loop quantization and the cosmology - iitg.ac.in€¦ · introduction quantum gravity loop quantum...
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IntroductionQuantum Gravity
Loop Quantum Cosmology
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......Loop Quantization and the Cosmology
Golam M Hossain
Department of Physical SciencesIndian Institute of Science Education and Research Kolkata
Golam M Hossain 1/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Pillars of Modern PhysicsGeneral RelativityClassical FRW Cosmology
.. Pillars of Modern Physics
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Golam M Hossain 2/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Pillars of Modern PhysicsGeneral RelativityClassical FRW Cosmology
.. Hand-full of General Relativity
Golam M Hossain 3/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Pillars of Modern PhysicsGeneral RelativityClassical FRW Cosmology
.. Hand-full of General Relativity
Schwarzschild geometry aroundthe Earth :ds2 = −
!1− rs
r
"dt2 + d l2
rs = 2GM ≈ 9 mmrGPS ≈ 26600 km
GPS accuracy of say 6m on ground requires time accuracy of20 nano-seconds.
GPS device will give wrong reading for above in about 2minutes if time dilation effect from GR is not included.
Golam M Hossain 3/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Pillars of Modern PhysicsGeneral RelativityClassical FRW Cosmology
.. Cosmology: FRW spacetime
Large scale universe is described by spatially flat FRWspacetime ds2 = −dt2 + a(t)2dx2. Einstein-Hilbert actionSg =
#d4x
√−gR reduces to
Sg =:
$dtLg ; Lg = −3aa2
8πG
A generalized coordinate Q := a2 and the conjugatemomentum P = ∂Lg
∂Qwith {Q,P} = 1
Gravitational Hamiltonian
Hg = PQ − Lg = −8πG
3P2%Q
Golam M Hossain 4/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Pillars of Modern PhysicsGeneral RelativityClassical FRW Cosmology
.. Classical FRW dynamics
Hamilton’s equation Q = {Q,Hg + Hm} implies
∂Hg
∂P→ P = − 3a
8πG
Hamiltonian constraint
H = Hg + Hm = −8πG
3P2%
Q + a3ρ = 0
leads to Friedmann equation
3 (a/a)2 = 8πGρ
→ Big Bang singularity, horizon problem, ...
Golam M Hossain 5/18
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IntroductionQuantum Gravity
Loop Quantum CosmologyQuantum Gravity
.. Quantum Gravity?
Gravitation: Is it classical or quantum?
“. . . Because of the intra-atomic movement of electrons, the atom
must radiate not only electromagnetic but also gravitational energy,
if only in minute amounts. Since, in reality, this cannot be the case
in nature, then it appears that the quantum theory must modify not
only Maxwell’s electrodynamics but also the new theory of
gravitation.” – Albert Einstein (1916, p. 696).
Golam M Hossain 6/18
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IntroductionQuantum Gravity
Loop Quantum CosmologyQuantum Gravity
.. Wheeler deWitt Quantum Cosmology
Hamiltonian and Poisson bracket
H = −8πG
3P2%
Q + a3ρ = 0 , {Q,P} = 1
Wheeler deWitt equation:
Schrodinger quantization
{Q,P} = 1 → [Q, P] = i! ; H |ψ⟩ = 0
→ does not resolve singularity problem in general
Golam M Hossain 7/18
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IntroductionQuantum Gravity
Loop Quantum CosmologyQuantum Gravity
.. Wheeler deWitt Quantum Cosmology
Hamiltonian and Poisson bracket
H = −8πG
3P2%
Q + a3ρ = 0 , {Q,P} = 1
Wheeler deWitt equation:
Schrodinger quantization
{Q,P} = 1 → [Q, P] = i! ; H |ψ⟩ = 0
→ does not resolve singularity problem in general
Golam M Hossain 7/18
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IntroductionQuantum Gravity
Loop Quantum CosmologyQuantum Gravity
.. Theories of Quantum Gravity
String Theory:→ Fundamental building blocks of our universe are extendedobjects such as strings, branes→ aims to unify the forces of nature
Loop Quantum Gravity:→ a non-perturbative approach to quantum gravity→ uses a background-independent quantization method known aspolymer quantization or loop quantization
???:→ ???
Golam M Hossain 8/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Basic Variables
Simple Harmonic Oscillator
Hamiltonian and Poisson bracket
H =p2
2m+
1
2mω2x2 , {x , p} = 1 ; x = {x ,H}
Schrodinger quantization:
{x , p} = 1 → [x , p] = i!
Loop (polymer) quantization: Uλ = e iλp
{x ,Uλ} = iλUλ → [x , Uλ] = −!λUλ
λ is a dimension-full parameter.
Golam M Hossain 9/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Basic Variables
Simple Harmonic Oscillator
Hamiltonian and Poisson bracket
H =p2
2m+
1
2mω2x2 , {x , p} = 1 ; x = {x ,H}
Schrodinger quantization:
{x , p} = 1 → [x , p] = i!
Loop (polymer) quantization: Uλ = e iλp
{x ,Uλ} = iλUλ → [x , Uλ] = −!λUλ
λ is a dimension-full parameter.
Golam M Hossain 9/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Basic Variables
Simple Harmonic Oscillator
Hamiltonian and Poisson bracket
H =p2
2m+
1
2mω2x2 , {x , p} = 1 ; x = {x ,H}
Schrodinger quantization:
{x , p} = 1 → [x , p] = i!
Loop (polymer) quantization: Uλ = e iλp
{x ,Uλ} = iλUλ → [x , Uλ] = −!λUλ
λ is a dimension-full parameter.
Golam M Hossain 9/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Elementary Operators
The basis states: ψµ(p) = e iµp (µ ∈ R)Ashtekar, Fairhurst, Willis (2002); Halvorson (2001)
Basic actions:
xψµ = i! ∂∂p
e iµp = −!µψµ
Uλψµ = e iλpe iµp = ψµ+λ
In Dirac notation:
x |µ⟩ = −!µ|µ⟩ ; Uλ|µ⟩ = |µ+ λ⟩
Golam M Hossain 10/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Elementary Operators
The basis states: ψµ(p) = e iµp (µ ∈ R)Ashtekar, Fairhurst, Willis (2002); Halvorson (2001)
Basic actions:
xψµ = i! ∂∂p
e iµp = −!µψµ
Uλψµ = e iλpe iµp = ψµ+λ
In Dirac notation:
x |µ⟩ = −!µ|µ⟩ ; Uλ|µ⟩ = |µ+ λ⟩
Golam M Hossain 10/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Elementary Operators
The basis states: ψµ(p) = e iµp (µ ∈ R)Ashtekar, Fairhurst, Willis (2002); Halvorson (2001)
Basic actions:
xψµ = i! ∂∂p
e iµp = −!µψµ
Uλψµ = e iλpe iµp = ψµ+λ
In Dirac notation:
x |µ⟩ = −!µ|µ⟩ ; Uλ|µ⟩ = |µ+ λ⟩
Golam M Hossain 10/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Inner Product
Inner product:
!ψµ′ ,ψµ
":= lim
T→∞
1
2T
$ T
−Tdpψ∗
µ′ψµ
⟨µ′|µ⟩ = δµ′,µ
→ rhs is the Kronecker delta→ even position eigenstates are normalizable.→ position eigenvalues are “discrete”
Golam M Hossain 11/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Momentum Operator
How to define momentum operator?
One possible way to define p could be to use classical relationUλ = e iλp as
p = −i
&dUλ
dλ
'
λ=0
Inner product ⟨µ′|µ⟩ = δµ′,µ implies
limλ→0
⟨µ|Uλ|µ⟩ = limλ→0
⟨µ|µ+ λ⟩ = 0 = ⟨µ|Uλ=0|µ⟩ = 1
→ p does not exist
Golam M Hossain 12/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Momentum Operator
How to define momentum operator?
One possible way to define p could be to use classical relationUλ = e iλp as
p = −i
&dUλ
dλ
'
λ=0
Inner product ⟨µ′|µ⟩ = δµ′,µ implies
limλ→0
⟨µ|Uλ|µ⟩ = limλ→0
⟨µ|µ+ λ⟩ = 0
= ⟨µ|Uλ=0|µ⟩ = 1
→ p does not exist
Golam M Hossain 12/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Momentum Operator
How to define momentum operator?
One possible way to define p could be to use classical relationUλ = e iλp as
p = −i
&dUλ
dλ
'
λ=0
Inner product ⟨µ′|µ⟩ = δµ′,µ implies
limλ→0
⟨µ|Uλ|µ⟩ = limλ→0
⟨µ|µ+ λ⟩ = 0 = ⟨µ|Uλ=0|µ⟩ = 1
→ p does not exist
Golam M Hossain 12/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Momentum Operator
How to define momentum operator?
One possible way to define p could be to use classical relationUλ = e iλp as
p = −i
&dUλ
dλ
'
λ=0
Inner product ⟨µ′|µ⟩ = δµ′,µ implies
limλ→0
⟨µ|Uλ|µ⟩ = limλ→0
⟨µ|µ+ λ⟩ = 0 = ⟨µ|Uλ=0|µ⟩ = 1
→ p does not exist
Golam M Hossain 12/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Actions of elementary operators
Golam M Hossain 13/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Actions of elementary operators
| x >
| x + λ>
λ |x> = |x+ λ>
PolymerQuantization
SchrodingerQuantization
U^ ^
P |x> = −− d−hi dx |x>
Golam M Hossain 13/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Polymer Momentum Operator
Using the classical relation Uλ = e iλp one can define
p⋆ :=1
2iλ⋆
(Uλ⋆ − U†
λ⋆
)
In the limit λ⋆ → 0, p⋆ → p
In polymer quantization, this limit doesn’t exist. So λ⋆ istaken to be a small but finite scale
Golam M Hossain 14/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Energy Spectrum of SHO
Energy eigenvalue equation: Hψ = EψHossain, Husain, Seahra (2010)
For small g limit
E2n ≈ E2n+1
=
*+n +
1
2
,−O (g)
-ω
Golam M Hossain 15/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Loop Quantum Cosmology
Momentum operator
P → P⋆ =(Uλ⋆ − U†
λ⋆
)/2iλ = sin(λP)/λ
Gravitational Hamiltonian operator
Hg → H⋆g = −8πG
3λ2sin(λP)2
%Q
Golam M Hossain 16/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Quantum modified dynamics
Effective Friedmann equation
3 (a/a)2 = 8πGρ
+1− ρ
ρc
,, ρc ∼ ρpl
Ashtekar, Pawlowski and Singh (2006)
→ implies a non-singular, bouncing universeBojowald (2001); Date and Hossain (2004)
→ has built-in super accelerating phase→ can lead to favourable initial conditions for a standardinflationary phase Ashtekar and Sloan (2011)
→ similar result holds even for anisotropic cosmologicalmodels
Golam M Hossain 17/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Open Issues
Quantum dynamics of the inhomogeneous modes in LQC
Understanding relation of LQC dynamics and its embedding infull Loop Quantum Gravity
Thank you
Golam M Hossain 18/18
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IntroductionQuantum Gravity
Loop Quantum Cosmology
Loop QuantizationMomentum OperatorOpen Issues
.. Open Issues
Quantum dynamics of the inhomogeneous modes in LQC
Understanding relation of LQC dynamics and its embedding infull Loop Quantum Gravity
Thank you
Golam M Hossain 18/18