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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

ghossain@iiserkol.ac.in

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

�.PEFSO�1IZTJDT

�*HQHUDO�5HODWLYLW\���'\QDPLFDO�6SDFHWLPH

���6WDQGDUG�0RGHO��4XDQWXP�)LHOG�7KHRU\

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