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### Particle physics fromquantum gravityLee Smolin

I Basic assumptions and results

II Quantum information and quantum gravity

F.Markopoulou hep-th/0604120

D. Krebs and F. Markopoulou gr-qc/0510052

III Unification of quantum geometry with matter

S. Bilson-Thompson, F. Markopoulou, LS hep-th/0603022

IV Dark energy from quantum gravity

F. Markopoulou, C. Prescod-Weinstein, LS

V. Doubly special relativity as the low energy limit

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

C Beetle

B Dettrich

P. Sungh

H, Sahlmann,

Fleishhack

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

M Varadarajan

S Speciale

A. Toporeski

G. Datta

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

Loop quantum gravity community- 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

- 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 }> 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

These theories realize 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 T[g,A] = Tr exp ∫gAg
- 3) Diffeomorphism invariance and background independence

The gravitational field can be described as a gauge theory:

- Spacetime connection = Gauge field.
- Spacetime metric = Electric field
- 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.
- T[g,A] = Trexp ∫gA E[S]= ∫SE

[

] = h G Int

,

Consider a background independent gauge theory, compact Lie group G on a spatial manifold S of dim >1. No metric!! (G=SU(2) for 3+1 gravity)

There is a unique cyclic representation of the loop/surface algebra in which the Hilbert space carries a unitary rep of the diffeomorphism group.

Lewandowski, Okolo, Sahlmann, Thiemann+ Fleishhack(LOST theorem)

This means that there is a unique diffeomorphism invariant quantum quantum theory for each G.

It is the one just defined! (up to fine print)

Ashtekar: GR is a diffeomorphism invariant gauge theory!!

Hence this class of theories includes loop quantum gravity and spin foam models

Role of the cosmological constant:

Requires quantum deformation of SU(2)

q=e2pi/k+2 k= 6p/GL

To represent this the spin network graphs must be framed:

Some old and new results in LQG:

- There exist semiclassical states.
- Excitations of semiclassical states include long wavelength gravitons
- Sums over labels in 3+1d and 4d spin foam models are convergent.
- analytically and numerically (Crane, Perez,Rovelli,Baez,Christensen..)
- Graviton propagator, and Newton’s law derived in 4d (Rovelli et al 05)
- 2+1 gravity with matter solved, gives an effective field theory on
- k-Poincare non-commutative manifold, implying DSR. (Freidel-Livine)
- Reduced models for cosmology and black hole interiors solved:
- Spacelike singularities eliminated and replaced by bounce. (Bojowald …)

Predictions for corrections to CMB (Hoffman-Winkler)

Black hole entropy understood including corrections and radiation

- (Krasnov, Baez, Ashtekar, Corichi, Dreyer, Ansari,….)

If LQG really unifies gravity and QM, shouldn’t it

automatically tell us about unifying the rest of physics?

S. Bilson-Thompson, F. Markopoulou, ls, hep-th/0603022

Some questions for LQG theories:

- The geometric observables such as area and volume measure the
- combinatorics of the graph. But they don’t care how the
- edges are braided or knotted. What physical information
- does the knotting and braiding correspond to?
- What is an observable in quantum gravity?
- A conserved quantity.
- But what are the conserved quantities?
- How do we describe the low energy limit of the theory?
- How do we define local 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?

Some answers: (Markopoulou et al )

- 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.

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.

A large class of causal spinnet theories have noiseless subsystems that can be interpreted as local excitations.

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.

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

group…or none.

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?

Invariance under the exchange moves:

The topology of the embedding remain

unchanged

All ribbon invariants are constants of the motion.

Invariance under the exchange moves:

The topology of the embedding:

All ribbon invariants

For example: the link of the ribbon:

Invariance under the exchange moves:

The topology of the embedding:

All ribbon invariants

For example: the link of the ribbon:

Invariance under the exchange moves:

The topology of the embedding:

All ribbon invariants

For example: the link of the ribbon:

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:

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

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:

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.

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.

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:

Braids on three ribbons and preons (Bilson-Thompson)

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.

In the future we should find a dynamical justification, for the

moment 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.

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

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.

Straightforward to prove them distinct.

eL+

1+ twist

dLr

dLb

dLg

2+ twists

uLr

uLb

uLg

The right handed states come from parity inversion:

No twists:

3 - twists

nR

eR-

1- twist

dRr

dRb

dRg

2- twists

uRr

uRb

uRg

So the emergent symmetries include:

- SU(2)L + SU(2)R + U(1) acts on twistings, keeps place fixed
- SU(3) exchanges place in braid, in cases where they are distinct.
- P: parity, left to right exchange
- C: invert braid vertically (top vrs bottom)
- Only 2 neutrino states P: nL = C: nL = nR
- 4 of each charged state eL,R+ eL,R-
- Fractionally charged states also have color.

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?

- 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
- But work is underway addressed to these and other questions.

A large class of causal spin network theories have

coherent (noisefree) subsystems which are emergent particle like

excitations.

SO THESE ARE ALREADY UNIFIED THEORIES!

In a large subclass the simplest excitations correspond to the

standard model fermions.

A large class of causal spin network theories have

coherent (noisefree) subsystems which are emergent particle like

excitations.

SO THESE ARE ALREADY UNIFIED THEORIES!

In a large subclass the simplest excitations correspond to the

standard model fermions.

“The most important lesson I’ve learned in my career is to

trust coincidences.”

-John Schwarz.

So the standard model fermions naturally occur as coherent

- conserved excitations in a large class of quantum gravity models.
- Many open questions….
- Mass matrix: must be a graph invariant. Kauffman bracket??
- Gauge invariance (symmetries are local)??
- Cabibbo and other mixing?
- CP violation? Natural source is Imirzi parameter, Soo, Alexander
- Emergent effective dynamics??
- But the minimal conclusion is that LQG is an already
- unified theory.

Cosmology and disordered locality

F. Markopoulou, C. Prescod Weinstein, LS

The problem of non-locality (F. Markopoulou, hep-th/0604120 )

Two kinds of locality:

Microlocality: 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?

The inverse problem for discrete spacetimes:

Its easy to approximate smooth fields with discrete structures.

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,

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.

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?

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 don’t.

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 don’t.

N2ways 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.

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 don’t.

N2ways 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?

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.

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.

R=non-local links/local links

= 20/800=1/40

The critical phenomena is the same, but the Curie

temperature increases slightly.

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
- NNL(a)= the number of non-local links in a co-moving volume V=a3

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

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

P[x,t;y,s] is the probability density of identification between

spacetime points. In a preferred slicing given by a T=constant

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.

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

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

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

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

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.

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”

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:

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.

The key observational question is:

What is the symmetry of the ground state?

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 L=0.
- Hence, the symmetry of the quantum ground state is
- a dynamical question.

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 L=0.
- Hence, the symmetry of the quantum ground state is
- a dynamical question.
- Three possibilities for lorentz invariance:
- 1 broken…there is a preferred frame
- 2 nothing new: realized as in ordinary QFT
- 3 Doubly special relativity (DSR)

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 E<<Ep.

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 E<<Ep.
- 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)
- DSR is realized precisely in 2+1 gravity with matter hep-th/0307085

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.

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

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

- see anomalous events:
- AGASA, Sugar
- Some don’t: 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.

Tests of energy dependence of the speed of light:

- Energy dependent speed of light. v=c(1+ a lp E + b lp2 E2 + …)
- Accumulates for long distances
- Observable in Gamma Ray bursts.
- present limits have a < 1000
- next satellite, GLAST will put limits a < 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, 415–417 (2003).
- Mitrofanov, Nature, VOL 426 13 Nov 2003
- These imply no helicity dependent speed of light at O(lP)

Does loop quantum gravity predict DSR?

- does so cleanly for 2+1 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

Classically, WhenL=1/L2 >0, the ground state is deSitter spacetime.

Its symmetry group is SO(1,4).

In the limit L-->0 the symmetry group contracts to the Poincare

group, with a scaling M0a = 1/L Pa

- Classically, WhenL=1/L2 >0, the ground state is deSitter spacetime.
- Its symmetry group is SO(1,4).
- In the limit L-->0 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)

- Classically, WhenL=1/L2 >0, the ground state is deSitter spacetime.
- Its symmetry group is SO(1,4).
- In the limit L-->0 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)

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
- r>1 singular
- r<1 Poincare
- r=1 k Poincare
- This argument is confirmed by
- The example of 2+1
- calculation of excitations of the Kodama state

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.

.

For the last two years, one reads papers in Nature

in which experimental results are used to rule out

predictions which follow from ansatz’s for the

ground state of quantum gravity.

Hence quantum gravity has become experimental

science.

- 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.

What about string theory?hep-th/0303185

- Assumptions:
- Duality
- Oscillations of loops (strings) give all fields (not just gauge fields)
- Background dependent
- Perturbation theory should be invariant under classical symmetries of background

All together they imply “the package”: SUSY+ 6 or 7 higher dimensions.

This leads to many difficulties:

- Many many versions (10100 ??) hence, doubts about falsifiability
- Hard to prove finiteness past genus 2 (boundaries of supermoduli space are hard)
- Stability (of compactified extra dimensions, of time dependent backgrounds....)

What is the remedy?

1 is compelling, 2 is appealing. Perhaps drop 3 and 4.

Hence, search for a background independent approach.

- Strings do arise from low energy limit of spin foams hep-th/9801022
- Cubic (CS) matrix models?? hep-th/0104050, 0006137, 0002009
- loop quantize 11d supergravity?? hep-th/0003285, 9703174

other???

We need an evolution rule for cancelling twist:

Canceling twists on a node:

Left and Right twists on a single node cancel each other:

So charge can annihilate anticharge.

We need one final rule: No states with Left and Right

twists in the same braid. Possibly because this costs a

lot of mass.

So all light states have some number of Left or Right twists,

but not both.

One strategy to choose the spin foam amplitudes:

- Recall the GR or SUGRA is classically a constrained
- topological field theory.
- Construct an exact finite path integral representation
- for the topological theory in terms of a state sum model.
- Impose the constraints that lead to GR or SUGRA
- in the measure of the state sum model.
- Leads to the Barrett Crane model G=SO(4), (jL,jR) constraint is jL=jR
- Sums over labels for a given history have been proved uv finite.
- This has also been confirmed in numerical studies.
- The strategy has been carried out in all dimensions. (Krasnov,Freidel, Putzio)
- works in all dimensions

The sum over spin foams is like

- a Feynman diagram computation.
- Rather than initial and final
- momentum eigenstates we have
- initial and final spin networks.
- Rather than diagrams we have
- spin foam histories.
- Rather than summing over
- momenta on edges
- we sum over spins on faces.
- Everything is one dimension up

For each spin foam model there

is a matrix model such that the

spin foams are the Feynman

diagrams of that model.

Some results on spin foam models:

- Sums over spins shown numerically and analytically finite for 3+1 and 4d Barrett-Crane model (Crane, Perez,Rovelli,Baez,Christensen..)
- Graviton propagator, and Newton’s law derived in 4d (Rovelli et al 05)
- Effective field theory for GR+matter solved in 2+1, yields DSR
- Freidel-Livine
- Physical inner product expressed as spin foam representation for projection operator (Rovelli,Reisenberger)
- For large labels Regge Calculus (hence the Einstein-action) is recovered (Oriti,Williams)
- Renormalization group for spin foams formulated in terms of Hopf algebra (Markopoulou...)
- Spin foam supergravity (Livine and Oeckl)
- Yang-Mills included (Barrett, Oriti, Speziale)
- Reviews: Perez gr-qc/0301113, and Oriti, gr-qc/0106091.

Gravitational theories are constrained or perturbed

topological field theories

We are familiar with gauge theories in which not

all components of a field are physically meaningful.

The rest are called pure gauge. dAa ~ af

A topological field theory is a theory whose local fields

are entirely pure gauge.

All the physical degrees of freedom and observables live

on the boundaries of spacetime.

Their quantum observables define topological invariants of space.

4) Gravitational theories are constrained topological field theories

The dynamics is given by

S= Topological theory + quadratic constraints

All the derivatives are in the topological term

So commutators, path integral measure and boundary terms

are those of the topological theory.

On top of these we just impose quadratic operator equations.

This is true of GR and supergravity in all dimensions, including 11.

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