Scalar field in the cosmological sector of loop quantum gravity

Scalar field in the cosmological sector of loop quantum gravity

Scalar field in the cosmological sector of loop quantum gravity
Jakub Bilski
Department of Physics
Fudan University
The Planck Scale II
XXXV Max Born Symposium
Wrocław, September 7th , 2015
Jakub Bilski (Fudan University)
Scalar field in QRLG
Wrocław, Sep 7th , 2015
1 / 15
Outline
1
Background-independent description of matter fields in the loop framework
Classical fields minimally coupled to gravity
Scalar field in LQG
2
Regularization of Hamiltonian constraint
Regularization in LQG
Hamiltonian constraint
3
Quantization of the scalar field Hamiltonian constraint
Quantization in QRLG
Semiclassical limit
4
Conclusions and open problems
5
References
Jakub Bilski (Fudan University)
Scalar field in QRLG
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Classical fields minimally coupled to gravity
Quantum gravity as the natural regulator of matter quantum field theories
F̂ 2 (x) −→ lim
ε→0
Z
1
d 3 y χε (x − y)F̂ (x)F̂ (y)
ε3
Matter fields in general relativity
S
(EH )
+S
(φ)
1
+ ... =
κ
Z
Jakub Bilski (Fudan University)
√
1
d x −gR +
2λ
M
4
Z
√
d 4 x −g g µν (∂µ φ)(∂ν φ) − V (φ) + ...
M
Scalar field in QRLG
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Scalar field in LQG
Gravitational field couples to matter through the co-triad eai
Polymer representation (point-holonomy representation) of scalar field on Σt
Diffeomorphisms of Σt act naturally as unitary operators in the Hilbert space
Hamiltonian of scalar field:
H (φ) =
Z
d 3 x N a π∂a φ + N
Σt
Z
=
h
√
√
q ab
q
λ 2
q ∂a φ∂b φ +
V (φ)
√ π +
2 q
2λ
2λ
(φ)
d 3 x N a Va(φ) + NHsc
i
Σt
Jakub Bilski (Fudan University)
Scalar field in QRLG
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Scalar field in LQG
Total kinematical Hilbert space:
R
(tot)
(gr)
(φ)
Hkin = RHkin ⊗ RHkin
(φ)
Hkin := a1 Uψ1 + ... + an Uψn : ai ∈ C, n ∈ N, ψi ∈ R
Uψ := e
i
P
v∈Σ
ψv φv
:= |Uψ i
hUψ |Uψ0 i := δψ,ψ0
(φ)
Hkin := L2 R̄Bohr Σ is obtained from the single-point one L2 R̄Bohr ;
Bohr measure:
Z
dµBohr (φ)e iψv φv = δ0,v
R̄Bohr
Basic variables:
Ûψ |Uψ0 i = |Uψ+ψ0 i
Π̂(V ) |Uψ i = ~
X
ψv |Uψ i
v∈V
Jakub Bilski (Fudan University)
Scalar field in QRLG
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Scalar field in LQG
Single-point states:
|v; Uψ i := e iψv φv
hw; Uψ |v; Uψ0 i := δw,v δψ,ψ0
Action of diffeomorphism:
ϕ∗ |v; Uψ i = |ϕ(v); Uψ i
Action of basic operators:
e iψw φ̂w |v; Uψ i = e iψw φw |v; Uψ i = |v ∪ w; Uψ i
Π̂(v) |v; Uψ i = −i~
∂
|v; Uψ i = ~ψv |v; Uψ i
∂φ(v)
Smeared momentum around a point v:
Z
Π(v) :=
d 3 uχε (v, u)π(u)
Canonical Poisson brackets:
φ(x), Π(y) = χε (x, y)
φ(x), φ(y) = Π(x), Π(y) = 0
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Scalar field in QRLG
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Regularization in LQG
(φ)
Hsc
[N ] :=
Z
d 3 xN
Σt
(φ)
√
√
q ab
q
λ 2
q ∂a φ∂b φ +
V (φ) :=
√ π +
2 q
2λ
2λ
(φ)
(φ)
:=Hkin [N ] + Hder [N ] + Hpot [N ]
Regularization via Thiemann’s method:
eai (x)
δ V(R)
δV(R)
2
=2
=
n−1
δEia
δEia
n V(R)
(φ)
Hkin =
n
4
=
nγκ V(R)
X
X X
221 λ
0
0
lim
N
(v)Π(v)Π(v
)
χ
(v,
v
)
ijk pqr lmn stu ×
ε
32 (γκ)6 ε→0 0
0 0
v,v ∈V(Γ)
hl−1
p (∆)
n
× tr τ k hl−1
r (∆)
n
× tr τ
i
n
n o
i
n−1 Aa (x), V(R)
× tr τ m hl−1
0
t (∆ )
n
∆(v) ∆ (v )
12
V(∆(v), ε)
12
V(∆(v), ε)
V(v 0(∆0 ), ε)
Jakub Bilski (Fudan University)
, hlp (∆)
o , hlr (∆)
21
hl−1
q (∆)
n
tr τ l hl−1
0
s (∆ )
n
tr τ
j
o , hlt (∆0 )
o tr τ n hl−1
0
u (∆ )
Scalar field in QRLG
21
V(∆(v), ε)
, hlq (∆)
12
o
V(v 0(∆0 ), ε)
, hls (∆0 )
n
21
V(v 0(∆0 ), ε)
×
o
, hlu (∆0 )
Wrocław, Sep 7th , 2015
×
o
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Regularized Hamiltonian constraint
(φ)
Hkin =
221 λ X
N (v) Π2 (v)
32 (γκ)6
n
1
o n
1
o 213
X
×tr τ i hl−1
V(v) 2, hlp (∆)
p (∆)
(φ)
e
34 λ (γκ)4
φv+~ep −φv
−e
2
n
1
o n
1
n
o
tr τ k hl−1
V(v) 2, hlr (∆)
r (∆)
1
o tr τ mhl−1
V(v) 2, hlt (∆0 )
0
t (∆ )
X
N (v)
v∈V(Γ)
n
tr τ j hl−1
V(v) 2, hlq (∆)
q (∆)
×tr τ l hl−1
V(v) 2, hls (∆0 )
0
s (∆ )
Hder =
ijk pqr lmn stu ×
∆(v)=∆0 (v 0 )=v
v∈V(Γ)
×
X
×
1
n
o
tr τ n hl−1
V(v) 2, hlu (∆0 )
0
u (∆ )
ijk pqr ilm stu ×
∆(v)=∆0 (v)=v
φv −φv−~ep
×
43
3
e φv+~es −φv − e φv −φv−~es
tr τ j hl−1
V(v) 4, hlq (∆)
q (∆)
2
o ×tr τ k hl−1
V(v) , hlr (∆)
r (∆)
n
34
n
o tr τ l hl−1
V(v) , hlt (∆0 )
0
t (∆ )
where we used the following expansion: ∂p φ(v) ≈
(φ)
Hpot =
o
n
×
34
o
tr τ mhl−1
V(v) , hlu (∆0 )
0
u (∆ )
−φv
φv −φv−~
φ
ep
1 e v+~ep
−e
ε
2
1 X
N (v)V (φv )V(v)
2λ
v∈V(Γ)
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Scalar field in QRLG
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Quantization in QRLG
(φ)
(φ)
(φ)
Ĥ |Γ; ml , iv ; Uψ iR = Ĥkin + Ĥder + Ĥpot |Γ; ml , iv ; Uψ iR .
tr τ
i
ĥl−1
p
n
V̂ (R), ĥlp
= tr τ
i
ĥl−1
V̂n (R) ĥlp
p
(φ)
Ĥkin |Γ; ml , iv ; Uψ iR =
−
215 λ
32 (8πγlP2 )6
X
N (v)Π̂2 (v)
X
ijk pqr lmn stu ×
∆(v)=∆0 (v)=v
v∈V(Γ)
12
ĥlp (∆) tr τ j ĥl−1
V̂(v)
q (∆)
12
ĥlr (∆) tr τ l ĥl−1
V̂(v)
0
s (∆ )
× tr τ i ĥl−1
V̂(v)
p (∆)
× tr τ k ĥl−1
V̂(v)
r (∆)
× tr τ m ĥl−1
V̂(v)
0
t (∆ )
Jakub Bilski (Fudan University)
12
21
12
ĥlq (∆) ×
12
ĥlt (∆0 ) tr τ n ĥl−1
V̂(v)
0
u (∆ )
Scalar field in QRLG
ĥls (∆0 ) ×
ĥlu (∆0 ) |Γ; ml , iv ; Uψ iR
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Quantization in QRLG
Action of the trace operator
tr τ i ĥl−1
V̂n (v) ĥlz = −
z
X
(τi )ab (ĥl−1
)bd V̂n (v) (ĥlz )da , along z direction
z
abd
(ĥlz )da = e iaθ δda , in the basis that diagonalizes τz
tr τ i ĥl−1
V̂n (v) ĥlz = − 8πγlP2
z
32 n (y)
Σ(x)
v Σv
1/2
n2 X
(τi )ab e −idθ δbd Σ(z)
v +a
n2
e iaθ δda =
abd=−1/2
n
(y) 2 i
3 2 2n
= 8πγlP
Σv(x) Σv
4
δ iz
Σv(z) −
1
2
n2
− Σ(z)
v +
1
2
n2 where
Σ(p)
=
v
1 (p)
(p)
(jv + jv−~ep ),
2
∆(p),n
=
v
V̂n (v) ĥlp |Γ; ml , iv ; Uψ iR =
tr τ i ĥl−1
p
Jakub Bilski (Fudan University)
1
2n
i
h
(p) n
(p)
(p) n
(p)
jv − 1 + jv−~ep − jv + 1 + jv−~ep
3 n
i
(r)
8πγlP2 2 Σ(q)
v Σv
4
Scalar field in QRLG
n2
(p), n
2
∆v
δ ip |Γ; ml , iv ; Uψ iR
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Quantization in QRLG
Action of the total scalar constraint operator
Ĥ (φ) |Γ; ml , iv ; Uψ iR =
X
Nv
211 λ
(8πγlP2 )
v
1
+
×
+
+
+
Jakub Bilski (Fudan University)
(y) (z)
Σ(x)
∆v
v Σv Σv
211 (8πγlP2 ) 2
(y) (z)
Σ(x)
v Σv Σv
34 λ
Σ(x)
v
Σv(y)
Σv(z)
34 34 34 8πγlP2
2λ
(y), 3
8
∆v
2 (z), 1
4
∆v
2
Π̂2v +
×
2 e φ̂v −φ̂v−~ex − e φ̂v+~ex −φ̂v
2
2
+
2
e φ̂v −φ̂v−~ey − e φ̂v+~ey −φ̂v
2
!2
(y), 3
∆v 8
2 e φ̂v −φ̂v−~ez − e φ̂v+~ez −φ̂v
2
2 2 (x), 3
∆v 8
2 Σ(x)
v
(z), 3
8
∆v
34
(y), 1
4
∆v
(z), 3
∆v 8
(x), 3
∆v 8
23 (x), 1
4
3
2
Σ(y)
v
Σv(z)
Scalar field in QRLG
12
V̂ φv
+
+
|Γ; ml , iv ; Uψ iR
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Semiclassical limit
Expansion for j 12 :
∆(p),n
= −n Σv(p)
v
n−1
+O
Σv(p)
n−3 h (φ) := hΓ; ml , iv ; Uψ | Ĥ (φ) |Γ; ml , iv ; Uψ iR ≈
≈
X
v
Nv
λ −1 2
1
Vv Πv +
Vv
2
2λ
(y)
(x)
Σv
ˆ 2x φv i +
ˆ 2y φv i+
h∆
h∆
(y) (z)
(x) (z)
2
2
8πγlP Σv Σv
8πγlP Σv Σv
Σv
(z)
+
Σv
(x)
ˆ 2z φv i
h∆
(y)
8πγlP2 Σv Σv
1
Vv hV̂ (φv )i ,
+
2λ
where the eigenvalues of the volume and of the momentum operator read:
Vv :=
Jakub Bilski (Fudan University)
8πγlP2
3
Σv(p) Σv(q) Σ(r)
v
12
Scalar field in QRLG
Πv = ~ψv
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Semiclassical limit
h
(φ)
X 1Z
≈ lim
ε3
ε→0
v
p
d 3 u χε (v, u) N (v)
2
1
q(v)
p (v)
p2 (v)
p3 (v)
ˆ 2x φv i+
ˆ 2y φv i+
ˆ 2z φv i +
h
∆
h
∆
h∆
2λ
p2 (v) p3 (v)
p1 (v) p3 (v)
p1 (v) p2 (v)
+ε
!
p
+ε
Π2 (v)
+
3
q(v) ε
λ
p
3
q(v) V̂ (φv )
2λ
Gravitational momenta are related to spin-numbers by the relation:
pi (v) ε2 = 8πγlP2 Σv(i)
(φ)
hcl →
Z
√ 2
2
2
q
λ
2
11
22
33
d u N (u) √ π (u) +
q ∂x φ(u) + q ∂y φ(u) + q ∂z φ(u) +
2 q
2λ
√
q
+
V φ(u)
2λ
3
Jakub Bilski (Fudan University)
Scalar field in QRLG
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Conclusions and open problems
Conclusions
QRLG allows to construct diffeomorphism invariant action of scalar field
matrix elements of scalar field Hamiltonian constraint are analytic
large j limit approach the classical Hamiltonian at the leading order
and gives convergent series of next-to-the-leading-order quantum corrections
Open problems
construction of semiclassical states
studies of phenomenological applications in cosmology
scalar field as a clock in QRLG
fermion and vector field coupling
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Scalar field in QRLG
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References
1
J. Bilski, E. Alesci, F. Cianfrani, "Quantum reduced loop gravity: extension to scalar field",
arXiv:1506.08579 [gr-qc].
2
E. Alesci, F. Cianfrani, "Loop Quantum Cosmology from Loop Quantum Gravity",
arXiv:1410.4788 [gr-qc].
3
E. Alesci, F. Cianfrani, "Quantum Reduced Loop Gravity: Semiclassical limit",
arXiv:1402.3155 [gr-qc].
4
E. Alesci, F. Cianfrani, "Quantum-reduced loop gravity: Cosmology", Phys Rev D 87 (2013)
083521, arXiv:1301.2245 [gr-qc].
5
A. Ashtekar, S. Fairhurst, J. L. Willis, “ Quantum gravity, shadow states, and quantum
mechanics”, Class. Quant. Grav. 20, 1031 (2003), arXiv:0207106 [gr-qc].
6
A. Ashtekar, J. Lewandowski, H. Sahlmann, “ Polymer and Fock representations for a scalar
field”’, Class. Quant. Grav. 20, 1031 (2003), arXiv:0211012 [gr-qc].
7
W. Kamiński, J. Lewandowski, M. Bobieński, “Background independent quantizations: The
Scalar field. I”, Class. Quant. Grav. 23, 2761 (2006), arXiv:0508091 [gr-qc].
8
W. Kamiński, J. Lewandowski, A. Okołów, “Background independent quantizations: The
Scalar field. II”, Class. Quant. Grav. 23, 5547 (2006), arXiv:0604112 [gr-qc].
9
M. Domagała, M. Dziendzikowski, J. Lewandowski, “ The polymer quantization in LQG:
massless scalar field”, arXiv:1210.0849 [gr-qc].
10
T. Thiemann, “QSD 5: Quantum gravity as the natural regulator of matter quantum field
theories”, Class. Quant. Grav. 15, 1281 (1998) [gr-qc/9705019].
11
T. Thiemann, “Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories”,
Class. Quant. Grav. 15, 1487 (1998) [gr-qc/9705019].
12
T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge University Press,
2007.
Jakub Bilski (Fudan University)
Scalar field in QRLG
Wrocław, Sep 7th , 2015
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Appendix: LQG −→ QRLG
Canonical variables in the diagonal gauge fixing:
Eia = pi δia , Aia = ci δai
Cuboidal lattice:
Γ −→ ΓR
Representation of the reduced group elements:
j
Dm
n (hl ) −→
j
−1 j
j
j
−1 j
−→ Dm
)±j m 0 (ul ) Dm
)±j n (ul ) :=
0 n 0 (hl ) Dn 0 ±j (ul ) (D
±j (ul ) (D
jl
hl
jl
Basis elements in QRLG:
Y
Y l jl
jl , xv ml , ~ul
h Γ, ml , xv =
Dm
l
v∈Γ
ml (hl ),
l
jl
j
j
−1 j
where jl = |ml | and lDm
)±j m 0 (ul ) Dm
0 n 0 (hl ) Dn 0 ±j (ul )
l ml (hl ) := (D
Jakub Bilski (Fudan University)
Scalar field in QRLG
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