Download Loop Quantum Gravity in a Nutshell

Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Loop Quantum Gravity in a Nutshell
Dah-Wei Chiou
Center for Advanced Study in Theoretical Sciences
National Taiwan University
Department of Physics
National Sun Yat-sen University
May 24, 2012
Summary
Motivations
Canonical GR
LQG
Spin Foam
Outline
Motivations
Why quantum gravity?
Difficulties of quantum gravity
Canonical general relativity
Connection dynamics
Canonical/Hamiltonian formulation
Loop quantum gravity (LQG)
Loop algebra
Quantum kinematics
Quantum dynamics
A glimpse of spin foam theories
A glimpse of loop quantum cosmology (LQC)
A glimpse of loop quantum black holes
Summary
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Why quantum gravity?
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Why quantum gravity?
• Classical-quantum inconsistency. The fundamental principles collide in
the classical Einstein equations
Rµν −
1
R gµν = 8πG Tµν (g )
2
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Why quantum gravity?
• Classical-quantum inconsistency. The fundamental principles collide in
the classical Einstein equations
Rµν −
1
R gµν = 8πG Tµν (g )
2
• The “only way” to make sense out of the above equations while keeping
the classical and quantum nature of geometry and matter respectively is:
Rµν −
1
R gµν = 8πG hT̂µν (g0 )i
2
g0 : background metric
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Why quantum gravity?
• Classical-quantum inconsistency. The fundamental principles collide in
the classical Einstein equations
Rµν −
1
R gµν = 8πG Tµν (g )
2
• The “only way” to make sense out of the above equations while keeping
the classical and quantum nature of geometry and matter respectively is:
Rµν −
1
R gµν = 8πG hT̂µν (g0 )i
2
g0 : background metric
• However, iteration results in “run-away” solutions (related to
nonrenormalizabality).
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Why quantum gravity?
• Classical-quantum inconsistency. The fundamental principles collide in
the classical Einstein equations
Rµν −
1
R gµν = 8πG Tµν (g )
2
• The “only way” to make sense out of the above equations while keeping
the classical and quantum nature of geometry and matter respectively is:
Rµν −
1
R gµν = 8πG hT̂µν (g0 )i
2
g0 : background metric
• However, iteration results in “run-away” solutions (related to
nonrenormalizabality).
• Have to quantize both geometry and matter simultaneously!
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Why quantum gravity?
• Classical-quantum inconsistency. The fundamental principles collide in
the classical Einstein equations
Rµν −
1
R gµν = 8πG Tµν (g )
2
• The “only way” to make sense out of the above equations while keeping
the classical and quantum nature of geometry and matter respectively is:
Rµν −
1
R gµν = 8πG hT̂µν (g0 )i
2
g0 : background metric
• However, iteration results in “run-away” solutions (related to
nonrenormalizabality).
• Have to quantize both geometry and matter simultaneously!
• Comment on holographic principle. . .
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
• QFT:
All dynamical variables are quantized except spacetime.
Matters move on the background spacetime.
• GR:
Spacetime is dynamical.
Matters and spacetime are no different.
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
• QFT:
All dynamical variables are quantized except spacetime.
Matters move on the background spacetime.
• GR:
Spacetime is dynamical.
Matters and spacetime are no different.
• ⇒ QG? How to reconcile the paradox?
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
• QFT:
All dynamical variables are quantized except spacetime.
Matters move on the background spacetime.
• GR:
Spacetime is dynamical.
Matters and spacetime are no different.
• ⇒ QG? How to reconcile the paradox?
• string theory (strings instead of point fields. . . )
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
• QFT:
All dynamical variables are quantized except spacetime.
Matters move on the background spacetime.
• GR:
Spacetime is dynamical.
Matters and spacetime are no different.
• ⇒ QG? How to reconcile the paradox?
• string theory (strings instead of point fields. . . )
• supergravity (inclusion of supersymmetry. . . )
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
• QFT:
All dynamical variables are quantized except spacetime.
Matters move on the background spacetime.
• GR:
Spacetime is dynamical.
Matters and spacetime are no different.
• ⇒ QG? How to reconcile the paradox?
• string theory (strings instead of point fields. . . )
• supergravity (inclusion of supersymmetry. . . )
• holographic principle
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
• QFT:
All dynamical variables are quantized except spacetime.
Matters move on the background spacetime.
• GR:
Spacetime is dynamical.
Matters and spacetime are no different.
• ⇒ QG? How to reconcile the paradox?
•
•
•
•
string theory (strings instead of point fields. . . )
supergravity (inclusion of supersymmetry. . . )
holographic principle
gravity as entropic force
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
• QFT:
All dynamical variables are quantized except spacetime.
Matters move on the background spacetime.
• GR:
Spacetime is dynamical.
Matters and spacetime are no different.
• ⇒ QG? How to reconcile the paradox?
•
•
•
•
•
string theory (strings instead of point fields. . . )
supergravity (inclusion of supersymmetry. . . )
holographic principle
gravity as entropic force
and much more. . .
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Difficulties of quantum gravity
• QFT:
All dynamical variables are quantized except spacetime.
Matters move on the background spacetime.
• GR:
Spacetime is dynamical.
Matters and spacetime are no different.
• ⇒ QG? How to reconcile the paradox?
•
•
•
•
•
•
string theory (strings instead of point fields. . . )
supergravity (inclusion of supersymmetry. . . )
holographic principle
gravity as entropic force
and much more. . .
LQG: quantize matters and spacetime on the equal footing!
(background-independent, non-perturbative)
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Metaphor: animals live on an island → animals live on the back of a whale
Summary
Motivations
Canonical GR
LQG
Spin Foam
Outline
Motivations
Why quantum gravity?
Difficulties of quantum gravity
Canonical general relativity
Connection dynamics
Canonical/Hamiltonian formulation
Loop quantum gravity (LQG)
Loop algebra
Quantum kinematics
Quantum dynamics
A glimpse of spin foam theories
A glimpse of loop quantum cosmology (LQC)
A glimpse of loop quantum black holes
Summary
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Connection dynamics
• geometrodynamics vs connection dynamics
• Einstein-Hilbert action (based on Riemannian geometry):
S[g ] =
1
16πG
Z
d 4x
p
|g | R
• Palatini action (based on Riemann-Cartan geometry):
S[e, ω] =
1
32πG
Z
IJKL e I ∧ e J ∧ ΩKL
tetrad (vierbein; “soldering form”): eµ I ,
so(1, 3) connection: ωµ I J
curvature: Ω := dω + ω ∧ ω
gµν = ηIJ eµ I eν J
• Equation of motion (in fact, as long as “torsion-free”) gives
de + ω ∧ e = 0
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
∂µ
∂ν
Riemannian geometry
metric: gµν
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
∂µ
∂ν
Riemannian geometry
metric: gµν
I,J
Riemann-Cartan geometry
so(1, 3) connection: ωµ I J
soldering form: eµ I
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Palatini action with Holst’s modification (based on Riemann-Cartan
geometry):
S[e, ω] =
1
32πG
Z
IJKL e I ∧ e J ∧ ΩKL −
1
16πγ
Z
γ: Barbero-Immirzi parameter
e I ∧ e J ∧ ΩIJ
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Palatini action with Holst’s modification (based on Riemann-Cartan
geometry):
S[e, ω] =
1
32πG
Z
IJKL e I ∧ e J ∧ ΩKL −
1
16πγ
Z
e I ∧ e J ∧ ΩIJ
γ: Barbero-Immirzi parameter
• cf. (non-abelian) gauge theory (Yang-Mills action with θ-term):
S=
1
2
Z
a
Faµν Fµν
+
θα
8π
Z
a
µνρσ Fµνa Fρσ
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Palatini action with Holst’s modification (based on Riemann-Cartan
geometry):
S[e, ω] =
1
32πG
Z
IJKL e I ∧ e J ∧ ΩKL −
1
16πγ
Z
e I ∧ e J ∧ ΩIJ
γ: Barbero-Immirzi parameter
• cf. (non-abelian) gauge theory (Yang-Mills action with θ-term):
S=
1
2
Z
a
Faµν Fµν
+
θα
8π
Z
a
µνρσ Fµνa Fρσ
• eµ I : tetrad (vierbein) as well as “soldering form”. (e I ∧ e J =: ΣIJ )
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Palatini action with Holst’s modification (based on Riemann-Cartan
geometry):
S[e, ω] =
1
32πG
Z
IJKL e I ∧ e J ∧ ΩKL −
1
16πγ
Z
e I ∧ e J ∧ ΩIJ
γ: Barbero-Immirzi parameter
• cf. (non-abelian) gauge theory (Yang-Mills action with θ-term):
S=
1
2
Z
a
Faµν Fµν
+
θα
8π
Z
a
µνρσ Fµνa Fρσ
• eµ I : tetrad (vierbein) as well as “soldering form”. (e I ∧ e J =: ΣIJ )
• comments on Poincaré gauge theories, teleparallel theories, metric-affine
connection theories, non-metric theories. . .
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Canonical/Hamiltonian general relativity
• ADM (3+1) foliation
N: lapse;
~ shift
N:
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• 3 + 1 splitting:
SO(1, 3)
eµ I (tetrad)
ωµ I J
I
de + ω
I
J
J
∧e =0
indices: I = (0, 1, 2, 3), µ
→
→
→
→
→
SO(3) ∼ SU(2)
ea i (triad), N, N a
Γia , Kai
de i + i jk Γi ∧ e k = 0
i = (1, 2, 3), a
Γia : spin connection associated to the triad,
(defined via de i + i jk Γi ∧ e k = 0)
Kai : extrinsic curvature of the constant time three surface
• Ashtekar variables:
(densitized) inverse triad: e a i (~x )
−→
densitize
Ẽia (~x ),
g g ab = Ẽia Ẽjb δ ij
Ashtekar connection:
Aia (~x ) = Γia (~x ) + γKai (~x )
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Starting with Palatini+Holst action, ADM (3+1) foliation and Legendre
transform give
• canonical pair
b
{Aa i (~x ), Ẽ j (~x 0 )} = 8πG γδji δab δ 3 (~x − ~x 0 )
b
Ẽ j (~x 0 ): “electric field”
Aa i (~x ): “magnetic potential”
• with
Z
S=
Z
dt
d 3 x Ẽia Lt Aia − h(Ẽia , Aia , N, N a , ω i · t)
h = (ω i · t)Gi + N a Ca + NC
Gi = 0
,C = 0
Ca = 0
i
Lagrange multipliers: ω · t, Na , N
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
“completely constrained” system:
Z
Z
S = dt d 3 x Ẽia Lt Aia − h(Ẽia , Aia , N, N a , ω i · t)
h = (ω i · t)Gi + N a Ca + NC ,
Gi = 0
Ca = 0
,C = 0
• Gauss (SU(2)) constraint:
Gi = Da Ẽia = ∂a Ẽia + ij k Aja Ẽka
• vector (Diff ) constraint:
i
Ca = Ẽib Fab
−
σ − γ2 i
Ka Gi
σγ
• scalar (Hamiltonian) constraint (“super-Hamiltonian”):
i
h
8πG γ 2
k
+ (σ − γ 2 )2K[ai Kb]j +8πG (γ 2 −σ)∂a
C = p Ẽia Ẽjb ij k Fab
2 |q|
K[ai
p
|q|
!
Gi
Motivations
Canonical GR
LQG
Spin Foam
Outline
Motivations
Why quantum gravity?
Difficulties of quantum gravity
Canonical general relativity
Connection dynamics
Canonical/Hamiltonian formulation
Loop quantum gravity (LQG)
Loop algebra
Quantum kinematics
Quantum dynamics
A glimpse of spin foam theories
A glimpse of loop quantum cosmology (LQC)
A glimpse of loop quantum black holes
Summary
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Loop algebra
One more step b/f quantization:
xf
• connection −→ holonomy (Wilson loop):
Aia (~x ) −→ hγ := Pe
R
γ
A
γ
,
xi
gauge transformation:
F (x)
A(x)
hγ
→
→
→
F 0 (x) = U(x)F (x)U † (x)
A0 (x) = U(x)A(x)U † (x) + U(x)dU † (x)
hγ0 = U(xf )hγ U † (xi )
• smear Ẽia :
Ẽia (~x )
−→ E [S, f ] :=
Z
dSa Ẽia fi
S
• loop algebra:
{hγ , hγ0 } = {E [S, f ], E [S 0 , f 0 ]} = 0
{hγ , E [S, f ]} = · · ·
Summary
Motivations
Canonical GR
Why (Wilson) loops?
LQG
Spin Foam
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Why (Wilson) loops?
• inspired by lattice gauge theory (nonperturbative approach)
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Why (Wilson) loops?
• inspired by lattice gauge theory (nonperturbative approach)
• loops transform “nicely” under both SU(2) and Diff .
SU(2) :
F (x)
A(x)
hγ
Diff :
···
→
→
→
F 0 (x) = U(x)F (x)U † (x)
A0 (x) = U(x)A(x)U † (x) + U(x)dU † (x)
hγ0 = U(xf )hγ U † (xi )
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Why (Wilson) loops?
• inspired by lattice gauge theory (nonperturbative approach)
• loops transform “nicely” under both SU(2) and Diff .
SU(2) :
F (x)
A(x)
hγ
Diff :
···
→
→
→
F 0 (x) = U(x)F (x)U † (x)
A0 (x) = U(x)A(x)U † (x) + U(x)dU † (x)
hγ0 = U(xf )hγ U † (xi )
• There exists a unique Diff -invariant measure (Ashtekar-Lewandowski
measure).
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Why (Wilson) loops?
• inspired by lattice gauge theory (nonperturbative approach)
• loops transform “nicely” under both SU(2) and Diff .
SU(2) :
F (x)
A(x)
hγ
Diff :
···
→
→
→
F 0 (x) = U(x)F (x)U † (x)
A0 (x) = U(x)A(x)U † (x) + U(x)dU † (x)
hγ0 = U(xf )hγ U † (xi )
• There exists a unique Diff -invariant measure (Ashtekar-Lewandowski
measure).
• Aharonov-Bohm effect. . .
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Why (Wilson) loops?
• inspired by lattice gauge theory (nonperturbative approach)
• loops transform “nicely” under both SU(2) and Diff .
SU(2) :
F (x)
A(x)
hγ
Diff :
···
→
→
→
F 0 (x) = U(x)F (x)U † (x)
A0 (x) = U(x)A(x)U † (x) + U(x)dU † (x)
hγ0 = U(xf )hγ U † (xi )
• There exists a unique Diff -invariant measure (Ashtekar-Lewandowski
measure).
• Aharonov-Bohm effect. . .
• more. . .
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Why (Wilson) loops?
• inspired by lattice gauge theory (nonperturbative approach)
• loops transform “nicely” under both SU(2) and Diff .
SU(2) :
F (x)
A(x)
hγ
Diff :
···
→
→
→
F 0 (x) = U(x)F (x)U † (x)
A0 (x) = U(x)A(x)U † (x) + U(x)dU † (x)
hγ0 = U(xf )hγ U † (xi )
• There exists a unique Diff -invariant measure (Ashtekar-Lewandowski
measure).
• Aharonov-Bohm effect. . .
• more. . .
Quantization scheme:
K
−→
SU(2)
K0
−→
Diff ∗
KDiff
−→
H
H
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Kinematic Hilbert space K
K
−→
SU(2)
K0
−→
Diff ∗
KDiff
−→
H
H
• cylindrical functions:
ψΓ,f (A) = f (U1 (A), . . . , Un (A))
instead of arbitrary functional ψ[A]
Γ : graph
• arbitrary Γ’s ⇒ all information captured (trough “projective limit”).
• inner product (Ashtekar-Lewandowski measure):
Z
(ψΓ,f , ψΓ,h ) =
dg1 . . . dgn f (g1 , . . . , gn ) h(g1 , . . . , gn )
SU(2)2n
(ψΓ1 ,f , ψΓ2 ,h ) := (ψΓ,f , ψΓ,h ),
Γ1 , Γ2 ⊂ Γ,
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
(Oriented) graphs: links (+ nodes)
(a)
(c)
(b)
(d)
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• an orthonormal basis (Peter-Weyl theorem):
|Γ, jl , αl , βl i = |Γ, j1 , . . . , jL , αl , . . . , αL , βl , . . . , βL i
hA|Γ, jl , αl , βl i = R (j1 )
α1
β1 (U1 (A)) . . . R
(jL ) αL
j1
α1
j2
β1
β2
α2
α3
β3
j3
βL (UL (A))
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
SU(2)-invariant Hilbert space K0
K
−→
K0
SU(2)
−→
Diff ∗
KDiff
−→
H
H
• spin network (graph Γ + “coloring” jl , in )
S = (Γ, jl , in ),
in : intertwiner (Clebsch-Gordan coefficients)
j1 ⊗ j2 ⊗ · · · = 0 ⊕ 0 ⊕ . . .
|Si
:=
X
αl ,βl
βn +1 ,...,βn
β1 ,...,βn
2
1
v 1
1
α1 ,...,αn i1
αn +1 ,...,αn
1
2
1
vi
βn
,...,βn
N−1
L
n
αn
,...,αn
N−1
L
×vi
j1
|Γ, jl , αl , βl i
j1
α1
j2
β1
···
α2
β2
α3
β3
j2
i2
⇒
i1
j3
j3
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Diff -invariant Hilbert space KDiff
K
−→
SU(2)
K0
−→
Diff ∗
KDiff
−→
H
H
• equivalent relation:
iff
|Si = |Γ, jl , in i ∼ |S 0 i = |Γ0 , jl0 , in0 i
jl0 = jl , in0 = in , and Γ0 = φ(Γ) for some φ ∈ Diff ∗
• An equivalent class K of graphs Γ under diffeomorphisms is called a
“knot”.
• s-knot: K + “coloring”(jl and in ) ∈ KDiff
• Inner product of KDiff can be defined in the natural way:
Ashtekar-Lewandowski measure.
• KDiff is separable!
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Equivalence under φ ∈ Diff ∗ :
−→
φ∈Diff ∗
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
s-knots (spin networks)
coloring/labels:
• link jl : SU(2) half-integers
• node in : intertwiners (Clebsch-Gordan coefficients)
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Discreteness of area and volume
• Both “area” and “volume” operators are well defined in K0 (and thus in
KDiff ).
A(S) =
Z q
na Ẽia nb Ẽ ib d 2 σ
S
∵
δ
δAia (~x )
⇒
Â(S) = lim
Xq
N→∞
E 2 (Sn )|Si = (8πG γ~)2 jn (jn + 1)|Si
Z
hγ = ds γ̇ a (s) δ 3 (γs, ~x ) hγ1 τi hγ2 ,
E 2 (Sn )
n
hγ = Pe
R
Aj τj
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• |Si is an eigenstate of Â(S):
Â(S)|Si = 8πG γ~
X p
jp (jp + 1)|Si
p∈S∩Γ
• spectrum of the area operator (associated to the link l) is discrete:
p
Al = 8πG γc −3 ~ jl (jl + 1)
• similar to “volume” operator (but much more complicated). . .
• Both “area” and “volume” are quantized!!
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• |Si is an eigenstate of Â(S):
Â(S)|Si = 8πG γ~
X p
jp (jp + 1)|Si
p∈S∩Γ
• spectrum of the area operator (associated to the link l) is discrete:
p
Al = 8πG γc −3 ~ jl (jl + 1)
• similar to “volume” operator (but much more complicated). . .
• Both “area” and “volume” are quantized!!
• comments on diffeomorphism invariance and the discreteness. . .
(cf. lattice theory of gravity)
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quanta of space: s-knots
• s-knot: ensemble of “chunks of space”
• node in — quanta of volume
• link jl — quanta of area:
p
Al = 8πG γc −3 ~ jl (jl + 1)
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quanta of space: s-knots
• s-knot: ensemble of “chunks of space”
• node in — quanta of volume
• link jl — quanta of area:
p
Al = 8πG γc −3 ~ jl (jl + 1)
• comments on inclusion of matters. . .
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Quanta of space: s-knots
• s-knot: ensemble of “chunks of space”
• node in — quanta of volume
• link jl — quanta of area:
p
Al = 8πG γc −3 ~ jl (jl + 1)
• comments on inclusion of matters. . .
• comments on background independence (quantum states are excitations
out of nothing, i.e. |∅i). . .
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Semiclassical (low-energy) limit
• However, what is the smooth macroscopic space we are familiar with?
• Can we find a state in KDiff that approximates a given macroscopic
solution?
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Semiclassical (low-energy) limit
• However, what is the smooth macroscopic space we are familiar with?
• Can we find a state in KDiff that approximates a given macroscopic
solution?
• Coherent state?
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Semiclassical (low-energy) limit
• However, what is the smooth macroscopic space we are familiar with?
• Can we find a state in KDiff that approximates a given macroscopic
solution?
• Coherent state?
• Analogy:
a piece of (smooth) cloth −→ interwoven with (irregular) threads
zoom in
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Semiclassical (low-energy) limit
• However, what is the smooth macroscopic space we are familiar with?
• Can we find a state in KDiff that approximates a given macroscopic
solution?
• Coherent state?
• Analogy:
a piece of (smooth) cloth −→ interwoven with (irregular) threads
zoom in
• It is still a big challenge to recover the low-energy physics!
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Semiclassical (low-energy) limit
• However, what is the smooth macroscopic space we are familiar with?
• Can we find a state in KDiff that approximates a given macroscopic
solution?
• Coherent state?
• Analogy:
a piece of (smooth) cloth −→ interwoven with (irregular) threads
zoom in
• It is still a big challenge to recover the low-energy physics!
• top-down approach is difficult!
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Semiclassical (low-energy) limit
• However, what is the smooth macroscopic space we are familiar with?
• Can we find a state in KDiff that approximates a given macroscopic
solution?
• Coherent state?
• Analogy:
a piece of (smooth) cloth −→ interwoven with (irregular) threads
zoom in
• It is still a big challenge to recover the low-energy physics!
• top-down approach is difficult!
• bottom-up approach available:
LQG ←→ midi-superspace ←→ mini-superspace (e.g. LQC)
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Dynamics
K
−→
SU(2)
K0
−→
Diff ∗
KDiff
−→
H
H??
• Hamiltonian operator:
Z
H
=
Ĥ
=
d 3x
···
i
8πG γ 2 N a b h ij k
p
Ẽi Ẽj k Fab + (σ − γ 2 )2K[ai Kb]j
2 |q|
Fab → h ,
Thiemann’s trick,
regularization
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Physical state |ψi ∈ H satisfies the quantum Hamiltonian constraint:
Ĥ|ψi = 0
H : physical Hilbert space
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Physical state |ψi ∈ H satisfies the quantum Hamiltonian constraint:
Ĥ|ψi = 0
H : physical Hilbert space
• inner product of H?
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Physical state |ψi ∈ H satisfies the quantum Hamiltonian constraint:
Ĥ|ψi = 0
H : physical Hilbert space
• inner product of H?
• meaning of evolution? (e.g. |multiple loopsi are solutions to Ĥ|ψi = 0)
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Physical state |ψi ∈ H satisfies the quantum Hamiltonian constraint:
Ĥ|ψi = 0
H : physical Hilbert space
• inner product of H?
• meaning of evolution? (e.g. |multiple loopsi are solutions to Ĥ|ψi = 0)
• dynamics looks “frozen” (timeless formalism)
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Physical state |ψi ∈ H satisfies the quantum Hamiltonian constraint:
Ĥ|ψi = 0
H : physical Hilbert space
• inner product of H?
• meaning of evolution? (e.g. |multiple loopsi are solutions to Ĥ|ψi = 0)
• dynamics looks “frozen” (timeless formalism)
• So far, kinematics of LQG is well understood, but little is known about
the dynamics.
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Physical state |ψi ∈ H satisfies the quantum Hamiltonian constraint:
Ĥ|ψi = 0
H : physical Hilbert space
• inner product of H?
• meaning of evolution? (e.g. |multiple loopsi are solutions to Ĥ|ψi = 0)
• dynamics looks “frozen” (timeless formalism)
• So far, kinematics of LQG is well understood, but little is known about
the dynamics.
• related to “the problem of time”.
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Physical state |ψi ∈ H satisfies the quantum Hamiltonian constraint:
Ĥ|ψi = 0
H : physical Hilbert space
• inner product of H?
• meaning of evolution? (e.g. |multiple loopsi are solutions to Ĥ|ψi = 0)
• dynamics looks “frozen” (timeless formalism)
• So far, kinematics of LQG is well understood, but little is known about
the dynamics.
• related to “the problem of time”.
• Inclusion of matters may be inevitable. (The notion of time emerges
through relational correspondence.)
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• Physical state |ψi ∈ H satisfies the quantum Hamiltonian constraint:
Ĥ|ψi = 0
H : physical Hilbert space
• inner product of H?
• meaning of evolution? (e.g. |multiple loopsi are solutions to Ĥ|ψi = 0)
• dynamics looks “frozen” (timeless formalism)
• So far, kinematics of LQG is well understood, but little is known about
the dynamics.
• related to “the problem of time”.
• Inclusion of matters may be inevitable. (The notion of time emerges
through relational correspondence.)
• alternative approach: spin foam formalism (covariant/path integral
approach, in contrast to the canonical/Hamiltonian approach of LQG)
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
A glimpse of spin foam theories
Summary
Motivations
Canonical GR
LQG
Spin Foam
Outline
Motivations
Why quantum gravity?
Difficulties of quantum gravity
Canonical general relativity
Connection dynamics
Canonical/Hamiltonian formulation
Loop quantum gravity (LQG)
Loop algebra
Quantum kinematics
Quantum dynamics
A glimpse of spin foam theories
A glimpse of loop quantum cosmology (LQC)
A glimpse of loop quantum black holes
Summary
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
A glimpse of loop quantum cosmology (LQC)
LQC: simplified version (“toy model”, bottom-up approach ) of LQG
(mini-superspace formalism)
• Ashtekar variables:
homogeneity & isotropy :
Aa i (~x ) → c,
a
Ẽ i (~x ) → p
b
{Aa i (~x ), Ẽ j (~x 0 )} = 8πG γδji δab δ 3 (~x − ~x 0 )
8πG γ
3
• classical Hamiltonian constraint for the k = 0 FRW model:
→ {c, p} =
C = Cgrav + Cmatt = −
p
Npφ2
3N
2
|p|
+
c
8πG γ 2
2 |p|3/2
• inclusion of matter (scalar field), serving as internal clock:
{φ, pφ } = 1
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
• “holonomization”:
c −→
sin(µ̄c)
e µ̄c − e −µ̄c
=
µ̄
2i µ̄
Introduce finite µ̄ to impose the discreteness of quantum geometry of
LQG by hand.
• “improved” quantization:
s
µ̄ =
∆
,
|p|
√
∆ = area gap = 2 3πγ`2Pl
• “holonomized” Hamiltonian:
Hµ̄ = −
Npφ2
3N sin2 µ̄c p
|p| +
2
2
8πG γ
µ̄
2 |p|3/2
• quantization. . .
• SU(2): trivial
• Diff : not an issue
• Ĥµ̄ |ψi = 0: can be done thanks to internal clock!
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
The formulation of LQC has been constructed rigorously, revealing that the big
bang singularity is resolved and replaced by the quantum bounce (big bounce).
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
The formulation of LQC has been constructed rigorously, revealing that the big
bang singularity is resolved and replaced by the quantum bounce (big bounce).
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
“Cosmological inflation driven by holonomy corrections of loop quantum
cosmology”, DWC and K. Liu, Phys. Rev. D 81, 063526 (2010),
arXiv:1002.2035 .
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
slo
w
ro
ll
“Cosmological inflation driven by holonomy corrections of loop quantum
cosmology”, DWC and K. Liu, Phys. Rev. D 81, 063526 (2010),
arXiv:1002.2035 .
reheating
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
slo
w
ro
ll
no
(v n-in
iol la
ati tio
on na
of ry
slo
w-
ro
ll)
“Cosmological inflation driven by holonomy corrections of loop quantum
cosmology”, DWC and K. Liu, Phys. Rev. D 81, 063526 (2010),
arXiv:1002.2035 .
reheating
superinflation
Motivations
Canonical GR
LQG
Spin Foam
“Footprints of LQC on inflation and CMB?
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
A glimpse of loop quantum black holes
• Is the black hole singularity also resolved by the loop quantum geometry?
And How?
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
A glimpse of loop quantum black holes
• Is the black hole singularity also resolved by the loop quantum geometry?
And How?
• The simplest step is to consider the interior of a Schwarzschild black hole,
in which the metric component is homogeneous with the Kantowski-Sachs
symmetry R × SO(3) (thus mini-superspace approach applicable).
The metric of the Schwarzschild black hole:
interior:
ds
2
=
ds
2
=
2GM
2GM −1 2
2
2
2
1−
dt + 1 −
dr + r dΩ
r
r
−1
2GM
2GM
2
−2
2
2
2
−1
− 1 dx + t dΩ
−
dt + L
t
t
−
where we rename (r , t) as (t, L−1 x) with t ∈ [0, 2GM), x ∈ R
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
A glimpse of loop quantum black holes
• Is the black hole singularity also resolved by the loop quantum geometry?
And How?
• The simplest step is to consider the interior of a Schwarzschild black hole,
in which the metric component is homogeneous with the Kantowski-Sachs
symmetry R × SO(3) (thus mini-superspace approach applicable).
The metric of the Schwarzschild black hole:
interior:
ds
2
=
ds
2
=
2GM
2GM −1 2
2
2
2
1−
dt + 1 −
dr + r dΩ
r
r
−1
2GM
2GM
2
−2
2
2
2
−1
− 1 dx + t dΩ
−
dt + L
t
t
−
where we rename (r , t) as (t, L−1 x) with t ∈ [0, 2GM), x ∈ R
• The loop quantum geometry of the Schwarzschild interior can be studied
at the level of heuristic effective dynamics, even though the fundamental
quantum theory remains to be done.
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
(Conjectured) Penrose diagrams
(b)
..
.
..
.
(c)
→ M(M(M )), 32 Kc
→ M ′ , −Kc
(a)
+
i
I+
i
i
I
0
i
III
I−
−
i
→ M, −Kc
I+
II
IV
0
→ M(M ), 3Kc
→ M, Kc
+
→ M ′ , Kc
I−
−
i
→ M, Kc
I+
0
i
I−
→ M ′ , −Kc
classical
→ M−1 (M ), Kc /3
µ̄-scheme
0
µ̄ -scheme
DWC, Phys. Rev. D 78, 064040 (2008), arXiv:0807.0665 .
..
.
..
.
..
..
.
.
→ M, Kc
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Resolution of the black singularity and extension of the quantum spacetime
may resolve the information loss paradox.
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Resolution of the black singularity and extension of the quantum spacetime
may resolve the information loss paradox.
i+
r =0
I+
ra
di
at
io
n
r =0
r =0
i−
I−
i0
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Resolution of the black singularity and extension of the quantum spacetime
may resolve the information loss paradox.
i+
i+
r =0
I+
I+
i−
io
di
ra
di
ra
r =0
n
r =0
at
i0
at
io
n
r =0
I−
I−
i−
i0
Motivations
Canonical GR
LQG
Spin Foam
Outline
Motivations
Why quantum gravity?
Difficulties of quantum gravity
Canonical general relativity
Connection dynamics
Canonical/Hamiltonian formulation
Loop quantum gravity (LQG)
Loop algebra
Quantum kinematics
Quantum dynamics
A glimpse of spin foam theories
A glimpse of loop quantum cosmology (LQC)
A glimpse of loop quantum black holes
Summary
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
Summary
LQC
Quantum BHs
Summary
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Summary
Wikipedia:
• Loop quantum gravity (LQG), also known as loop gravity and quantum
geometry, is a proposed quantum hypothesis of spacetime which attempts
to reconcile the theories of quantum mechanics and general relativity.
• Loop quantum gravity postulates that space can be viewed as an
extremely fine fabric or network “woven” of finite quantised loops of
excited gravitational fields called spin networks. When viewed over time,
these spin networks are referred to as spin foam.
• The theory of LQG is considered a major quantum gravity contender,
along with string theory, but has the perceived advantage of consistently
incorporating general relativity without requiring the use of “higher
dimensions”. [minimalist approach]
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Summary
Wikipedia:
• Loop quantum gravity (LQG), also known as loop gravity and quantum
geometry, is a proposed quantum hypothesis of spacetime which attempts
to reconcile the theories of quantum mechanics and general relativity.
• Loop quantum gravity postulates that space can be viewed as an
extremely fine fabric or network “woven” of finite quantised loops of
excited gravitational fields called spin networks. When viewed over time,
these spin networks are referred to as spin foam.
• The theory of LQG is considered a major quantum gravity contender,
along with string theory, but has the perceived advantage of consistently
incorporating general relativity without requiring the use of “higher
dimensions”. [minimalist approach]
• A lot remains to be done. . .
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Summary
Wikipedia:
• Loop quantum gravity (LQG), also known as loop gravity and quantum
geometry, is a proposed quantum hypothesis of spacetime which attempts
to reconcile the theories of quantum mechanics and general relativity.
• Loop quantum gravity postulates that space can be viewed as an
extremely fine fabric or network “woven” of finite quantised loops of
excited gravitational fields called spin networks. When viewed over time,
these spin networks are referred to as spin foam.
• The theory of LQG is considered a major quantum gravity contender,
along with string theory, but has the perceived advantage of consistently
incorporating general relativity without requiring the use of “higher
dimensions”. [minimalist approach]
• A lot remains to be done. . .
• LQC. . .
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary
Summary
Wikipedia:
• Loop quantum gravity (LQG), also known as loop gravity and quantum
geometry, is a proposed quantum hypothesis of spacetime which attempts
to reconcile the theories of quantum mechanics and general relativity.
• Loop quantum gravity postulates that space can be viewed as an
extremely fine fabric or network “woven” of finite quantised loops of
excited gravitational fields called spin networks. When viewed over time,
these spin networks are referred to as spin foam.
• The theory of LQG is considered a major quantum gravity contender,
along with string theory, but has the perceived advantage of consistently
incorporating general relativity without requiring the use of “higher
dimensions”. [minimalist approach]
• A lot remains to be done. . .
• LQC. . .
• Loop quantum black holes. . .
Motivations
Canonical GR
LQG
Spin Foam
LQC
Quantum BHs
Summary