Quantum Field Theory on LQC Bianchi Spacetimes

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Quantum Field Theory on LQC Bianchi Spacetimes

ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

ong>Quantumong> ong>Fieldong> ong>Theoryong>

on

LQC Bianchi Spacetimes

Andrea Dapor, Jerzy Lewandowski, Yaser Tavakoli

Stockholm, 6th July 2012


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

1 Introduction

2 QFT and Effective Geometry

3 Next Order

Outline


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

1 Introduction

2 QFT and Effective Geometry

3 Next Order

Outline


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Deformed mass-shell constraint:

p 2 = E 2



⎢⎣ 1 + ξ

Eα ⎤

⎥⎦

Lorentz Violation

pl


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Deformed mass-shell constraint:

p 2 = E 2



⎢⎣ 1 + ξ

Eα ⎤

⎥⎦

Measured velocity:

v = dE

dp

Lorentz Violation

pl

= 1 − ξ Eα

E α

Pl

Deviation from ”conventional” speed of light (c = 1): GRB?


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Deformed mass-shell constraint:

p 2 = E 2



⎢⎣ 1 + ξ

Eα ⎤

⎥⎦

Measured velocity:

v = dE

dp

Lorentz Violation

pl

= 1 − ξ Eα

E α

Pl

Deviation from ”conventional” speed of light (c = 1): GRB?

QFT on QFRW [Ashtekar, Kaminski, Lewandowski (2009)]


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Deformed mass-shell constraint:

p 2 = E 2



⎢⎣ 1 + ξ

Eα ⎤

⎥⎦

Measured velocity:

v = dE

dp

Lorentz Violation

pl

= 1 − ξ Eα

E α

Pl

Deviation from ”conventional” speed of light (c = 1): GRB?

QFT on QFRW [Ashtekar, Kaminski, Lewandowski (2009)]

• Compare QFT on ECS and semiclassical limit of QFT on QS


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Deformed mass-shell constraint:

p 2 = E 2



⎢⎣ 1 + ξ

Eα ⎤

⎥⎦

Measured velocity:

v = dE

dp

Lorentz Violation

pl

= 1 − ξ Eα

E α

Pl

Deviation from ”conventional” speed of light (c = 1): GRB?

QFT on QFRW [Ashtekar, Kaminski, Lewandowski (2009)]

• Compare QFT on ECS and semiclassical limit of QFT on QS

• Effective metric ¯gµν, independent of k: no L-violation


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Deformed mass-shell constraint:

p 2 = E 2



⎢⎣ 1 + ξ

Eα ⎤

⎥⎦

Measured velocity:

v = dE

dp

Lorentz Violation

pl

= 1 − ξ Eα

E α

Pl

Deviation from ”conventional” speed of light (c = 1): GRB?

QFT on QFRW [Ashtekar, Kaminski, Lewandowski (2009)]

• Compare QFT on ECS and semiclassical limit of QFT on QS

• Effective metric ¯gµν, independent of k: no L-violation

• FRW is conformally flat


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Deformed mass-shell constraint:

p 2 = E 2



⎢⎣ 1 + ξ

Eα ⎤

⎥⎦

Measured velocity:

v = dE

dp

Lorentz Violation

pl

= 1 − ξ Eα

E α

Pl

Deviation from ”conventional” speed of light (c = 1): GRB?

QFT on QFRW [Ashtekar, Kaminski, Lewandowski (2009)]

• Compare QFT on ECS and semiclassical limit of QFT on QS

• Effective metric ¯gµν, independent of k: no L-violation

• FRW is conformally flat

Solution: consider more gravitational dof’s: Bianchi I.


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

ong>Quantumong> Bianchi I Spacetime

gµνdx µ dx ν = −dt 2 +

3

i=1

a 2 i (t)(dxi ) 2


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

ong>Quantumong> Bianchi I Spacetime

gµνdx µ dx ν = −dt 2 +

3

i=1

Ashtekar variables: A a i = ciδ a i and Ei a = piδ i a.

a 2 i (t)(dxi ) 2

p1 = a2a3, p2 = a3a1, p3 = a1a2


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

ong>Quantumong> Bianchi I Spacetime

gµνdx µ dx ν = −dt 2 +

3

i=1

Ashtekar variables: A a i = ciδ a i and Ei a = piδ i a.

a 2 i (t)(dxi ) 2

p1 = a2a3, p2 = a3a1, p3 = a1a2

⇒ Hgeo spanned by pi-eigenstates |λ〉 := |λ1, λ2, λ3〉


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

ong>Quantumong> Bianchi I Spacetime

gµνdx µ dx ν = −dt 2 +

3

i=1

Ashtekar variables: A a i = ciδ a i and Ei a = piδ i a.

a 2 i (t)(dxi ) 2

p1 = a2a3, p2 = a3a1, p3 = a1a2

⇒ Hgeo spanned by pi-eigenstates |λ〉 := |λ1, λ2, λ3〉

−i∂TΨ0(T, λ) = H0Ψ0(T, λ)


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

1 Introduction

2 QFT and Effective Geometry

3 Next Order

Outline


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Real scalar field φ

Real Scalar ong>Fieldong>

Lφ = 1

2 (gµν ∂µφ∂νφ − m 2 φ 2 )


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Real scalar field φ

Mode decomposition:

where Hk is Hamiltonian of a h.o.

Real Scalar ong>Fieldong>

Lφ = 1

2 (gµν ∂µφ∂νφ − m 2 φ 2 )


Hφ(T) = Hk (T)

k∈L


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Single mode k:

QFT on QS

H (k)

kin = Hk ⊗ Hgeo, C (k) = Hk + Cgeo


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Single mode k:

QFT on QS

H (k)

kin = Hk ⊗ Hgeo, C (k) = Hk + Cgeo


−i∂TΨ(T, λ, qk ) = H0 − H 0

1 − 2

1 −

Hk

H 2

0 Ψ(T, λ, qk )


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Single mode k:

QFT on QS

H (k)

kin = Hk ⊗ Hgeo, C (k) = Hk + Cgeo


−i∂TΨ(T, λ, qk ) = H0 − H 0

1 − 2

We want to extract an equation for matter only.

1 −

Hk

H 2

0 Ψ(T, λ, qk )


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Single mode k:

QFT on QS

H (k)

kin = Hk ⊗ Hgeo, C (k) = Hk + Cgeo


−i∂TΨ(T, λ, qk ) = H0 − H 0

1 − 2

We want to extract an equation for matter only.

⇒ Take the ”classical geometry” limit.

1 −

Hk

H 2

0 Ψ(T, λ, qk )


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Single mode k:

QFT on QS

H (k)

kin = Hk ⊗ Hgeo, C (k) = Hk + Cgeo


−i∂TΨ(T, λ, qk ) = H0 − H 0

1 − 2

We want to extract an equation for matter only.

⇒ Take the ”classical geometry” limit.

Test field approximation:

1 −

Hk

H 2

0 Ψ(T, λ, qk )

Ψ(T, λ, qk ) = Ψ0(T, λ) ⊗ ψ(T, qk ), −i∂TΨ0 = H0Ψ0


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Single mode k:

QFT on QS

H (k)

kin = Hk ⊗ Hgeo, C (k) = Hk + Cgeo


−i∂TΨ(T, λ, qk ) = H0 − H 0

1 − 2

We want to extract an equation for matter only.

⇒ Take the ”classical geometry” limit.

Test field approximation:


i∂Tψ = 〈H 0

= 1


⎢⎣ 2

1 −

Hk

H 2

0 Ψ(T, λ, qk )

Ψ(T, λ, qk ) = Ψ0(T, λ) ⊗ ψ(T, qk ), −i∂TΨ0 = H0Ψ0

1 − 2

1 −

Hk

H 2

0 〉 Ψ0(T,λ)

−1 2

〈H 0 〉0p

k +


3

⎜⎝ 〈H

i=1

1 − 2 ψ = 〈H 0

1 1 − 2 −

0 pi H 2

0 〉20 k2 i

Hk (T)H − 1

2

0 〉0ψ =

1

1

− 2

− 2 2

+ 〈H 0 |p1p2p3|H 0 〉0m


⎟⎠ q2 ⎥⎦ ψ

k


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

QFT on ECS

Effective ”classical” spacetime (ECS) of the Bianchi I form:

¯gµνdx µ dx ν = − ¯N 2 dT 2 + |¯p1 ¯p2 ¯p3|

3

i=1

(dx i ) 2

¯p 2 i


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

QFT on ECS

Effective ”classical” spacetime (ECS) of the Bianchi I form:

Single mode k:

¯gµνdx µ dx ν = − ¯N 2 dT 2 + |¯p1 ¯p2 ¯p3|

Hk = L2 (R, dqk )

3

i=1

(dx i ) 2

¯p 2 i


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

QFT on ECS

Effective ”classical” spacetime (ECS) of the Bianchi I form:

Single mode k:

¯gµνdx µ dx ν = − ¯N 2 dT 2 + |¯p1 ¯p2 ¯p3|

Hk = L2 (R, dqk )

3

i=1

(dx i ) 2

¯N

i∂Tψ(T, qk ) =

2 ⎡

⎢⎣

|¯p1 ¯p2 ¯p3|

p2 k +


3

⎜⎝ (¯piki) 2 + |¯p1 ¯p2 ¯p3|m 2


⎟⎠ q2


k ⎥⎦ ψ(T, qk )

i=1

¯p 2 i


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

QFT on QS:

i∂Tψ = 1


⎢⎣ 2

−1 2

〈H 0 〉0p

k +


3

⎜⎝ 〈H

i=1

1 1 − 2 −

0 pi H 2

0 〉20 k2 i

Comparison

1

1

− 2

− 2 2

+ 〈H 0 |p1p2p3|H 0 〉0m


⎟⎠ q2 ⎥⎦ ψ

k


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

QFT on QS:

i∂Tψ = 1


⎢⎣ 2

−1 2

〈H 0 〉0p

k +


3

⎜⎝ 〈H

i=1

1 1 − 2 −

0 pi H 2

0 〉20 k2 i

Comparison

1

1

− 2

− 2 2

+ 〈H 0 |p1p2p3|H 0 〉0m

QFT on ECS:

¯N

i∂Tψ =

2 ⎡

⎢⎣

|¯p1 ¯p2 ¯p3|

p2 k +


3

⎜⎝ (¯piki) 2 + |¯p1 ¯p2 ¯p3|m 2


⎟⎠ q2


k ⎥⎦ ψ

i=1


⎟⎠ q2 ⎥⎦ ψ

k


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

QFT on QS:

i∂Tψ = 1


⎢⎣ 2

−1 2

〈H 0 〉0p

k +


3

⎜⎝ 〈H

i=1

1 1 − 2 −

0 pi H 2

0 〉20 k2 i

Comparison

1

1

− 2

− 2 2

+ 〈H 0 |p1p2p3|H 0 〉0m

QFT on ECS:

¯N

i∂Tψ =

2 ⎡

⎢⎣

|¯p1 ¯p2 ¯p3|

p2 k +


3

⎜⎝ (¯piki) 2 + |¯p1 ¯p2 ¯p3|m 2


⎟⎠ q2


k ⎥⎦ ψ

⇒ effective metric:

¯gµνdx µ dx ν = 〈H −1


0 〉1/2

0

× ⎢⎣ −dT2 +


⎜⎝

i=1

3

〈H

i=1

3

i=1

〈H −1

0

1 − 2

0 p2 i

1 − 2 (T)H 0 〉0

1

1 − 2 〉0〈H 0 p2 i


⎟⎠

1

2

×

1 − 2 (T)H 0 〉0

(dx i ) 2


⎥⎦


⎟⎠ q2 ⎥⎦ ψ

k


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

QFT on QS:

i∂Tψ = 1


⎢⎣ 2

−1 2

〈H 0 〉0p

k +


3

⎜⎝ 〈H

i=1

1 1 − 2 −

0 pi H 2

0 〉20 k2 i

Comparison

1

1

− 2

− 2 2

+ 〈H 0 |p1p2p3|H 0 〉0m

QFT on ECS:

¯N

i∂Tψ =

2 ⎡

⎢⎣

|¯p1 ¯p2 ¯p3|

p2 k +


3

⎜⎝ (¯piki) 2 + |¯p1 ¯p2 ¯p3|m 2


⎟⎠ q2


k ⎥⎦ ψ

⇒ effective metric:

¯gµνdx µ dx ν = 〈H −1


0 〉1/2

0

× ⎢⎣ −dT2 +


⎜⎝

i=1

3

〈H

i=1

3

i=1

〈H −1

0

1 − 2

0 p2 i

1 − 2 (T)H 0 〉0

1

1 − 2 〉0〈H 0 p2 i


⎟⎠

1

2

×

1 − 2 (T)H 0 〉0

(dx i ) 2

Dispersion relation for mode k on the background ¯gµν gives

v = 1


⎥⎦


⎟⎠ q2 ⎥⎦ ψ

k


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

1 Introduction

2 QFT and Effective Geometry

3 Next Order

Outline


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

−i∂tΨ = HΨ = [Tn + He]Ψ

Atomic B-O


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

−i∂tΨ = HΨ = [Tn + He]Ψ

• heavy dof’s: nucleus position, n

Atomic B-O


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

−i∂tΨ = HΨ = [Tn + He]Ψ

• heavy dof’s: nucleus position, n

• light dof’s: electron position, e

Atomic B-O


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

−i∂tΨ = HΨ = [Tn + He]Ψ

• heavy dof’s: nucleus position, n

• light dof’s: electron position, e

Atomic B-O

On the (Coulomb) background, solve the eigenequation for He:

Heχi(e) = ɛi(n)χi(e)


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

−i∂tΨ = HΨ = [Tn + He]Ψ

• heavy dof’s: nucleus position, n

• light dof’s: electron position, e

Atomic B-O

On the (Coulomb) background, solve the eigenequation for He:

Heχi(e) = ɛi(n)χi(e)

Substitute back, and solve the eigenequation for H:


Φα = ϕi,α(n)χi(e), [Tn + ɛi(n)]ϕi,α(n) = Eαϕi,α(n)

i


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

−i∂tΨ = HΨ = [Tn + He]Ψ

• heavy dof’s: nucleus position, n

• light dof’s: electron position, e

Atomic B-O

On the (Coulomb) background, solve the eigenequation for He:

Heχi(e) = ɛi(n)χi(e)

Substitute back, and solve the eigenequation for H:


Φα = ϕi,α(n)χi(e), [Tn + ɛi(n)]ϕi,α(n) = Eαϕi,α(n)

i

The ”corrected” state of the system:


Ψ0 =

α

cαΦ 0 α −→ Ψ =


cαΦα =

α

i,α

cαϕi,αχi


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

Cosmological B-O


1

−i∂T Ψ =

2 H 2 0 −

Hk Ψ


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

• heavy dof’s: geometry, λ

Cosmological B-O


1

−i∂T Ψ =

2 H 2 0 −

Hk Ψ


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

• heavy dof’s: geometry, λ

• light dof’s: matter, qk

Cosmological B-O


1

−i∂T Ψ =

2 H 2 0 −

Hk Ψ


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

• heavy dof’s: geometry, λ

Cosmological B-O


1

−i∂T Ψ =

2 H 2 0 −

Hk Ψ

• light dof’s: matter, qk

On the background Ψ0, solve the eigenequation for Hk :

Hk

χi(qk ) = ɛi(p)χi(qk )


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

• heavy dof’s: geometry, λ

Cosmological B-O


1

−i∂T Ψ =

2 H 2 0 −

Hk Ψ

• light dof’s: matter, qk

On the background Ψ0, solve the eigenequation for Hk :

Substitute back:

Φα =


ϕi,α(λ)χi(qk ),

i

Hk

χi(qk ) = ɛi(p)χi(qk )


1

2 H 2

0 + ɛi(ˆp) ϕi,α(λ) = Eαϕi,α(λ)


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

• heavy dof’s: geometry, λ

Cosmological B-O


1

−i∂T Ψ =

2 H 2 0 −

Hk Ψ

• light dof’s: matter, qk

On the background Ψ0, solve the eigenequation for Hk :

Substitute back:

Φα =


ϕi,α(λ)χi(qk ),

i

The ”corrected” state of the system:


Ψ0 =

α

Hk

χi(qk ) = ɛi(p)χi(qk )

cαΦ 0 α −→ Ψ =


1

2 H 2

0 + ɛi(ˆp) ϕi,α(λ) = Eαϕi,α(λ)


cαΦα =

α

i,α

cαϕi,αχi


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

where

δΨ =


i,α,βα

Ψ = Ψ0 ⊗ ψ + δΨ


〈Φ 0 β |ɛi(ˆp)|Φ 0 α〉

E 0 β − E0 α

Lorentz violation

Φ 0 β χi


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

where

δΨ =


i,α,βα

Ψ = Ψ0 ⊗ ψ + δΨ


〈Φ 0 β |ɛi(ˆp)|Φ 0 α〉

E 0 β − E0 α

Lorentz violation

Φ 0 β χi ∝ k


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

where

δΨ =


i,α,βα

Ψ = Ψ0 ⊗ ψ + δΨ


⇒ ”corrected” effective metric:

〈Φ 0 β |ɛi(ˆp)|Φ 0 α〉

E 0 β − E0 α

Lorentz violation

Φ 0 β χi ∝ k

¯gµνdx µ dx ν = −(1 + ξkℓPl) 2 〈p 2 〉 3/2 dT 2 + 〈p 2 〉


1/2

(dx i ) 2

i


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

where

δΨ =


i,α,βα

Ψ = Ψ0 ⊗ ψ + δΨ


⇒ ”corrected” effective metric:

〈Φ 0 β |ɛi(ˆp)|Φ 0 α〉

E 0 β − E0 α

Lorentz violation

Φ 0 β χi ∝ k

¯gµνdx µ dx ν = −(1 + ξkℓPl) 2 〈p 2 〉 3/2 dT 2 + 〈p 2 〉


1/2

(dx i ) 2

Dispersion relation for mode k on the background ¯gµν gives

v = 1 + ξ

2 kℓPl

i


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

where

δΨ =


i,α,βα

Ψ = Ψ0 ⊗ ψ + δΨ


⇒ ”corrected” effective metric:

〈Φ 0 β |ɛi(ˆp)|Φ 0 α〉

E 0 β − E0 α

Lorentz violation

Φ 0 β χi ∝ k

¯gµνdx µ dx ν = −(1 + ξkℓPl) 2 〈p 2 〉 3/2 dT 2 + 〈p 2 〉


1/2

(dx i ) 2

Dispersion relation for mode k on the background ¯gµν gives

v = 1 + ξ

2 kℓPl

Lorentz violation around E ∼ EPl (GRB bound: ∼ 10 −2 EPl).

i


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions

• first steps toward QFT on Bianchi LQC spacetimes


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions

• first steps toward QFT on Bianchi LQC spacetimes

• concept of ”effective geometry” felt by quanta of matter


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions

• first steps toward QFT on Bianchi LQC spacetimes

• concept of ”effective geometry” felt by quanta of matter

• no L-violation in Bianchi I case at 0th order (test field approx.)


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions

• first steps toward QFT on Bianchi LQC spacetimes

• concept of ”effective geometry” felt by quanta of matter

• no L-violation in Bianchi I case at 0th order (test field approx.)

• possible L-violation when backreaction is taken into account


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions

• first steps toward QFT on Bianchi LQC spacetimes

• concept of ”effective geometry” felt by quanta of matter

• no L-violation in Bianchi I case at 0th order (test field approx.)

• possible L-violation when backreaction is taken into account

What next:


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions

• first steps toward QFT on Bianchi LQC spacetimes

• concept of ”effective geometry” felt by quanta of matter

• no L-violation in Bianchi I case at 0th order (test field approx.)

• possible L-violation when backreaction is taken into account

What next:

• try to include massive fields


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions

• first steps toward QFT on Bianchi LQC spacetimes

• concept of ”effective geometry” felt by quanta of matter

• no L-violation in Bianchi I case at 0th order (test field approx.)

• possible L-violation when backreaction is taken into account

What next:

• try to include massive fields

• refine the QFT part: consider an infinite number of modes


ong>Quantumong> ong>Fieldong>

ong>Theoryong>

on

LQC Bianchi

Spacetimes

Andrea Dapor,

Jerzy

Lewandowski,

Yaser Tavakoli

Introduction

QFT and Effective

Geometry

Next Order

What we saw:

Conclusions

• first steps toward QFT on Bianchi LQC spacetimes

• concept of ”effective geometry” felt by quanta of matter

• no L-violation in Bianchi I case at 0th order (test field approx.)

• possible L-violation when backreaction is taken into account

What next:

• try to include massive fields

• refine the QFT part: consider an infinite number of modes

• check the validity of B-O scheme, and explicitely compute ξ

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