Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G...
Transcript of Quantum Gravity - University of California, Irvineaeneas.ps.uci.edu/bh.pdf · Running Newton’s G...
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Quantum Gravity on a
Lattice
A Picture Book Description
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Outline
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Perturbative Quantum Gravity
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Bad high energy behavior
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Non-Renormalizability in Four Dimensions
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� 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 =
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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: ��� � ).
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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).
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� 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:
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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]
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Perturbatively Non-Renorm. Interactions
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• 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.• …
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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
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(two loops, manifestly covariant, gauge independent)
A phase transition…
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� 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 …
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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
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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]…
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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
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� 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.
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Key question:
What is left of the above q. gravity scenario in 4 dimensions?
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Strongly coupled gravity
“Hic sunt leones”
The Roman’s description of unknown territory…
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Lattice Theory
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Lattice Quantum Gravity
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� 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.
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Proto: Wilson’ Lattice Gauge Theory
Local gauge invariance
� exact lattice Ward identities
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Lattice Gauge Theory Works
[Particle Data Group LBL, 2008]
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Quantum Continuum Limit
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Wilson Loop in SU(N) Gauge Theories
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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.
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Curvature - Described by Angles
2=d
Curvature determined by edge lengths3=d
2=d
4=dT. Regge 1961J.A. Wheeler 1964
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Lattice Rotations
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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
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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:
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Lattice Measure
CMS, 1982 ; T.D.Lee, 1982J.Hartle, 1984 ; H. & Williams, 1984 ;B. Berg, 1985 .
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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 …
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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,
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… 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:
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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,
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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:
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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
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Wilson Loop vs. Loop correlations
Giddings, Hartle & Marolf PRD 2006
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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
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Correlations
… of invariant operators at fixed geodesic distance.
Distance is a function of metric, which fluctuates:
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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:
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Dynamical Triangulations
� 0 �� � �&����- ������5�11��)�� �*� �&1�����1��-��� ���� ����4���*� �3 &��3���� BCD� &�A ��4�EF
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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
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� 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
&���������'����
��������������������$(�
H & Williams, PRD 2006
Early work in continuum by A. Strominger (1984, λ=0), ...
compared to
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Large D Limit - Exponent ν
� At large d, characteristic size ) of random surface diverges logarithmically as #� #� (D. Gross PLB 1984).
� Suggests universal correlation length exponent H !�I.
Known results from random surface theory then imply:
D. Litim PRL 2004, PLB 2007
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CM5 at NCSA, 512 processors
Numerical Evaluation of Z
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Dedicated Parallel Supercomputer
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Edge length/metric distributions
�
�
�
�
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Two Phases of L. Quantum Gravity
Smooth phase: R � 0
Rough phase :branched polymer, d � 2
Lattice manifestation of conformal instability
Unphysical
Physical
Similar two-phase structure also found later in some d=4 DTRS models [Migdal, …]
Earliest studies of Regge lattice theories found evidence for :
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Invariant Averages
� Divergent local averages provide information about non-trivial exponents.
� Finite Size Scaling (FSS) theory useful.
� Correlations are harder to compute directly (geodesic distance).
0 �1$�� � �-� ���������������1*�� ���&���� ��&����������� � 3 � �- �� ��� �����3�����1�-�
!"��������� ������# ����$��%
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Determination of Scaling Exponents
Find value close to 1/3:
?5D�HA�����
Scaling assumption:
H J K
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(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+���������������<-
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Exponent H compared
�
��!�" ��!�M *H!I+
∞∞∞∞
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�
����
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):
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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…
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New question then :
Is this new scenario physically acceptable?
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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
]
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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�������� ���������-������ ���������
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