I Basic assumptions and results

1
Emergence of chiral matter from quantum
gravityLee Smolin

• I Basic assumptions and results
• II Gerards holographic principle and quantum
  gravitational dynamics
• III Quantum information and quantum gravity
• F.Markopoulou hep-th/0604120
• D. Krebs and F. Markopoulou gr-qc/0510052
• IV Unification of quantum geometry with matter
• S. Bilson-Thompson, F. Markopoulou, LS
  hep-th/0603022

2
Loop quantum gravity community

• A.Sen
• A.Ashtekar
• L. Smolin
• C. Rovelli
• P. Renteln
• T. Jacobson
• V. Husain
• R Gambini
• J Pullin
• B Bruegmann
• R. Loll
• F.Markopoulou
• J. Samuels
• T. Newmann
• P Morao
• A. Perez
• J Iwasaki
• A Mikovic
• J Wisniewski

Y Ling S Major D. Longais A Stradobodsov M
Arnsdorf C Isham R Garcia S Alexandrapov H
Kodama J. Dell R. Capovilla, J. Romano S
Alexander M Shepard G Amelino-Camelia J.
Magueijo J Kiwalski-Glickman L.N. Chang B
Krishnan G Egan M Ansari S. Hoffman J.
Brunnemann M. Okolo

• T. Thiemann
• J. Lewandowski
• J. Morao
• E. Hawkins
• H. Sahlman
• O. Winkler
• M. Reisenberger
• L. Crane
• J. Baez
• J. Barrett
• R. de Pietro
• L. Freidel
• K. Krasnov
• R. Livine
• L. Kauffman
• H. Morales-Tecotl
• O. Dreyer
• C. Soo
• S Fairhurst

D. Oriti R. Williams D.Christensen S. Gupta J.
Ambjorn K.Anagastopolo K. Christensen R. Tate L.
Mason O. Dreyer M. Bojowald D. Yetter A Corichi J
Zapata J Malecki M. Varadarajan L. F. Urrutia J.
Alfaro K. Noui P. Roche M Bojowald F. Girelli T.
Konopka
3
LQG is not , so far, a candidate for the
fundamental theory.
4
LQG is not , so far, a candidate for the
fundamental theory. It is A well defined and
consistent framework for defining and studying a
large class of diffeomorphism invariant gauge
theories. These include GR, SUGRA, in any
spatial dimension, dgt1, with varied matter
fields. There is a large body of results of
different kinds, some rigorous, some heuristic,
some from study of related models of black holes
and quantum cosmologies. It is a well developed
framework for approaching background independent
quantum theories.including perhaps string/M
theory. It is based on 4 principles
5

• Three principles
• 1) The Gauge principle All forces are described
  by gauge fields
• Gauge fields Aa valued in an algebra G
• Gravity Aa valued in the lorentz group of SU(2)
  subgroup
• p form gauge fields
• Supergravity Ym is a component of a connection.
• 2) Duality equivalence of gauge and loopy
  (stringy) descriptions
• observables of gauge degrees of freedom are
  non-local
• described by measuring parallel transport around
  loops
• Wilson loop Tg,A Tr exp ?gA g
• 3)

6

• Three principles
• 1) The Gauge principle All forces are described
  by gauge fields
• Gauge fields Aa valued in an algebra G
• Gravity Aa valued in the lorentz group of SU(2)
  subgroup
• p form gauge fields
• Supergravity Ym is a component of a connection.
• 2) Duality equivalence of gauge and loopy
  (stringy) descriptions
• observables of gauge degrees of freedom are
  non-local
• described by measuring parallel transport around
  loops
• Wilson loop Tg,A Tr exp ?gA g
• Developed on a background with fixed metric, this
  leads to string theory!

7

• Three principles
• 1) The Gauge principle All forces are described
  by gauge fields
• Gauge fields Aa valued in an algebra G
• Gravity Aa valued in the lorentz group of SU(2)
  subgroup
• p form gauge fields
• Supergravity Ym is a component of a connection.
• 2) Duality equivalence of gauge and loopy
  (stringy) descriptions
• observables of gauge degrees of freedom are
  non-local
• described by measuring parallel transport around
  loops
• Wilson loop Tg,A Tr exp ?gA g
• 3) Diffeomorphism invariance and background
  independence

8

• The gravitational field can be described as a
  gauge theory
• Spacetime connection Gauge field
  configuration variable
• Spacetime metric Electric field
  momentum
• Quantum gauge fields can be described in terms of
  operators that
• correspond to Wilson loops and electric flux.
  These have a natural
• algebra that can be quantized
• The loop/surface algebra.
• Tg,A Trexp ?gA ES ?SE

h G Int
,
9
The fundamental theorem Consider a background
independent gauge theory, compact Lie group G on
a spatial manifold S of dim gt1. No metric!!
(GSU(2) for 31 gravity) There is a unique
representation of the loop/surface algebra in
which the Hilbert space carries a unitary rep of
the diffeomorphism group of S, called
Hkin. Lewandowski, Okolo, Sahlmann, Thiemann
Fleishhack (LOST theorem)
This means that there is a unique diffeomorphism
invariant quantum quantum theory for each G and
S. The Hilbert space of diffeo invariant
states, H, is a subspace of Hkin Ashtekar GR
is a diffeomorphism invariant gauge theory!!
The dynamics of GR have been expressed in
closed form in terms of finite operators and
evolution amplitudes on H.
10

• General structure causal spin network theories
  
• Pick an algebra G
• Def G-spin network is a graph G with edges
  labeled by representations of G and vertices
  labeled by invariants.
• Pick a differential manifold S.
• G an embedding of G in S, up to
  diffeomorphisms
• Define a Hilbert space H
• G gt Orthonormal basis element for each G
• Define a set of local moves and give each an
  amplitude
• A history is a sequence of moves from an in state
  to an out state
• Each history has a causal structure

11
Role of the cosmological constant Requires
quantum deformation of SU(2) qe2pi/k2 k
6p/GL
To represent this the spin network graphs must be
framed
12

• Gerards Holographic principle
• The physical degrees of freedom live on spatial
  boundaries
• There is one degree of freedom per unit Planck
  area

13

• Gerards Holographic principle
• The physical degrees of freedom live on spatial
  boundaries
• There is one degree of freedom per unit Planck
  area
• This is automatically implemented in the LQG
  framework
• (Anti) desitter boundary conditions Fab L
  Sab selfdual 2-form
• Boundary Hilbert space HB
• conformal blocks on punctured surface
• Given the area operator A S
• Bekenstein bound satisfied
• S Ln Dim HB A/4Gh with choice of g

14
How is the holographic principle implemented
dynamically? Use the topological QFT ladder of
dimensions D 2 Gq rational conformal field
theory
15
How is the holographic principle implemented
dynamically? Use the topological QFT ladder of
dimensions D 2 Gq WZW conformal field
theory D3 Chern-Simons theory in G
G
16
How is the holographic principle implemented
dynamically? Use the topological QFT ladder of
dimensions D 2 Gq WZW conformal field
theory D3 Chern-Simons theory in G D4 BF
theory S IBF Sboundary
TQFT is automatically holographic
Boundary condition Fab L Sab k 6p/GL
17
How is the holographic principle implemented
dynamically? Use the topological QFT ladder of
dimensions D 2 Gq WZW conformal field
theory D3 Chern-Simons theory in G D4 BF
theory S IBF Sboundary
TQFT is automatically holographic
Boundary condition Fab L Sab k 6p/GL
18
What about quantum gravity? GR, SUGRA are
constrained or perturbed TQFTs

• Since the modifications from TQFT are
  non-derivative,
• boundary conditions and terms, are unchanged.
• So the boundary Hilbert space is the same, with
  the same
• implications.
• The path integral measure for GR is constrained
  or perturbed
• from that of the TQFT and is well defined.
• This is the basis of the Barrett-Crane spin foam
  model.
• This is also the basis of the LQG description of
  horizons.

19
4) Gravitational theories are constrained or
perturbed topological field theories Proposal
The holographic principle is equivalent to
this 4th principle.
20
Example d11 supergravity as a constrained
Topological field theory Y. Ling ls
hep-th/0003285
The topological field theory gauges 11d
super-Poincare central terms
SPoincare gauge field
3 and 6 forms, a3 and b6 gauge central
terms Curvatures
21

• What has and has not been done?
• A well defined, and unique framework for
  formulating and studying diffeomorphism invariant
  gauge theories in any dim, with or without susy.
  Discreteness of area and
  volume operators.
• In d3,4 the hamiltonian constraint for quantum
  GR is known on H in closed form and is uv finite,
  including all usual matter fields.
• In d3 the theory coupled to interacting scalar
  fields has been solved and gives an effective QFT
  on k-Minkowski spacetime.
• In d4 the path integral involves a sum over
  diagrams and on each a sum over labels. The
  latter are known to be uv finite.
• In d4 there is evidence, not yet proof, for a
  good low energy limit which recovers GR QFT.
• Semiclassical states exist, excitations are low
  energy gravitons.
• Propagator (See Carlos talk)

22

• Indications of novel and testible O(lPl) effects
  including deformation of
• Poincare symmetry leading to an energy dependent
  speed of light. This
• is shown precisely in 21 but only
  semiclassically in 31.
• Reduced models for cosmology and black hole
  interiors solved
• spacelike singularities eliminated and replaced
  by bounce. (Bojowald )
• Predictions for corrections to CMB (Hoffman-Winkle
  r)
• Black hole entropy understood in terms of quantum
  geometry of
• horizons, recent results involve corrections to
  entropy and radiation
• (Krasnov, Baez, Ashtekar, Corichi, Dreyer,
  Ansari,.)

Ansari hep-th/0607081
23
If LQG really unifies gravity and QM, shouldnt
it automatically tell us about unifying the rest
of physics?
S. Bilson-Thompson, F. Markopoulou, ls,
hep-th/0603022
24

• Some hard questions for LQG theories
• The geometric observables such as area and volume
  measure the
• combinatorics of the graph. But they dont care
  how the
• edges are braided or knotted. What physical
  information
• does the knotting and braiding correspond to?
• How do we describe the low energy limit of the
  theory?
• How do we define local subsystems without a
  background?
• How do we recognize states that correspond to
  gravitons or
• other local excitations?
• What keeps them from loosing coherence by mixing
  into the
• quantum spacetime foam?

25

• Some answers (Markopoulou, Kribs )
• hep-th/0604120 gr-qc/0510052
• Define local as a characteristic of excitations
  of the graph states. To identify them in a
  background independent way look for noiseless
  subsystems, in the language of quantum
  information theory.
• Identify the ground state as the state in which
  these propagate
• coherently, without decoherence.
• 3) This can happen if there is also an emergent
  symmetry which
• protect the excitations from decoherence. Thus
  the ground state has symmetries because this is
  necessary for for excitations to persist as pure
  states.

26
Suppose we find, a set of emergent symmetries
which protect some local excitations from
decoherence. Those local excitations will be
emergent particle degrees of freedom.
27
Two results A large class of causal spinnet
theories have noiseless subsystems that can be
interpreted as local excitations.
28
Two results A large class of causal spinnet
theories have noiseless subsystems that can be
interpreted as local excitations. There is a
class of such models for which the simplest such
coherent excitations match the fermons of the
standard model.
29
We study theories based on framed graphs in three
spatial dimensions. The edges are framed
The nodes become trinions Basis States
Oriented, twisted ribbon graphs, embedded in
S3 topology, up to topological class.
Labelings any quantum groupor none.
30
The evolution moves Exchange
moves Expansion moves The amplitudes
arbitrary functions of the labels
Questions Are there invariants under the
moves? What are the simplest states preserved
by the moves?
31
Invariance under the exchange moves
32
Invariance under the exchange moves The
topology of the embedding remain unchanged All
ribbon invariants are constants of the motion.
33
Invariance under the exchange moves The
topology of the embedding All ribbon
invariants For example the link of the
ribbon
34
Invariance under the exchange moves The
topology of the embedding All ribbon
invariants For example the link of the
ribbon
35
Invariance under the exchange moves The
topology of the embedding All ribbon
invariants For example the link of the
ribbon
36
Invariance under the exchange moves The
topology of the embedding All ribbon
invariants For example the link of the
ribbon But we also want invariance under the
expansion moves
37
Invariance under the exchange moves The
topology of the embedding All ribbon
invariants For example the link of the
ribbon But we also want invariance under the
expansion moves The reduced link of the
ribbon is a constant of the motion Reduced
remove all unlinked unknotted circles
38
Definition of a subsystem The reduced link
disconnects from the reduced link of the whole
graph. This gives conserved quantities
labeling subsystems.
After an expansion move
39
Chirality is also an invariant
P

• Properties of these invariants
• Distinguish over-crossings from under-crossings
• Distinguish twists
• Are chiral distinguish left and right handed
  structures
• These invariants are independent of choice of
  algebra G and
• evolution amplitudes. They exist for a large
  class of theories.

40
What are the simplest subsystems with non-trivial
invariants?
Braids on N strands, attached at either or both
the top and bottom.
The braids and twists are constructed by
sequences of moves. The moves form the braid
group. To each braid B there is then a group
element g(B) which is a product of braiding and
twisting. Charge conjugation take the inverse
element. hence reverses twisting.
41
We can measure complexity by minimal crossings
required to draw them The simplest conserved
braids then have three ribbons and two crossings

Each of these is chiral
Other two crossing braids have unlinked circles.
P
42
Braids and preons (Bilson-Thompson)
hep-ph/0503213
preon ribbon Charge/3 twist P,C P,C triplet
3-strand braid Position?? Position in braid
In the preon models there is a rule about mixing
charges No triplet with both positive and
negative charges. This becomes No braid with
both left and right twists. This should have a
dynamical justification, here we just assume it.
The preons are not independent degrees of
freedom, just elements of quantum geometry. But
braided triplets of them are bound by
topological conservation laws from quantum
geometry.
43
Two crossing left handed invariant braids
44
Two crossing left handed invariant braids No
twists
45
Two crossing left handed invariant braids No
twists 3 twists
46
Two crossing left handed invariant braids No
twists 3 twists
1 twist
47
Two crossing left handed invariant braids No
twists 3 twists
1 twist
2 twists
48
Two crossing left handed twist braids No
twists 3 twists
Charge twist/3
nL
eL
1 twist
dLr
dLb
dLg
2 twists
uLr
uLb
uLg
49
Two crossing left handed twist braids No
twists 3 twists
Charge twist/3
nL
Including the negative twists (charge) these area
exactly the 15 left handed states of the first
generation of the standard model. Straightforwa
rd to prove them distinct.
eL
1 twist
dLr
dLb
dLg
2 twists
uLr
uLb
uLg
50
The right handed states come from parity
inversion No twists 3 - twists
nR
eR-
1- twist
dRr
dRb
dRg
2- twists
uRr
uRb
uRg
51
Higher generations come from braids with more
crossings generation crossings -1 Second
generation from three crossing braids
From all allowed twists we get a copy of the 1st
generation.
These give additional states which are SU(3)
SU(2) singlets but come in left and right
versions. Could these be the right handed
neutrinos?
52

• There is much we dont know yet
• That these excitations are fermions
• They are chiral but could be spinors or chiral
  vectors.
• How to best incorporate interactions.
• That there are candidate ground states in which
  these carry
• conserved energy and momentum.
• What the mass matrix is.
• Where P and CP breaking comes from
• Work is underway addressed to these and other
  questions.

53

• CONCLUSIONS
• LQG is based on 4 simple principles
• Gauge principle all interactions are based on
  gauge fields.
• Duality of quantum gauge theories and dynamics of
  loops
• Background and diffeomorphism invariance.
• Gravitational theories are constrained or
  deformed topological
• field theories.
• There are so far many results about quantum
  spacetime kinematics
• and dynamics.
• A large class of causal spin network theories
  have coherent (noisefree)
• subsystems which are emergent particle like
  excitations.
• In a subclass the simplest excitations have a
  pattern reminiscent of
• the standard model fermions.

54
The low energy limit and phenomenology Prediction
s for high energy astrophysics experiments?
55
The key observational question is What is the
symmetry of the ground state?
56
The key observational question is What is the
symmetry of the ground state? Global Lorentz and
Poincare invariance are not symmetries of
classical GR, they are only symmetries of the
ground state with L0. Hence, the symmetry of
the quantum ground state is a dynamical question.

57
The key observational question is What is the
symmetry of the ground state? Global Lorentz and
Poincare invariance are not symmetries of
classical GR, they are only symmetries of the
ground state with L0. Hence, the symmetry of
the quantum ground state is a dynamical question.
Three possibilities for lorentz invariance 1
brokenthere is a preferred frame 2 nothing
new realized as in ordinary QFT 3 Doubly
special relativity (DSR)
58

• Principles of deformed special relativity
  (DSR)
• Relativity of inertial frames
• The constancy of c, a velocity
• The constancy of an energy Ep
• 4) ,c is the universal speed of photons for
  EltltEp.

59

• Principles of deformed special relativity
  (DSR)
• Relativity of inertial frames
• The constancy of c, a velocity
• The constancy of an energy Ep
• 4) ,c is the universal speed of photons for
  EltltEp.
• Consequences
• Modified energy-momentum relations
• Momentum space has constant curvature given by Ep
• Spacetime geometry is non-commutative.
• metric becomes scale dependent gab (E)
• Usual energy-momentum conservation non-linear
• Linear conservation of new 5d momentum.
  (Girelli-Livine)

60

• The three possibilities are experimentally
  distinguishable
• Poincare invariance
• Lorentz invariance broken prefered frame
• Poincare deformed (DSR)
• There are two basic low energy QG effects
• 1) Corrections to energy momentum relations
• E2 p2 m2 a lp E3 b lp2 E4
• v c(1 a lp E ...)
• 2) Modifications in the conservation laws.

61

• Broken lorentz invariance gives modified
  dispersion relations
• but unmodified conservation laws
• GZK threshold moves appreciably
• helicity dependent energy dependent speed of
  light
• Deformed lorentz invariance gives both.
• GZK threshold as in ordinary special relativity
• helicity independent energy dependent speed of
  light
• To distinguish the three possibilities we need
  two experiments
• AUGER tests GZK
• GLAST tests energy dependence of photons

62
One effect of a modified energy momentum relation
alone E2 p2 m2 a lp E3 b lp2 E4
is to move the threshold for pion production
from protons scattering from microwave
photons. The threshold is predicted to be at 3
1019 ev. There is evidence from AGASA the cutoff
is not seen.
AGASA
63

• Some experiments
• see anomalous events
• AGASA, Sugar
• Some dont HIRES
• We wait for AUGER.

Tev photons from blazars, have a similar cutoff,
coming from scattering off the infrared
background. It is presently controversial
whether anomalies exist.
64

• Tests of energy dependence of the speed of light
• Energy dependent speed of light. vc(1 a lp
  E b lp2 E2 )
• Accumulates for long distances
• Observable in Gamma Ray bursts.
• present limits have a lt 1000
• next satellite, GLAST will put limits a lt 1
• A helicity dependent v(E) can be tested for
  measuring polarization
• from radio galaxies and other distant sources
• The effect is energy dependent, so polarization
  washes out
• Radio galaxies Not seen!!!
• Polarization observed in Gamma Ray Burst 021206
• Colburn,W. Boggs, S. E. Nature 423,
  415417 (2003).
• Mitrofanov, Nature, VOL 426 13 Nov 2003
• These imply no helicity dependent speed of
  light at O(lP)

65

• Does loop quantum gravity predict DSR?
• does so cleanly for 21 gravity coupled to matter
• semiclassical argument hep-th/0501091
• gab (x,t) gab (x,t, E)
• Algebraic argument (next slide)
• Naive argument why
• Discreteness implies dispersion
• Diffeomorphism and background independence
  implies
• there can be no preferred frame.
• Together these imply DSR

66
Why DSR?
67
Why DSR? Classically, When L1/L2 gt0, the
ground state is deSitter spacetime. Its
symmetry group is SO(1,4). In the limit L --gt0
the symmetry group contracts to the
Poincare group, with a scaling M0a 1/L
Pa
68

• Why DSR?
• Classically, When L 1/L2 gt0, the ground state
  is deSitter spacetime.
• Its symmetry group is SO(1,4).
• In the limit L --gt0 the symmetry group contracts
  to the Poincare
• group, with a scaling M0a 1/L Pa
• Quantum mechanically, when L is non-zero, the
  symmetry group becomes quantum deformed to
  SOq(1,4) Starodubtsev
• (i when lorentzian)

69

• Why DSR?
• Classically, When L 1/L2 gt0, the ground state
  is deSitter spacetime.
• Its symmetry group is SO(1,4).
• In the limit L --gt0 the symmetry group contracts
  to the Poincare
• group, with a scaling M0a 1/L Pa
• Quantum mechanically, when L is non-zero, the
  symmetry group becomes quantum deformed to
  SOq(1,4) Starodubtsev
• (i when lorentzian)
• DSR arises as the small L limit of this quantum
  deformed DeSitter symmetry
• (Amelino-Camelia, Starodubtsev, ls,
  hep-th/0306134)

70
Assume we start with symmetry group
SOq(1,4) When we take the contraction, L is
now in two places, in the definition of Pa and
in the quantum deformation parameter. We also
have to renormalize the matter energy-momentum

• When we take the limit now we get
• rgt1 singular
• rlt1 Poincare
• r1 k Poincare
• This argument is confirmed by
• The example of 21
• calculation of excitations of the Kodama state

71
Cosmology and disordered locality F.
Markopoulou, C. Prescod Weinstein, LS
72
The problem of non-locality (F. Markopoulou,
hep-th/0604120 ) Two kinds of locality Microloc
ality connectivity of a single spin net
graph causal structure of a single spin foam
history. Macrolocality nearby in the classical
metric that emerges Issues Semiclassical
states may involve superpositions of large
numbers of graphs. In addition being
semiclassical is a coarse grained, low energy
property. Could there not be mismatches
between micro and macrolocality? What if
these are rare, but characterized by the
cosmological rather than the Planck scale?
73
The inverse problem for discrete spacetimes
Its easy to approximate smooth fields with
discrete structures.
74
The inverse problem for discrete spacetimes
Its easy to approximate smooth fields with
combinatoric structures.
But generic graphs do not embed in manifolds of
low dimension, preserving even approximate
distances.
?
Those that do satisfy constraints unnatural in
the discrete context,
75
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
76
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
Lets add one more link randomly. Does it
conflict with the locality of the
embedding?
77
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
Lets add one more link randomly. Does it
conflict with the locality of the embedding? d N
ways that dont.
78
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
Lets add one more link randomly. Does it
conflict with the locality of the embedding? d N
ways that dont. N2 ways that do. Thus, if the
low energy definition of locality comes from a
coarse graining of a combinatorial graph, it will
be easily violated in fluctuations.
79
If locality is an emergent property of graphs, it
is unstable G a graph with N nodes that has
only links local in an embedding (or whose dual
is a good manifold triangulation) in d dimensions.
Lets add one more link randomly. Does it
conflict with the locality of the embedding? d N
ways that dont. N2 ways that do. Thus, if the
low energy definition of locality comes from a
coarse graining of a combinatorial graph, it will
be easily violated in fluctuations.
Might there then be dislocations or disordering
of locality?
80
Hypothesis the low eneregy limit of QG is
characterized by a small worlds network
Dislocations in locality are scale invariant up
to the Hubble scale Numerical studies of
evolving spin networks by H. Finkel show that
this is a generic outcome of evolution of
random initial graphs by local moves.
81
Suppose the ground state is contaminated by a
small proportion of non-local links (locality
defects)?? What is the effect of a small
proportion of non-local edges in a regular
lattice field theory? If this room had a small
proportion of non-local link, with no two nodes
in the room connected, but instead connecting to
nodes at cosmological distances, could we
tell? Yidun Wan studied the Ising model on a
lattice contaminated by random non-local links.
Rnon-local links/local links 20/8001/40
82
The critical phenomena is the same, but the
Curie temperature increases slightly.
83
(No Transcript)
84

• Cosmology with disordered locality a simple
  model
• Start with standard flat FRW
• Disorder locality by choosing a random
  distribution of
• pairs of points in the spatial manifold that are
  identified.
• Microscopically these are nodes in an underlying
  spin-network
• which are connected by a single link.
• P(x,y,a) is the probability that there is a
  non-local-connection
• between a point in a unit physical volume around
  x and a point
• in a unit physical volume around y, as a
  function of scale a.
• Scale invariant plus random implies P(x,y,a)
  NNL(a) /V2

85

• We assume for the continuum approximation the
  following
• annealing approximation A random
  distribution of identified
• points with probability P(x,y,a) has the same
  effect on the
• energetics in the thermodynamic limit as a small
  non-local
• coupling between pairs of points of strength
• b P(x,y,a)/V2

86
The action is the standard gravity matter
action plus non-local term.

• refers to any degree of freedom with non-local
  couplings.
• Microscopically

antiferromagnetic Ansari-Markopoulou
The nearest neighbor interactions across a
non-local link give a non-local term in the
action
In a continuum approximation this becomes
Px,ty,s is the probability density of
identification between spacetime points. In a
preferred slicing given by a Tconstant
87

• The evolution of NNL(a), the number of non-local
  connections.
• There are microscopic processes by which
  non-local links split into
• two and processes in which pairs annihilate.
• We assume these come to equilibrium. This gives
  us the dependence
• of NNL with scale factor a.

88
Exchange moves can increase the non-local edges.
Perform a 2 to 2 move
1
1/2
1/2
1/2
1/2
1/2
1/2
89
Exchange moves can increase the non-local edges.
Perform a 2 to 2 move
1
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1
1/2
1/2
1/2
90
Exchange moves can increase the non-local edges.
Perform a 2 to 2 move
1
1/2
1/2
1/2
1/2
1/2
1/2
The two left and two right edges can now
evolve away from each other, leading to two
non-local edges.
1/2
1/2
1/2
1
1/2
1/2
1/2
91
Exchange moves can increase the non-local edges.
Perform a 2 to 2 move
1
1/2
1/2
1/2
1/2
1/2
1/2
The probability for this to happen on each local
move is a N/V
1/2
1/2
1/2
1
1/2
1/2
1/2
92
Exchange moves that decrease the non-local edges.
This requires the inverse move on two non-local
edges both of whose ends are coincident
1
1/2
1/2
1/2
1/2
The probability is the probability that there
are are two non-local edges that coincide on
each end times the probability that the move acts
on one of them b N2/V3
1/2
1/2
1/2
1/2
1/2
1
1/2
1/2
1/2
93
dN/dt a N/V - b N2/V3
So there is a stable equilibrium when
N (a/b) V2
This in turn implies the probabilities are
time independent, P(x,y,a) N(a) /V2 (a/b)
N0/V0
constant and metric independent.
94
Putting this back into our continuum
approximation, we find a correction from the
non-local interactions to the Tab
Since P is metric independent we have
So a coarse grained field at each point has on
average a tiny interaction with its averaged
value
If we expand in modes
We find a quintessence model
95
Estimating the number of non-local
connections If s is a gravitational degree of
freedom dimensionless, hence order unity
If s is a matter degree of freedom, there is an
extra G
From naturality
96
Spectrum of fluctuations from disordered
locality The non-local links give a non-local
contribution to the two point function for a
thermal bath of radiation
Sum i is over non-local links
Prob. to jump across a NL link
Fluctuations in thermal spectrum
Fourier transform gives the power spectrum
This is scale invariant because the distribution
of non-local links is scale invariant.
97
LQG is a precise framework for quantum
gravity. LQG is a precisely defined,
mathematically consistent framework for quantum
theories of gravity. It provides both a
hamiltonian and a path integral framework. It
incorporates the basic principles of quantum
mechanics and GR. It is based on an
understanding that gravity is closely related to
gauge theories and topological field theory.
It is finite because it predicts that space
and spacetime are discrete. It is already
unified and a large class of models have the
standard model fermion spectrum. Applications
to black holes and cosmology are in progress.
Results so far agree with and extend
semiclassical expectations. It has the
possibility to make genuine predictions for real
experiments that probe Planck scale physics. .
98
For the last two years, one reads papers in
Nature in which experimental results are used to
rule out predictions which follow from ansatzs
for the ground state of quantum gravity. Hence
quantum gravity has become experimental science.
99

• What about strings?
• Could LQG provide the background independent
  framework
• for string/M theory?
• Quantize 11d supergravity (Ling, ls...)
• Quantize 7d Topological M theory hep-th/0503140
• Chern-Simons matrix models
• All seem to work, much more to do....for string
  theorists
• who want to do physics rather than gardening.
• complex structure, symplectic structure i h
• So Calabi-Yau only meaningful in classical
  approximation.