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

[email protected]

Golam M Hossain 1/18

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Pillars of Modern PhysicsGeneral RelativityClassical FRW Cosmology

.. Pillars of Modern Physics

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.. Hand-full of General Relativity


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.. 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.


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.. 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


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.. 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, ...


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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).


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.. 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


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.. 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

???:→ ???


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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.


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.. 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⟩


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.. 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”


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.. 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


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p = −i

&dUλ

dλ

'

λ=0


limλ→0


⟨µ|µ+ λ⟩ = 0

= ⟨µ|Uλ=0|µ⟩ = 1



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p = −i

&dUλ

dλ

'

λ=0


limλ→0


⟨µ|µ+ λ⟩ = 0 = ⟨µ|Uλ=0|µ⟩ = 1



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.. Actions of elementary operators


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.. Actions of elementary operators

| x >

| x + λ>

λ |x> = |x+ λ>

PolymerQuantization

SchrodingerQuantization

U^ ^

P |x> = −− d−hi dx |x>


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.. 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


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.. 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


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.. 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


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.. 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


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.. Open Issues

Quantum dynamics of the inhomogeneous modes in LQC

Understanding relation of LQC dynamics and its embedding infull Loop Quantum Gravity

Thank you


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