seemingly leads to a dead end.

� Non-perturbative methods ? �non-perturbative regularization, search for a new vacuum …

I =

Page 5: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Feynman Path Integral

Reformulate QM amplitudes in terms of discrete Sum over Paths

• non-commuting operators ��� replaced by randomWiener paths.

• In complex time ������ττττ probabilities are real (as in statistical mechanics: ��� � ).

Page 6: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Path Integral for Quantum Gravitation

DeWitt approach to measure: Volume element in function space obtained from super-metric over metric deformations.

Euclidean E-H action unbounded below (conformal instability).

Page 7: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

� In the absence of matter, only one dimensionless coupling:

Similar to the �of QCD !

Only One Coupling

Rescale metric (edge lengths):

Pure gravity path integral:

Page 8: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Functional Measure cont’d

Skeptics should systematically investigate (on the lattice) effectsdue to the addition of an ultra-local term of the type

Add volume term to functional measure (Misner 1955) ;

coordinate transformation

Due to it’s ultra-local nature, such a term would not be expectedto affect the propagation properties of gravitons (which are det. by R-term).

[Faddeev & Popov, 1973]

Page 9: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Perturbatively Non-Renorm. Interactions

������������ �����

• K.G. Wilson, Quantum Field Theory Models in D < 4, PRD 1973. • K. Symanzik, Renormalization of Nonrenormalizable Massless �� Theory, CMP 1975.• G. Parisi, Renormalizability of not Renormalizable Theories, LNC 1973.• G. Parisi, Theory Of Nonrenormalizable Interactions - Large N, NPB 1975.• E. Brézin and J. Zinn-Justin, Nonlinear � Model in 2+� Dimensions, PRL 1976.• …

Page 10: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Gravity in 2.000001 Dimensions

• Wilson expansion: formulate in 2+� dimensions…

G becomes dimensionless in d = 2 ... “Kinematic singularities” as d � 2 make limit very delicate.

But G is dim-less and theory is pert. renormalizable,

cG

�� ��� �� ��

�� � � �� � ��

� �� � � � �� � � � �� � �

� � �� �� � � � � � �� � �

(two loops, manifestly covariant, gauge independent)

A phase transition…

Page 11: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

� Expansion parameter !�" not small …

� Singularity structure in d > 2 unclear (Borel)…

� <$��nalytical control of UV fixed point at #� .

Nontrivial scaling determined by UV FP.

More on 2.000001 dim’s …

Page 12: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Detour : Non-linear Sigma model

$ Field theory description [O(N) Heisenberg model] :

Coupling g becomes dimensionless in d = 2.For d > 2 theory is not perturbatively renormalizable, but in the 2+ � expansion one finds:

Phase Transition = non-trivial UV fixed point; new non-perturbative mass scale.

E. Brezin J. Zinn-Justin 1975F. Wegner, 1989N.A. Kivel et al, 1994E. Brezin and S. Hikami, 1996

Page 13: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Renormalization Group Equations

In the framework of the double (g and ��) expansion the modellooks just like any other renormalizable theory, to every order…

Callan-Symanzik Eq.

… but the price one pays is that now one needs �� �

Similar result are obtained in large N limit [Parisi]…

Page 14: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Experimental test: O(2) non-linear sigma model describesthe 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 FP)

But is it correct ?

One of the most accurate predictions of QFT -Theory value reviewed in J. Zinn-Justin, 2007

Page 15: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

%������&� ������ ���������� ��'������ ������ �(�� � ��(������������ ���������� � � ����

� Is not perturbatively renormalizable in d=3 .

� Nevertheless leads to detailed, calculablepredictions in the scaling limit r » a ( q² « �² ) .

� Involves a new non-perturbative scale ξ, essential in determining the scaling behavior in the vicinity of the FP.

� Whose non-trivial, universal predictions agree withexperiments.

Page 16: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Key question:

What is left of the above q. gravity scenario in 4 dimensions?

Page 17: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Strongly coupled gravity

“Hic sunt leones”

The Roman’s description of unknown territory…

Page 18: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Theory

Page 19: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Quantum Gravity

�������� �����������������

�������������������� �����

� Regularized theory is finite, allows non-perturbative treatment.

� Methods of statistical field theory.

� Multi-year experience with lattice QCD.

� Numerical evaluation feasible. � Continuum limit requires UV

fixed point.

Page 20: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Proto: Wilson’ Lattice Gauge Theory

Local gauge invariance

� exact lattice Ward identities

Page 21: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Gauge Theory Works

[Particle Data Group LBL, 2008]

)��� �������������������� ����*������+�������������� �� ����� ������������������������������������������� ���,-

Page 22: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Quantum Continuum Limit

� . �/������� ������ � ���

� ����������� ������ � ��

*����������#+��

0 ���������������*1 2 �� (����� ��+� ����3� ��� ���������( ���������������&�� � ������� ������ � ��45 ����6�789:;-

���� �����

Page 23: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Wilson Loop in SU(N) Gauge Theories

� 5 ���������� ��)��� ���#�����%���� ��6

# ����� ��������� ������ 4��(������������6�<��� � =������������-�9>';

? !��������������� ���������

� @���������������������� ������������ � ��������-�

%���������� �*��������� � ����+����������������������� !" A �����������-

Page 24: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Simplicial Lattice Formulation

� Based on a dynamical lattice.

� Incorporates continuous local invariance.

� Puts within the reach of computationproblems which in practical terms arebeyond the power of normal analyticalmethods.

� It affords any desired level of accuracyby a sufficiently fine subdivision of thespace-time region under consideration.

�� �� ��� �� ��� �� � �� �� � � ��� � �� �� � ��� �� � ��� ��

Page 25: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Curvature - Described by Angles

2=d

Curvature determined by edge lengths3=d

2=d

4=dT. Regge 1961J.A. Wheeler 1964

Page 26: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Rotations

,����������� ���B�� ��� �� ���� ����� ��6������������������������������ ������������������� �����������������������

C(�������� ���@ ���� � ���� ��6

Page 27: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Action

More than one way to finite-difference a continuum expression…

� Alternate actions can be a useful device for analytical estimates (i.e. large d)� Should exhibit same continuum limit (universality)

rotation matrixhinge bivector

J. Fröhlich 1980T.D. Lee 1984Caselle, d’Adda Magnea 1989

Page 28: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Choice of Lattice Structure

Timothy Nolan,

Carl Berg Gallery, Los Angeles

Regular geometric objects (hypercubes) can be stacked -to form a regularly coordinated lattice of infinite extent.

A not so regular lattice …

… and a more regular one:

Page 29: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Measure

CMS, 1982 ; T.D.Lee, 1982J.Hartle, 1984 ; H. & Williams, 1984 ;B. Berg, 1985 .

D���� �� ������B������B�� ������������ �� ����� ����(6

0 ������� ����6�������������������� �����������������,�5 ��B��*�����+����� ����������� ���������� ���6��������� ��������������4��;�

��� ���������� ����� ������������������# �����������

Page 30: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Measure is Non-Trivial

There are important nontrivial constraints on the lattice gravitational measure,

which is generally subject to the “triangle inequality constraints” :

Generally these are implied in the continuum functional measure as well, but are normally not spelled out in detail …

Page 31: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Path Integral

Without loss of generality, one can set bare λλλλ $%;

Besides the cutoff, the only relevant coupling is E (or #).

Lattice path integral follows from edge assignments,

Page 32: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

… then Fourier transform, and express result in terms of metric

deformations :

Lattice Weak Field Expansion• Exhibits correct nature of gravitational degrees of freedoms in the lattice weak field limit.• Allows clear connection between lattice and continuum operators.

… start from Regge lattice action 5�=�>�&�� �� -4�?@<�A �A

… call small edge fluctuations “�” :

… obtaining in the vacuum gauge precisely the familiar TT form in k limit:

Page 33: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Lattice Higher Derivative Terms

� HDQG is perturbatively renormalizable, asymptotically free, but contains s=0 and s=2ghosts,

� Lattice higher derivative terms… involve deficit angles squared, as well as coupling between hinges,

Page 34: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Scalar Matter

Make use of lattice metric to correctly define lattice field derivatives [Ninomiya 1985] …

… and obtain a simple geometric form, involving dual (Voronoi) volumes

…which also allows correct definition of lattice Laplacian:

Page 35: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Fermionic Matter

Start from continuum Dirac action

Discrete action [Drummond 1986] involves lattice spin connection :

ψ(s)ψ(s’)

Potential problems with fermion doubling (as in ordinary LGT)…

S

Page 36: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Wilson Loop vs. Loop correlations

Giddings, Hartle & Marolf PRD 2006

Page 37: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Wilson Loop does not give Potential

In ordinary LGT, Wilson loop gives ����

In lattice regularized gravity, potential is computed fromthe correlation of geodesic line segments, associated withthe particle’s world line:

G. Modanese, PRD 1994;NPB 1995

Page 38: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Correlations

… of invariant operators at fixed geodesic distance.

Distance is a function of metric, which fluctuates:

Page 39: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Hypercubic Lattice Gravity

� ������������� ������ ���& ������� ������������������ ����e.g. Mannion &Taylor PLB 1982 ; see also Smolin 1978; Das Kaku Townsend 1982.

� )��� ���� ����� F�� ���������������� �����������������)*"6�+�������G *'67+��������G *:+

Local gauge invariance:

Path integral over U’s (Haar) and E’s:

Page 40: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Dynamical Triangulations

� 0 �� � �&����- ������5�11��)�� �*� �&1�����1��-��� ���� ����4���*� �3 &��3���� BCD� &�A ��4�EF

���-�� ��-����� ������������ ��$�3���4�- �3�������*��-�����B�� 3 ����� 3�

0�� �1�*�������1 3������ �������$3� &��������*�[Loll et al]

�$�� 3��@�����% ������ ���1���:7 ���*���$���-����&�3�����1��3�� --$�-;�

� 6�� �& ������ ������3��� �$�$-���� 34����3��� �$�$-�& ����-�� �$���$����� �-� ��& -3�����-���-�� 6��3��� �$�$-���� 3�&����� ��-�G ���3���� 7�����4��&���

1�� ���-�:����-������ �����B�� 3 ��7*;�

an integer

Page 41: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

� Conformal mode instability disappears, O(1/d).

� At large d, partition function at large G dominated by closed surfaces, tiled with elementary parallel transport polygonal loops. Very large surfaces are important as k � kc .

Large D Limit

N-cross polytope, homeomorphic to a sphere

&���������'����

��������������������$(�

H & Williams, PRD 2006

Early work in continuum by A. Strominger (1984, λ=0), ...

compared to

Page 42: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Large D Limit - Exponent ν

� At large d, characteristic size ) of random surface diverges logarithmically as #� #� (D. Gross PLB 1984).

� Suggests universal correlation length exponent H !�I.

Known results from random surface theory then imply:

D. Litim PRL 2004, PLB 2007

Page 43: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

CM5 at NCSA, 512 processors

Numerical Evaluation of Z

Page 44: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Dedicated Parallel Supercomputer

Page 45: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Edge length/metric distributions

Page 46: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Two Phases of L. Quantum Gravity

Smooth phase: R � 0

Rough phase :branched polymer, d � 2

Lattice manifestation of conformal instability

Unphysical

Physical

Similar two-phase structure also found later in some d=4 DTRS models [Migdal, …]

Earliest studies of Regge lattice theories found evidence for :

Page 47: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Invariant Averages

� Divergent local averages provide information about non-trivial exponents.

� Finite Size Scaling (FSS) theory useful.

� Correlations are harder to compute directly (geodesic distance).

0 �1$�� � �-� ���������������1*�� ���&���� ��&����������� � 3 � �- �� ��� �����3�����1�-�

!"��������� ������# ����$��%

Page 48: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Determination of Scaling Exponents

Find value close to 1/3:

?5D�HA�����

Scaling assumption:

H J K

Page 49: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

(Lattice) Continuum Limit � � �

Bare # must approach UV fixed point at #�

UV cutoff � � �

(average lattice spacing � 0)RG invariant correlation length I is kept fixed

?

�������� *�����"����� ���� �������������+

%�������������� ���� ������� �#����� ������#*L+���������������<-

Page 50: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Exponent H compared

��!�" ��!�M *H!I+

∞∞∞∞

Page 51: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

����

g

��g�

RG Running Scenarios

g

��g�

G

��G�

� Coupling gets weaker at large r� … approaches an IR FP at large r.� … gets weaker at small r : UV FP� Both possibilities can coexist:

nontrivial UV fixed point.Wilson-Fisher FP in d<4“Triviality” of lambda phi 4

Asymptotic freedom of YM Ising model, �-model, Gravity (2+�, lattice)

Callan-Symanzik. beta function(s):

Page 52: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

G

��G�

Only One Phase?

Gravity

*���� ������������������ ���������

�������������"�

,���-������������

� Lattice results appear to exclude theweak coupling phase as physicallyrelevant…

� Leads to a gravitational coupling Gthat increases with distance…

/////////

Page 53: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

New question then :

Is this new scenario physically acceptable?

Page 54: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

Running Newton’s G

� � is a new invariant scale of gravity.� Newton’s constant G must run.� Cutoff dependence determines �-function :

[ In fact, one can be quite specific …

Running of � det. largely by � and � :

and

]

Page 55: Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G is a new invariant scale of gravity. Newton’s constant G must run. Cutoff dependence

So, what value to take for ξ ?

In Yang-Mills m = glueball mass

" ξ is an RG invariant.

" m=1/ξ has dimensions of a mass-

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Three Theories Compared

Suggests

RG invariants Running couplings

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Gravitational Wilson Loops

� <������������������������������������ ������ �������� ������� (

������������������ ��� ������ ��*���������- � � �����������������������������

����� ����� ������ ������������������ ���������������� �� [Caselle, d’Adda, Magnea PLB 1989]

- Stokes theorem -

N0 ������� O ������ ������������� � ��A A =� --5 �� ���6�<�,�9P6�"II9

$ � ��������������������-$ � �#� ���� ���-$ ���� �� ��������� � ��

�������� ������������Q

� R������ ��6�5 ������������� ��������������� �������� ��[G. Modanese PRD 1993; PRD 1994]

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Vacuum Condensate Picture of QG?

� Lattice Quantum Gravity: Curvature condensate

� Quantum Chromodynamics: Gluon and Fermion condensate

� Electroweak Theory: Higgs condensate

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Effective Theory

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Graviton Vacuum Polarization Cloud

Picture: Source mass M surrounded by virtual graviton cloud

Need a covariant running of G.

.����������# ������+

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Relative Scales in the Cutoff Theory

Pl ξξ<<<< rlP ξ>>r

cm3310−≈ cm2810≈

�������?��3>����1�����7����-�����/�

���������������������&�����������������&����������������������� �����'

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Cosmological Solutions

… for RW metric

Explore possible effective field equations…generally covariant

… and perfect fluid

Form of D’Alembertian depends on object it acts on …

Consistency condition:

G. VenezianoG.A. Vilkovisky ..

initially for simplicity

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Solution of Effective Field Equation

� Full effective field equation involves D’Alembertian on tensor

Repeated action of D’Alembertian ,

existence of solution requires as before,

and ,���������������'��/�������0$1�

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Cosmological Solutions – Cont’d� Modified FRW solution acquires a significant radiation-like (vac. pol.)

component at large times,

t-t eq.

r-r eq.

Effective pressure term

At (very) large times, G is further modified to:

IR regulator

Similarities to:

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Modified cosmological expansion rate

Λ-dominated expansion at later times

Standard FRW expansion at early times

Running G effects are maximal “now”[T. D’Amour 2007]

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Static Isotropic Solution

Start again from fully covariant effective field equations

Search solution for a point source, or vacuum solution for r�0.

General static isotropic metric 0

�����

H. & Williams, PLB 2006; PRD 2007

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Static Isotropic Solution

Non-relativistic solution can be obtained from vacuum density:

Relativistic field equations become:

Promote S*�+�to a covariantly conserved, relativistic perfect fluid, with a � �"+

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Relativistic Fluid cont’d

And finally …

��� � ���������C,�*1 ��� ��+����� ���

…which can be consistently interpreted as a �����

a 42.

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Outlook

� More Work is Needed– 2 + � expansion to three loops is a clear, feasible goal.– Systematic careful investigation of 4d s. gravity should be pursued– Status of weak coupling phase unclear– Connection with other lattice models, eg hypercubic?

� Covariant Effective Field Equations– Formulation of fractional operators.– Further investigation on nature of solutions (horizons).– Possible Cosmological (observable) ramifications.

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The End

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“Herb, as far as I know you are the only one that still believes in this non-trivial ultraviolet fixed pointscenario [for gravity] � ”

Howard Georgi, January 30, 2008

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Large D and Strong Coupling

At large d, strong coupling (large #) expansion simplifies considerably, as excluded volume effects can be neglected in this limit …

� Strong coupling expansion for gravity,

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Galactic Rotation Curves

� Straightforward, calculable relationship between potential modification and deviations in galactic rotation curves

Capoziello, Cardone,Troisiastro-ph0602349 (2006)

Very large values of �� *�� make effects tiny on kp scales.

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Relation to �� Gravity Models

� Superficial resemblance of running 2 �" model to �� = scalar-tensor gravity theories (within FRW cosmology framework).

� Obtained - from running 2 �" models - by simply replacing scale factor a(t) with scalar curvature R :

S. Capoziello, A.Troisi et alS. Carroll et al ; E. Flanagan

3 4" models generally lack justification as to whyonly Ricci scalar 4 should be considered in action.

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Based on same arguments as in d = 4 would expect solution to exists only if ,consistent with the result found on the lattice at d = �. Problem not fully worked out yet.

Static Isotropic Solution in d Dim’s

Covariant effective field equations in d space-time dimensions

In the absence of a running G, static isotropic solutions in d dimensions are given by:

I

Myers and Perry, Ann. Phys 1986Xu, Class Q Grav 1988

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Effects of small gauge breaking

� N,���� �������� � ��������������������������O

,-���������6�A -�@-�. �����6�-�. ��� �� ��<��� ���)�������@�8:6�7'T�*78>I+

5*���/���������������������������� ���������������'/�����������������'/���������������������� ����-������������������������������������6

� #-�<�� � ���������& ������������������ *��-�788P6������� ����+��

N%������������������������&����� ��������������������F���O

<��������������.���� �7� ����� �������R����� � � ���G ����������������@���� ��)������������ ����*<�,�7"6�'89>6789T+6�������������������������-

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Quantum “Gravity” in two dimensions ?

• KPZ formula predicts H!'U"����-�*���&G ��������(�������+������!7U"�*R� �� �� ��+���������������� ��� 5�����������������������������6"�

• A flat space realization of same KPZ exponents is found instead: The change in the exponents appears due to the randomness of the interaction.

Vekic, Liu & H, PLB 1994; PRD 1994

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Beirl and Berg, NPB 1995

Follow up: dynamically triangulated Ising spins (c=1/2) onfixed curved geometry(sphere) also give KPZ exponents.

Some lattice re-linkings

Is there Q. Gravity in two dimensions ?

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Three Approaches Compared

G

��G�

� Simplicial Lattice QG:

� E-H “truncation” (Reuter/Litim):

� Continuum 2+� expansion (1 loop):

� � generally complex � Scale � Q�<������Q� Formulation of #*�+ not covariant� C�������� ���������-������ ���������

/////////