Loop Quantum Gravity and Recent Progress

Yi Ling ( 凌 意）ITP, Chinese Academy of SciencesDec.27, 2003

Loop Quantum Gravity and Recent Progress

Brief history of loop quantum gravity Ashtekar-Sen variablesSpin networks

Applications and recent progress Isolated horizon and entropy of black holesLoop quantum cosmologyPositive cosmological constant and Chern-Simons st

ates

Einstein’s Dream

Quantum mechanics +special relativity Quantum field theory Electromagnetic interaction (QED) Weak interaction Strong interaction (QCD)

Renormalizable Quantum field theory +general relativity Quantum gravity Gravitational interaction Non-renormalizable

(3) (2) (1)SU SU U

Advices From Dirac

How then do they manage with these incorrect equation? These equations lead to infinities when one tries to solve them; These infinities ought not to be there. They remove them artificially…

------ The inadequacies of quantum filed theory.

General Relativity +Quantum Mechanics

MS gR matter

12

R Rg kT

[ ]ijg H

12

R Rg k T

General Relativity +Quantum Mechanics 3+1 decomposition

: ijh g n n h

2 2 2 ( )

( )

i iij

j j

ds N dt h dx N dt

dx N dt

0 it Nn N Nn N

t

t

iN

0n

General Relativity +Quantum Mechanics

Wheeler-Dewitt equation

1 1 ( ) [ ] 02

ij kl ik jl

ik jl

h h h h hR h hh hh

( , ) ( , )ijij ij

ij

h hh

3 ...ijij WDS dt dx h NC

ˆ0 0WD WDC C

Ashtekar-Sen Variables Ashtekar-Sen variables

i ja ijaE E hh( / 2)a a a

i i iA K

( , ) ( , )ij a iij i ah A E

1 ;a aj k li ij ij i j k lK K E K h h n

h

[ ] 0ai je

a: SU(2) index

Constraints 3+1 decomposition

3 ( )i a a ia i t a iS dt dx E A A N N

G H H

:aG 0ii aD E

:iH 0j aa ijE F

:H 0abc i ja b ijcE E F

Solutions to These Equations Three steps

SU(2) Diff(M) HH 0H DiffH physH

[ ]A [ ]h A0[ ]A ?

Loop States Holonomy

Multi-loop states

[ , ] : exp ( ( ))

exp (2)

i ai aU A ds A s

A SU

P

P

[ ] : [ , ]A TrU A

[ ] : [ , ]ii

A TrU A

Spin Networks Spin networks

m nΓ, j ,v

j1j2 j3v1

Γ, ,n l m

Discreteness of Quantum Geometry

Microscopic version of space

Discreteness of Quantum Geometry

Area spectrum

2ˆ( ) , 8 ( 1) ,pA j l j j j S

( ) 2 1Dim j j

j

2 2

2

( ) det( )

ˆ ˆ ( ) ( )i aja i j

A d h

d E E n n

S

S

S

S

, j

Dynamics Causal spin networks and spin foams

t1

t2

t3

Recent Progress in Loop Quantum Gravity

Isolated horizons and dynamical black holes

Quantum gravity with positive cosmological constant

The absence of singularity in loop quantum cosmology

Extending it to supergravity and M theory

Isolated Horizons and Statistical Entropy of Black Holes

Bekenstein-Hawking entropy

4ASG

ln(# )S of microstates

ˆA A

A Quick Review on Isolated Horizons

Motivationsa. Origin of black hole entropy b. Physical laws of general black holes

The zeroth and first law: Stationary black holes

c. (Quasi)local quantities on the horizon

The mass, angular momentum and charges: Global concepts

A Quick Review on Isolated Horizons

Weakly isolated horizons1. SO(3) (non-rotating) 2. Axi-symmetry(rotating)3. Distortion

Dynamical horizons

1

2

0i

Boundary Theory on Isolated Horizons

Self-dual action

[ , ] ( ) ( )8 M M

iS e A Tr B F Tr B AG

'':AB AA A

AB e e

:F dA A A

''

AAab a bAAg e e

( / 2)AB AB ABa a aA K

Consequences of boundary conditions on horizon

Boundary Theory on Isolated Horizons

2AB ABab ab

H

F BA

2HATrB A TrF A

( ) ( )[ , ] 28 ( )

4 3

MTr B F Tr B A

iS e A AG Tr A dA A A A

Boundary Theory on Isolated Horizons

Boundary Theory on Isolated Horizons

Quantization

B S A A AB S H H H

2exp( ) exp( )B S B SH

iF i BA

Boundary Theory on Isolated Horizons

From quantum general relativity

From quantum Chern-Simons theory

2 2

1

8 ( , )N

ab B p i i ab Bi

B l j x p

1

2exp( ) exp( )N

iS S

i

i niFk

2i in j

Boundary Theory on Isolated Horizons

Hilbert space

bh physH H

i iB S

i

Boundary Theory on Isolated Horizons

Topological quantum field theory

( , , )S y j ( , , )S y j

H

j (2)qSU2

2i

kq e

( , , )dim (2 1)S y j j

H

The most probable distribution

The area of discrete horizon

min min min, ,...,j j j

2min min

( )

8 ( 1)p

A j

N l j j

Discrete Horizons From Quantum Geometry

Counting the Number of Microstates of Quantum Gravity The entropy of discrete horizon

minln(2 1)S N j

min2

min min

ln(2 1)8 ( 1)p

j Al j j

min1 ln 2;2 3

j

4AS

minln 31;

2 2j

4AS

Fixing the Immirzi Parameter Quasinormal modes of black holes

Bohr’s correspondence principle

ln 38QNM M

32QNMAMM

min( )A A j

Fixing the Immirzi Parameter

Quantum GR

Loop quantization of N=1 supergravity

minln 31;

2 2j

4AS

(2) (3)SU SO

min1 ln 3;2 2

j

Some Progress in Loop Quantum Cosmology

The absence of singularity in loop quantum cosmology Closed universe with k=1, Scale factor

Originated from a big-bang

3S ( )a t

32

1( )

Ra t

( ) 0a t 3classically R

Some Progress in Loop Quantum Cosmology

Quantum Mechanically

We would expect

Thus, is a classical relation.

min max1

planckpl

L l El

3max 2

1 1. .pl

R const constl G

3 2 1R a

0

3 3

0256 1!81

a

R RG

Some Progress in Loop Quantum CosmologyQuantum geometry

2 3 2 3 ˆ ˆ ˆˆ, , ,a R a R V M

ˆ ˆ[ , ] 0V M

-2-10123450 1 2 3 4 5

n

1n ny V M

0

30

0256 1!81

V

M RG

Some Progress in Loop Quantum Cosmology

Positive cosmological and Chern-Simons states

03

abc i j abc i j ka b ijc ijk a b cE E F E E E

3k

ijc ijk cF E

Some Progress in Loop Quantum Cosmology

Kodama state

– Problem of time – Energy-momentum relations(Lorentz Violatio

n)

( )3

Kijc K ijk c

k

AFA

3 22 3( )

A dA A A A

K A Ne

Extend It to supergravity and M Theory

M theory String theory

D=10 SupergravityD=11 Supergravity

D=4,N=8 Supergravity

Loop Quantum GR

D=4,N=1,2 Supergravity

Thank You