QFT in Quantum Spacetimeadapor/GR20.pdfQFT in Quantum Spacetime A Dapor, J Lewandowski, J Puchta...

QFT in Quantum Spacetime

A Dapor, J Lewandowski, J Puchta

University of Warsaw

Warsaw, 11 July 2013

Phys. Rev. D 87, 104038 (2013)arXiv:1302.3038

A Dapor, J Lewandowski, J Puchta QFT in Quantum Spacetime

outline

1 motivation

2 construction

3 conclusions

outline

1 motivation

2 construction

3 conclusions

motivation

the goal:

framework for QFT on quantum (cosmological) spacetime

point γ on phase space Γ: expansion

γ = γ[0] + εδγ[1] +12ε2δγ[2] + ...

it is dynamical expansion!

⇒ the P-algebra among these variables is messed up.

motivation

the goal:

γ = γ[0] + εδγ[1] +12ε2δγ[2] + ...

motivation

the goal:

γ = γ[0] + εδγ[1] +12ε2δγ[2] + ...

motivation

the goal:

γ = γ[0] + εδγ[1] +12ε2δγ[2] + ...

motivation

an example: Mukhanov-Sasaki variables

Q(k) := φ(k), P(k) := πφ(k) −πα2φ(k)

where {α, πα} = 1 and {φ(k), πφ(k′)} = δk,k′ .

⇒ non-trivial P-algebra between perturbations and background

{α,P(k)} = −κ

2Q(k) , 0

⇒ we need a different idea!

motivation

Q(k) := φ(k), P(k) := πφ(k) −πα2φ(k)

{α,P(k)} = −κ

2Q(k) , 0

motivation

Q(k) := φ(k), P(k) := πφ(k) −πα2φ(k)

{α,P(k)} = −κ

2Q(k) , 0

motivation

Q(k) := φ(k), P(k) := πφ(k) −πα2φ(k)

{α,P(k)} = −κ

2Q(k) , 0

motivation

a solution: replace dynamical expansion

γ = γ[0] + εδγ[1] +12ε2δγ[2] + ...

with kinematical splittingγ = γ(0) + δγ

γ(0) is the homogeneous isotropic component

δγ is the ”rest”

⇒ valid on the full phase space: it corresponds to a choice of coordinate system on Γ.

motivation

γ = γ[0] + εδγ[1] +12ε2δγ[2] + ...

motivation

γ = γ[0] + εδγ[1] +12ε2δγ[2] + ...

outline

1 motivation

2 construction

3 conclusions

full theory

action

∫d4x√−g

R −12

gµν∂µT∂νT]

canonical analysis⇒ Γ = ΓG × ΓT

gµν −→ (qab, πab)

T −→ (T , pT )

plus C and Ca.

full theory

action

∫d4x√−g

R −12

gµν∂µT∂νT]

T −→ (T , pT )

plus C and Ca.

full theory

action

∫d4x√−g

R −12

gµν∂µT∂νT]

T −→ (T , pT )

plus C and Ca.

full theory

action

∫d4x√−g

R −12

gµν∂µT∂νT]

T −→ (T , pT )

plus C and Ca.

kinematical splitting

cauchy surface Σ = T3 ⇒ cosmological coordinates (xa) ∈ [0, 1)3

α = 1

Σd3xqab

)πα = 2e2αδab

d3xπab

T (0) =∫

Σd3xT

p(0)T =

the ”rest” δqab(x) = qab(x) − e2αδab

δπab(x) = πab(x) − πα6 e−2αδab

δT(x) = T(x) − T (0)

δpT (x) = pT (x) − p(0)T

α = 1

Σd3xqab

)πα = 2e2αδab

d3xπab

T (0) =∫

Σd3xT

p(0)T =

δT(x) = T(x) − T (0)

δpT (x) = pT (x) − p(0)T

α = 1

Σd3xqab

)πα = 2e2αδab

d3xπab

T (0) =∫

Σd3xT

p(0)T =

δT(x) = T(x) − T (0)

δpT (x) = pT (x) − p(0)T

α = 1

Σd3xqab

)πα = 2e2αδab

d3xπab

T (0) =∫

Σd3xT

p(0)T =

δT(x) = T(x) − T (0)

δpT (x) = pT (x) − p(0)T

gauge-fixing

F-transform the rest (δγ(x) −→ δγ(k)); and projection of geometry modes in scalar,vector and tensor sectors

{α, πα} = 1, {T (0), p(0)T } = 1

{qm(k), pn(k′)} = δnmδk,k′ , {δT(k), δpT (k′)} = δk,k′

T-expansion of C and Ca (1st order):

C(0), E(k) := C(1)(k), M(k) := kaC(1)a (k), V(k) := vaC(1)

a (k), W(k) := waC(1)a (k)

reduction to ΓC:

C(0) ≈ 0 ⇒ p(0)T =

√κπ2

α/6E(k) ≈ 0, M(k) ≈ 0 ⇒ p1 = p1(γfree), p2 = p2(γfree)V(k) ≈ 0, W(k) ≈ 0 ⇒ p3 = p3(γfree), p4 = p4(γfree)

G-fixing to Γ:T (0) − τ = 0, q1 = q2 = q3 = q4 = 0

gauge-fixing

{α, πα} = 1, {T (0), p(0)T } = 1

a (k), W(k) := waC(1)a (k)

reduction to ΓC:

C(0) ≈ 0 ⇒ p(0)T =

√κπ2

gauge-fixing

{α, πα} = 1, {T (0), p(0)T } = 1

a (k), W(k) := waC(1)a (k)

reduction to ΓC:

C(0) ≈ 0 ⇒ p(0)T =

√κπ2

gauge-fixing

{α, πα} = 1, {T (0), p(0)T } = 1

a (k), W(k) := waC(1)a (k)

reduction to ΓC:

C(0) ≈ 0 ⇒ p(0)T =

√κπ2

gauge-fixing

{α, πα} = 1, {T (0), p(0)T } = 1

a (k), W(k) := waC(1)a (k)

reduction to ΓC:

C(0) ≈ 0 ⇒ p(0)T =

√κπ2

hamiltonian(s)

pull-back (γfree) along the τ-dependent embedding of Γ into ΓC

⇒ τ-dependent coordinates (γτfree) on Γ, with canonical P-algebra

(γfree) and T (0) uniquely define the dynamics:

F(γτfree) ={F(γτfree), h

hP = Hhom −∑

k,0,m=5,6

HGm,k −

∑k,0

hamiltonian(s)

hP = Hhom −∑

k,0,m=5,6

HGm,k −

∑k,0

hamiltonian(s)

hP = Hhom −∑

k,0,m=5,6

HGm,k −

∑k,0

hamiltonian(s)

hP = Hhom −∑

k,0,m=5,6

HGm,k −

∑k,0

quantization

√6κπ2

(δpT (k) −

κπα2δT(k)

e4αk2δT(k)2]

introduce M-S var’s⇒ not a canonical transfdefine creation/annihilation op’s⇒ a2 and (a†)2 terms [Fernandez-Mendez, MenaMarugan, Olmedo]

canonical transformation⇒

HTk = H−1

k + k2a4e−κQ2k Q2

]standard QM: Qk = Qk, Pk = −i∂/∂Qk ⇒H = H0 ⊗Hk = L2(R, dµBohr) ⊗ L2(R, dQk)dynamics:

Ψτ(ν,Qk) = HΨτ(ν,Qk)

H = Hhom −∑k,0

HTk = Hhom −

∑k,0

[−H−1

hom ⊗∂2

∂Q2k

+ k2(H−1/2

hom a4H−1/2hom

)⊗ e−κQ

quantization

√6κπ2

(δpT (k) −

κπα2δT(k)

e4αk2δT(k)2]

HTk = H−1

H = Hhom −∑k,0

HTk = Hhom −

∑k,0

[−H−1

hom ⊗∂2

∂Q2k

+ k2(H−1/2

hom a4H−1/2hom

)⊗ e−κQ

quantization

√6κπ2

(δpT (k) −

κπα2δT(k)

e4αk2δT(k)2]

HTk = H−1

standard QM: Qk = Qk, Pk = −i∂/∂Qk ⇒H = H0 ⊗Hk = L2(R, dµBohr) ⊗ L2(R, dQk)dynamics:

H = Hhom −∑k,0

HTk = Hhom −

∑k,0

[−H−1

hom ⊗∂2

∂Q2k

+ k2(H−1/2

hom a4H−1/2hom

)⊗ e−κQ

quantization

√6κπ2

(δpT (k) −

κπα2δT(k)

e4αk2δT(k)2]

HTk = H−1

]standard QM: Qk = Qk, Pk = −i∂/∂Qk ⇒H = H0 ⊗Hk = L2(R, dµBohr) ⊗ L2(R, dQk)

dynamics:

H = Hhom −∑k,0

HTk = Hhom −

∑k,0

[−H−1

hom ⊗∂2

∂Q2k

+ k2(H−1/2

hom a4H−1/2hom

)⊗ e−κQ

quantization

√6κπ2

(δpT (k) −

κπα2δT(k)

e4αk2δT(k)2]

HTk = H−1

H = Hhom −∑k,0

HTk = Hhom −

∑k,0

[−H−1

hom ⊗∂2

∂Q2k

+ k2(H−1/2

hom a4H−1/2hom

)⊗ e−κQ

outline

1 motivation

2 construction

3 conclusions

conclusions

What was shown:

M-S variables are not good for (canonical) QFT on QS

a good choice of variables exists

quantization can be performed (work in progress)

What comes next

complete the quantization

extend to inflaton field and repeat the analysis of [Agullo, Ashtekar, Nelson] aboutpre-inflationary effects on CMB

dressed metric: if Ψτ(ν,Qk) = Ψ(0)τ (ν) ⊗ ψτ(Qk), then one shows that

iddτψτ(Qk) =

[−〈Ψ(0)

τ |H−1hom|Ψ

(0)τ 〉

∂Q2k

+ k2〈Ψ(0)τ |

(H−1/2

hom a4H−1/2hom

)|Ψ(0)

τ 〉e−κQ2

]ψτ(Qk)

if it exists, find the classical (effective) metric gµν such that

iddτψτ(Qk) = HT

k ψτ(Qk) =12

[−A(g)

∂Q2k

+ k2B(g)e−κQ2k Q2

]ψτ(Qk)

conclusions

What was shown:

What comes next

iddτψτ(Qk) =

[−〈Ψ(0)

τ |H−1hom|Ψ

(0)τ 〉

∂Q2k

+ k2〈Ψ(0)τ |

(H−1/2

hom a4H−1/2hom

)|Ψ(0)

τ 〉e−κQ2

]ψτ(Qk)

iddτψτ(Qk) = HT

k ψτ(Qk) =12

[−A(g)

∂Q2k

]ψτ(Qk)

conclusions

What was shown:

What comes next

iddτψτ(Qk) =

[−〈Ψ(0)

τ |H−1hom|Ψ

(0)τ 〉

∂Q2k

+ k2〈Ψ(0)τ |

(H−1/2

hom a4H−1/2hom

)|Ψ(0)

τ 〉e−κQ2

]ψτ(Qk)

iddτψτ(Qk) = HT

k ψτ(Qk) =12

[−A(g)

∂Q2k

]ψτ(Qk)

Thank you!

QFT in Quantum Spacetimeadapor/GR20.pdfQFT in Quantum Spacetime A Dapor, J Lewandowski, J Puchta...

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