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Massive Scalar Field Quantum Cosmology
Sang Pyo Kim Kunsan National University
3rd International Workshop on
Dark Matter, Dark Energy and Matter-Antimatter Asymmetry
December 29, 2012
Outline • Motivations
– Why Quantum Cosmology? – Why Massive Scalar Field Quantum Cosmology?
• Quantum Cosmology • Quantum-Classical Transition • Second Quantized Universes • Third Quantization • Conclusion
Why Quantum Cosmology?
Big Bang as an Ingredient of Cosmology
• The singularity theorem implies the Big Bang (BB) [Hawking, Penrose, Proc. R. Soc. Lond. A 314 (‘70)].
• Inflationary spacetimes have the singularity [Borde, Guth, Vilenkin, PRL 90 (‘03)].
• What is the spacetime geometry including the BB? • How to quantize the spacetime as well as matter
fields, that is, what is quantum gravity and quantum cosmology?
• How do a classical universe and the unitary quantum field theory emerge from quantum cosmology?
Why Massive Scalar Field Quantum Cosmology?
Single-Field Inflation Models 7-Year WMAP
[Astrophys. J. Suppl. 192 (‘11)] SPT
[arXiv:1210.7231]
9-Year WMAP [arXiv:1212.5226]
R2 Inflation Model • Starobinsky’s R2 inflation model ( ) [PLB 91 (’80)]
: a de Sitter-type acceleration . • Mukhanov and Chibisov [JETP Lett 33 (‘81)]:
– approximately scale-invariant quantum fluctuations
– N = number of e-folds of expansion between the end of
inflation and the epoch at which a fluctuation with kWMAP left the horizon.
• Whitt [PLB 145 (‘84)] and Maeda [PRD 37 (‘87)]: equivalent to a scalar field under a conformal transformation
Nkdkdn R
s2
ln)(ln1
WMAP
2
−=∆
=−
2RR α+
)21ln(2/3,)21(~ RgRg αα µνµν +=Ψ+=
Quantum Corrections to EH Action • A charged scalar field in (D=d+1)-dimensional curved spacetime
• The effective action in Schwinger-DeWitt proper time integral
• The perturbative one-loop corrections to Einstein-Hilbert action
)(,)(,0)( 2 xiqADmDDxHxH µµµµµ −∂=+−==Φ
);',()4)((
)(21
'||)(
1)(2
2/)1(0
1
0
1
2
isxxFsis
eisdgxd
xexis
isdgxdiW
d
simd
isHd
+
−∞+
−∞+
∫∫
∫∫
−=
−−=
π
( )∫ −+−=
−++==
µνµν
µναβµναβ
µνµν
µναβµναβ
µµ
RRRRRg
RRRRRRfRf
4numberEuler 180
1180
1121
301,
2
2;;21
One-Loop Action for dS • de Sitter space with the metric
• Bogoliubov coefficients for a massive scalar field
22
222 )(cosh
ddH
Htdtds Ω+−=
4)1(,
)2/1()2/()()1(
,)2/1()2/(
)()1(
22
0
dR
mdddldl
ii
Zlidlidl
ii
l
l
−+
=−−Γ+Γ
Γ−Γ=
∈−−−Γ−+Γ
−Γ−Γ=
γγγβ
γγγγα
One-Loop Action for dS • Vacuum polarization and vacuum persistence from
in-out formalism [SPK, arXiv:1008.0577]
• One loop action is related to f(R)-gravity with all coefficients for higher curvature terms determined.
( )
−−+=
+=
+==
∫∑∞ −
)2/sin()2/cos()2/)12cos(()(
1ln)(Im2,)sinh(
)2/(sin||
0states
sceff
scsceff
22sc
sssdl
sedsPRW
NRLdlN
s
lll
γ
πγπβ
Hartle-Hawking No-Boundary Wave Function
Euclidean solutions for a FRW coupled to a massive field scalar [Hawking, NPB 239 (1984)]
Inflation with Negative Λ [Hartle, Hawking, Hertog, arXiv:1205.3807,1207.6653]
• Negative Λ and massive scalar field with negative mass
• Phase transition from AdS to dS
• The wave function is peaked around the classical
trajectories.
22
24
negative
222 1)2( φπφπa
kaama +−+Λ=
00 22 ≥⇒≤ aa ππ
What is Quantum Cosmology?
ADM Formalism • Arnwitt-Deser-Misner formalism: foliate a globally
hyperbolic spacetime manifold by spacelike 3-surfaces
• The Hamiltonian for gravity and matter fields
orshift vect&function lapse
2)( 222
==
++−−=
i
jiiji
ii
NNdxdxhdtdxNdtNNNds
[ ]
( ) form) lfundamenta second:(16
,2)(
)'()'()()'()()'()(2
1)',(
),,()2(16
)'()'()()',(16'
][
2/12
02/1|
32/1
003
232/123
0
03
ijijijPijiijj
i
klijjkiljlikijkl
ijPklij
ijklP
ii
KKhKhmThxH
xxxhxhxhxhxhxhh
xxG
hTRmxxhxxxxGmxdH
HNNHxdH
−−=−=
−−+=
+Λ+−−+=
+=
∫
∫−
πππ
δ
πφπ
δπππ φ
Wheeler-DeWitt Equation • The WDW equation from the super-Hamiltonian
constraint and/or the super-momentum constraints via Dirac quantization:
• Quantum cosmology necessarily includes quantum
fluctuations from spacetime and matter fields.
ij
ij
i
ijP
klijijkl
P
Thxhi
xH
hi
TRmxxh
xhxhxxG
mxdH
02/1
003
232/1
23
0
|)(
2)(
),,()2(16
)'(
)'()()',(16'
−=
+Λ+−−+
−= ∫
δδ
δφδφ
πδ
δδ
δδπ
Quantum-Classical Transition
From QG to SQG to CG
Classical Gravity
Inflationary models
Semiclassical Quantum Gravity
QFT in curved spacetime, Hawking radiation, pair
production
Quantum Gravity
WDW, HH wave function, tunneling wave function µνµν π TGG ˆ8ˆ =
µνµνµν π TGGGG ˆ8][QC =+
( )][8][ QCQC µνµνµνµν π TTGGGG +=+
0/1 2P →= mG
0→
de Broglie-Bohm Pilot-Wave Theory
• In the causal interpretation, a particle has a definite (suitable) path that is affected by the wave function
• In a semiclassical regime, where
ψψψ Vmt
i +∇−=∂∂ 2
2
2
( )
( )
∇
===⋅∇+∂∂
∇−==++∇+
∂∂
mSvFv
t
FF
mVVVS
mtS
,,0
QM) ofpart (imaginaryequation continuity2
,021
QM) ofpart (realequation Jacobi-Hamilton
22
22
QQ2
ρρρ
Si
eF =ψ
Quantum FRW Universe (minisuperspace model)
• The metric for Friedmann-Robertson-Walker universe
• The WDW equation for a FRW universe with a minimal
scalar field (inflaton) up to a factor ordering
23
2222 )( Ω+−= dtadtNds
( )
==Λ−=
=Ψ
+−
∂∂
−
π
φφπφ
43,1,
21)(
0,),(ˆ)(2
242
2
22
PG
mG
mMcakaaV
aHaMVaM
Wave Packet for FRW with a Minimal Scalar
A closed universe (k=1), m = 6, and n =120 (harmonic quantum number) [Fig. from Kiefer, PRD 38 (‘88)]
Pilot-Wave Theory and Born-Oppenheimer Idea
• The wave functions are peaked around some trajectories (wave packets) and allow the pilot-wave theory
• Apply the Born-Oppenheimer idea that separates a slow moving massive particle (M=Planck mass squared) from a fast moving light particle (matter field) and then expand the quantum state for the fast moving variable by a certain basis to be determined
( ) ( )ijaaaa hhhhiHhMVM
==Ψ
−+−∇− ,0,),,(ˆ)(
2 G2
2
φδφδφ
( ) ( )( ) ( )∑ Φ=Φ
Φ=Ψ
kakaka
aaa
hhch
hhh
,)(,
,)(,
φφ
φψφ
Semiclassical Quantum Gravity [SPK, PRD 52 (‘95); CQG 13 (‘96); PRD 55 (‘97)]
• Apply the de Broglie-Bohm pilot-wave theory to the gravity part only
• Then, in a semiclassical regime, the WDW equation is equivalent to
/)()()( ahiSaa ehFh =ψ
( ) ( )
( )
−
∇⋅
∇=
Φ∇Φ=ΦΦ=
=+∇⋅∇
+∇
=−∇
−+−∇
∑k n
knk
n
nann
akanankakanank
nn
nnnna
ccAi
cc
FFhQ
hhihAhHhhH
QSFFS
QMF
FM
HhMVSM
:)(
),(),(:)(;),(ˆ),(:)(
0Im21
0Re2
)(2
1
2
222
G2
φφφφ
Semiclassical Quantum Gravity
• In quantum gravity, time is NOT a priori given since the WDW equation is a constraint equation (problem of time). Thus, time should be defined from the wave function itself.
• In the semiclassical quantum gravity, time emerges from the wave packet and the cosmological time is defined as the directional derivative of the action, not necessarily a classical one, along the trajectory
∇⋅∇= )(1: ahS
Mδτδ
Semiclassical Quantum Gravity
• The matter field obeys the Heisenberg (matrix) equation
• The unitarity of the quantum state of matter field is preserved.
),(),(2:)(
),(),(:)(;),(ˆ),(:)(
2
22
2
akannknkank
akanankakanank
knk
nkkk
nkknk
nkn
hhAih
hhihAhHhhH
cM
cASM
cHci
φφδ
φφφφ
δτδ
Φ∇Φ+∇⋅−∇=Ω
Φ∇Φ=ΦΦ=
Ω−⋅∇−= ∑∑∑≠≠
==⋅+
)()(
)(,1)()( 2
1
ττ
τττ cc
CCC
Scalar Field Cosmology
• The extended superspace for a FRW with a minimal scalar and the cosmological time:
• The Heisenberg matrix equation for the scalar field
∂∂
−=∂
∂∂∂
∂∂
−=∂∂
+−=
aaS
Maa
aaaS
Ma
daadads)(1)(,)(1
2322
ττ
τ
φ
Φ
∂∂
−∂∂
Φ+∂∂
−
∂∂
−∂∂
−=Ω
Φ∂∂
Φ=ΦΦ=
Ω−−=∂∂
∑∑∑≠≠
knnknknk
knnkknnk
knk
nkkk
nkknk
nkn
aaiB
aa
aa
iaBHaH
cMa
cBcHci
τττδ
τττ
τττ
τ
2
2
2
2
2
2
21:))((
:))((;ˆ:))((
2
Scalar Field Cosmology
• The semiclassical Friedmann equation
• The effective energy density
( ))Im()3/4(2
1/:
132)Re(
34
38
22
22
2
2
2
322
2
nnPnn
k n
knk
n
nnn
nnnnP
nnnnP
nnP
Rmaaaa
FaFU
ccBi
ccR
Ua
Uam
RUaam
Hama
kaa
π
πππ
+−=
∂∂=
−=
++−=Λ−+
⋅
∑
++−= nnnn
Pnnnn
Pnnnn U
aU
amRU
aamH
132)Re(
34 2
2
2
2
2 ππρ
Massive Scalar Field Cosmology
• The semiclassical Friedmann equation with a massive scalar field at the lowest order of
( )
[ ] 03;21
21;0
132
38
22ˆ
2*2*3
2)0()0(
)0(2)0(2
2
322
2
232
2
2
3
2
=+++
+=
−==
++=Λ−+
+∂∂
−=
⋅
ϕϕϕϕϕϕϕ
ππ
φφ
maamnaH
aaaaUR
Ua
Uam
Hama
kaa
ama
H
nn
nnnn
nnnnP
nnP
M/
Back-reaction from Quantum Gravity • Explain the back-reaction of de Sitter radiation
[J-A. Gu, SPK, C-M. Shen, arXiv:1210.7902] – dS temperature increased by the ratio of the dS scale
to the Planck scale. – Equivalent to the back-reaction of Hawking radiation
in a de Sitter Schwarzschild black hole [Greene et al, JHEP04 (‘06)057].
( )][8][ QCQC µνµνµνµν π TTGGGG +=+
Second Quantized Universes
Scalar Field Quantum Cosmology • The ADM formalism for a FRW geometry
• The WDW equation (Dirac quantization of Hamiltonian
constraint) for a FRW universe minimally coupled to a single-field inflaton (scalar field); the WDW equation is already second quantized
23
2222 )( Ω+−= dtadtNds
( ) ( )
====Λ−=
=Ψ
−+
∂∂
−∂∂
116,2)(
0,)(21
2242
42
2
22
2
PPG
G
mlcakaaV
aaVVaaa
π
φφφ
Quantum Universes in the Superspace
• The supermetric for FRW geometry and a minimal scalar • The Hamiltonian constraint and the WDW equation
• A Cauchy initial value problem w.r.t. the scale factor a and
a prescription for the boundary condition.
2222 φdadads +−=
( ) ( )
[ ] ( )42
2
2
22
22
42
Hpart fieldscalar
622
Hpart gravity
2
2)(,1
0,)(2)(
0)(21)(),(MG
akaaVaa
aVaaV
Vaa
aVaH
G
G
Ga
Λ−=∂∂
+∂∂
−=∇
=Ψ+−∇−
=+++−=
φ
φφ
φππφ φ
Quantum Universes in the Superspace [SPK, Page, PRD 45 (‘92); SPK, PRD 46 (‘92)]
• The scalar field for single-field inflation model • The eigenfunctions and the Symanzik scaling law
• The coupling matrix among the energy eigenfunctions
)2/()( 22 pV p
pφλφ =
( )( ) ( )( )φλλφ
ελ
φφφ
)1/(162
)1(4/162
)1/(162
//),(
/)(
),()(),(),(
++
+
=Φ
=
Φ=Φ
ppn
ppn
np
pn
nnnM
paFpaa
paaE
aaEaaH
( )( ) 2)()()1(4/3)(
),()(),(
ςςςςεε
φφ
nmnmmn FFdapa
aaaa
∫−+=Ω
ΦΩ=Φ∂∂
Quantum Universes in the Superspace [SPK, Page, PRD 45 (‘92); SPK, PRD 46 (‘92)]
• The gravitational part of the WDW equation
• The transition matrix and the Cauchy problem
0)()(1)()(2)()(
)(),(),(
222
2
=
Ω−Ω−Ω−+−
⋅Φ=Ψ
aaa
adada
aaEaV
dad
aaa
G
T
ψ
ψφφ
−=
Ω=Φ=Ψ
−
∫
daada
TaEVT
I
daada
dad
adaaTaaTaa
G
aT
/)()(
0
0
/)()(
)'('exp)(;)()(),(),(
21
ψψ
ψψ
ψφφ
Quantum Universes in the Superspace [SPK, Page, PRD 45 (‘92); SPK, PRD 46 (‘92)]
• The two-component wave function
• The off-diagonal components are the gravitational part
equation only with . • The continuous transitions among energy eigenfunctions.
Ω−
Ω×
ΦΦ
=
∂Ψ∂
Ψ
∫00
02 /)(
)('
)'('/)'()'(
exp
),(00),(
/),(),(
daada
daaaEaVIa
T
aa
aaa
G
T
T
ψψ
φφ
φφ
2/)( aEaVG −
Third Quantization
Third Quantization in 3+1 Dimensions
Second quantization Third quantization Particle Universe
Interaction Vertex Topology Change Field Third Quantized Field
Spacetime Superspace of Three Geometries
Free Laplacian Wheeler-DeWitt Operator Vacuum Void
[Strominger, “Baby Universes,’’ in Quantum Cosmology and Baby Universes edited by S. Coleman et al (World Scientific, 1991)]
Hilbert Space for Quantum Universes • The inner product for the WDW equation [SPK, J. Kim,
K.S. Soh, NPB 406 (‘93); SPK, Y.H. Lee, JKPS 26 (‘93)]
• The Feshbach-Villars’ first order equation for the WDW
equation guarantees the inner product [Mostafazadeh, JMP 39 (‘98)].
0,,
),(),(*
=ΨΨ−=ΨΨ=ΨΨ
Ψ∂Ψ=ΨΨ ∫Σβααββααββα
βαβα
δδ
φφφ
IIIIIIIII
a aada
∂Ψ∂
−Ψ
∂Ψ∂
+Ψ
+−−−−+
=
∂Ψ∂
−Ψ
∂Ψ∂
+Ψ
∂∂
ai
ai
DDDD
ai
ai
a )1(1)1(1
21
Hilbert Space for Quantum Universes • The wave function is positive/negative w.r.t.
so that the universe is quantized [Banks, NPB 309 (‘88); McGuigan, PRD 38 (‘88); Giddings, Strominger, NPB 321 (‘89); Peleg, CQG 8 (‘91); Castagnino et al, 10 (‘93), SPK et al (‘93)]
• The operator creates a universe in the wave function , and so on. • The Hilbert space consists of the void of no-universe, one-
universe H and multi-universes.
+Ψ+Ψ=Ψ ∑ αααα
α φφφ IIII AaAaa ),(),(),( *
*/ αα II ΨΨ ai∂
⊕⊗⊕⊕= )()( HHHCHF
+αIA
*αI
Ψ
Third Quantization • The WDW equation from the third quantized Hamiltonian
• A massless field is a sum of a-dependent oscillators [Hosoya,
Morikawa, PRD 39 (‘89); Abe, PRD 47 (‘93); Horiguchi, 48 (‘93)] and is a tachyonic state for the closed universe [SPK, arXiv:1212.5355].
• The third quantization of a massive field is analogous to the second quantized charged KG in a time-dependent, homogeneous, magnetic field .
( ) ( )( )
Ψ−+
∂Ψ∂
+
∂Ψ∂
−= ∫ 242
2
2
21 aVVaaa
dadS Gφφ
φ
)2/)(),(( rtBrtA ×=
Third Quantization • Expand the wave function by the energy eigenfunctions of
Hamiltonian, , for the scalar field to obtain the third quantized Hamiltonian
• The massive scalar quantum cosmology can be solved in the sense that the coupling matrix Ω and the energy-eigenvalue matrix E are explicitly known.
)()(/)(
)(21)(
21)(
)O(a :universe late
2O(1/a) :universeearly very
1)2p)/(p-(4
aaaa
aEa
aaH TTT
ψψπ
ψψψπππ
Ω+∂∂=
+Ω−⋅=
+
)(),(),( aaa T ψφφ
⋅Φ=Ψ
WDW Equation vs KG Equation in B • The WDW equation for the FRW universe with a massive
scalar field
• The transverse motion of a charged scalar in a temporal,
homogeneous, magnetic field
( ) ( ) 0),(1)( 22622
2 =Ψ
+++− φφππ φ ama
aaVGa
( ) 0),()(2
)( 2222
22
2
=Φ
++
−
++
∂∂
⊥⊥⊥⊥ xtkmLtqBxtqBpt zz
2/)(),( rtBrtA ×=
Conclusion • Quantum cosmology may be a consistent framework for
studying quantum fluctuations of spacetime and matter fields.
• Semiclassical and classical cosmology with the back-reaction can be derived from quantum cosmology.
• The ΛCDM model with a massive scalar field seems to be viable with the current CMB data (WMAP 7, SPT, etc.).
• Can massive scalar field quantum cosmology predict all observational data (WMAP 9)?
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