View
76
Download
4
Category
Preview:
DESCRIPTION
Loop Quantum Gravity and Recent Progress. Yi Ling ( 凌 意) ITP, Chinese Academy of Sciences Dec.27, 2003. Loop Quantum Gravity and Recent Progress. Brief history of loop quantum gravity Ashtekar-Sen variables Spin networks Applications and recent progress - PowerPoint PPT Presentation
Citation preview
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
Recommended