Quantum Dilaton Gravity with Fermions

Quantum Dilaton Gravity with Fermions

René Meyer

Institute for Theoretical Physics University of Leipzig, Germany

March 23rd, 2006

Outline

1 First Order Gravity with Matter First Order Gravity

Fermions

Non-linear gauge theory

2 Quantizing Gravity

Three steps to Quantized Gravity The effective action

3 Matter perturbation theory 1-Loop effects

4-Point Vertices & Virtual Black Holes

4 Conclusions & Outlook

First Order Gravity with Matter

Outline

1 First Order Gravity with Matter First Order Gravity

Fermions

Non-linear gauge theory

2 Quantizing Gravity

Three steps to Quantized Gravity The effective action

3 Matter perturbation theory 1-Loop effects

4-Point Vertices & Virtual Black Holes

4 Conclusions & Outlook

First Order Gravity with Matter First Order Gravity

First Order Gravity

S_FOG= Z

X^a(De)_a+Xdω+ X⁺X⁻U(X) +V(X) Classically equivalent to 2D Dilaton Gravity

Exactly solvable (PσM)

Absolute conserved quantity: dC^(g)=0

C^(g)=e^Q(X)X⁺X⁻+w(X)

Q(X) = RX

U(y)dy,w(X) = RX

e^Q(y)V(y)dy

Local quantum triviality: [W. Kummer, H. Liebl, D. Vassilevich 1996-98]

Γ[<e^a, ω,X^a,X >] =S_FOG[<e^a, ω,X^a,X >]

Review: D. Grumiller, W. Kummer, D. Vassilevich, hep-th/0204253

First Order Gravity with Matter First Order Gravity

Model U(X) V(X)

Schwarzschild −_2X¹ −λ²

SRG (genericD>3) −_(D−2)X^D−3 −λ²X(D−4)/(D−2)

Jackiw-Teitelboim 0 −ΛX

Witten BH/CGHS −_X¹ −2λ²X

(A)dS ground state −_X^a −^B₂X Rindler ground state −_X^a −^B₂X^a

BH attractor 0 −^B₂X⁻¹

All above:ab-family −_X^a −^B₂X^a+b Reissner-Nordström −_2X¹ −λ²+^Q_X² Schwarzschild-(A)dS −_2X¹ −λ²−`X

Katanaev-Volovich α βX²−Λ

KK reduced CS 0 ¹₂X(c−X²) Symmetric kink generic −XΠⁿ_i=1(X²−X_i²) 2D type 0A/0B −_X¹ −2λ²X +^λ_8π^2q2

ESBH ⇒ solved

First Order Gravity with Matter Fermions

Fermions

S_χ= i 2

Z

F(X) (∗e^a)∧(χγ_a←→ dχ) +

Z

H(X)

mχχ+λ(χχ)² Spin connectionωdrops out in D=2.

Exact solutions for chiral fermions exist, but integrability is lost in general.

Absolute conservation law: 0=d C^(g)+C^(m) dC^(m)=e^Q(X)

X⁺δS_χ

δe⁺ +X⁻δS_χ δe⁻

4D minimally coupled Dirac fermions→2 Dirac Fermions in 2D + intertwiner term

First Order Gravity with Matter Non-linear gauge theory

FOG as Non-linear gauge theory

The system is invariant under: Local SO(1,1)γ, diffeosξ^µ Generated by three 1^st class constraintsG_i ≈0, i =1,2,3 Hamiltonian Constraint: H=−

3

P

i=1

qⁱG_i, ^{qi= (ω}0,e⁻₀,e⁺₀)

Symmetry algebra:

{G_i,G⁰_i}^∗=0 {G₁,G_2/3⁰ }^∗=∓G_2/3δ

{G₂,G⁰₃}^∗ =

"

−

3

X

i=1

dV dp_iG_i+

gH⁰−H

FF⁰g⁰·(χχ)

G₁

# δ

withg(χχ) =mχχ+λ(χχ)²,g⁰(χχ) = _∂(χχ)^∂g ,p_i= (ω₁,e₁⁻,e⁺₁).

2^ndclass constraints: Relateχto canonical conjugate momentum

First Order Gravity with Matter Non-linear gauge theory

FOG as Non-linear gauge theory

The system is invariant under: Local SO(1,1)γ, diffeosξ^µ Generated by three 1^st class constraintsG_i ≈0, i =1,2,3 Hamiltonian Constraint: H=−

3

P

i=1

qⁱG_i, ^{qi= (ω}0,e⁻₀,e⁺₀)

Symmetry algebra:

{G_i,G⁰_i}^∗=0 {G₁,G_2/3⁰ }^∗=∓G_2/3δ

{G₂,G⁰₃}^∗ =

"

−

3

X

i=1

dV dp_iG_i+

gH⁰−H

FF⁰g⁰·(χχ)

G₁

# δ

withg(χχ) =mχχ+λ(χχ)²,g⁰(χχ) = _∂(χχ)^∂g ,p_i= (ω₁,e₁⁻,e⁺₁).

2^ndclass constraints: Relateχto canonical conjugate momentum

First Order Gravity with Matter Non-linear gauge theory

FOG as Non-linear gauge theory

The system is invariant under: Local SO(1,1)γ, diffeosξ^µ Generated by three 1^st class constraintsG_i ≈0, i =1,2,3 Hamiltonian Constraint: H=−

3

P

i=1

qⁱG_i, ^{qi= (ω}0,e⁻₀,e⁺₀)

Symmetry algebra:

{G_i,G⁰_i}^∗=0 {G₁,G_2/3⁰ }^∗=∓G_2/3δ

{G₂,G⁰₃}^∗ =

"

−

3

X

i=1

dV dp_iG_i+

gH⁰−H

FF⁰g⁰·(χχ)

G₁

# δ

withg(χχ) =mχχ+λ(χχ)²,g⁰(χχ) = _∂(χχ)^∂g ,p_i= (ω₁,e₁⁻,e⁺₁).

2^ndclass constraints: Relateχto canonical conjugate momentum

First Order Gravity with Matter Non-linear gauge theory

FOG as Non-linear gauge theory

The system is invariant under: Local SO(1,1)γ, diffeosξ^µ Generated by three 1^st class constraintsG_i ≈0, i =1,2,3 Hamiltonian Constraint: H=−

3

P

i=1

qⁱG_i, ^{qi= (ω}0,e⁻₀,e⁺₀)

Symmetry algebra:

{G_i,G⁰_i}^∗=0 {G₁,G_2/3⁰ }^∗=∓G_2/3δ

{G₂,G⁰₃}^∗ =

"

−

3

X

i=1

dV dp_iG_i+

gH⁰−H

FF⁰g⁰·(χχ)

G₁

# δ

withg(χχ) =mχχ+λ(χχ)²,g⁰(χχ) = _∂(χχ)^∂g ,p_i= (ω₁,e₁⁻,e⁺₁).

2^ndclass constraints: Relateχto canonical conjugate momentum

First Order Gravity with Matter Non-linear gauge theory

FOG as Non-linear gauge theory

The system is invariant under: Local SO(1,1)γ, diffeosξ^µ Generated by three 1^st class constraintsG_i ≈0, i =1,2,3 Hamiltonian Constraint: H=−

3

P

i=1

qⁱG_i, ^{qi= (ω}0,e⁻₀,e⁺₀)

Symmetry algebra:

{G_i,G⁰_i}^∗=0 {G₁,G_2/3⁰ }^∗=∓G_2/3δ

{G₂,G⁰₃}^∗ =

"

−

3

X

i=1

dV dp_iG_i+

gH⁰−H

FF⁰g⁰·(χχ)

G₁

# δ

withg(χχ) =mχχ+λ(χχ)²,g⁰(χχ) = _∂(χχ)^∂g ,p_i= (ω₁,e₁⁻,e⁺₁).

2^ndclass constraints: Relateχto canonical conjugate momentum

Quantizing Gravity

Outline

1 First Order Gravity with Matter First Order Gravity

Fermions

Non-linear gauge theory

2 Quantizing Gravity

Three steps to Quantized Gravity The effective action

3 Matter perturbation theory 1-Loop effects

4-Point Vertices & Virtual Black Holes

4 Conclusions & Outlook

Quantizing Gravity Three steps to Quantized Gravity

Three steps to Quantized Gravity

1 Ghosts(cⁱ,p^c_j)for theG_i, BRST chargeΩ =cⁱG_i+¹₂cⁱc^jC_ij^kp_k^c terminates at Yang-Mills level

2 Fix EF gauge(ω₀,e⁻₀,e⁺₀) = (0,1,0):gµν =e⁺₁

0 1

1 2e₁⁻

3 Path integration:

DcⁱDp^c_i → DP_χδ(Φ_α) → D(ω₁,e⁻₁,e₁⁺)→ D(X,X⁺,X⁻) DetM trivial δ(EOM(X,X⁺X⁻)) ( ˆX,Xˆ⁺,Xˆ⁻)

(DetM)⁻¹

∂₀X =j₁+X⁺

∂0X⁺=j₂− i

√

2F(X)(χ^∗₁←→

∂0χ1) (∂0+U(X)X⁺)X⁻=j₃−V(X) + i

√

2F(X)(χ^∗₀←→

∂0χ0) +H(X)g(χχ)

Quantizing Gravity Three steps to Quantized Gravity

Three steps to Quantized Gravity

1 Ghosts(cⁱ,p^c_j)for theG_i, BRST chargeΩ =cⁱG_i+¹₂cⁱc^jC_ij^kp_k^c terminates at Yang-Mills level

2 Fix EF gauge(ω₀,e⁻₀,e⁺₀) = (0,1,0):gµν =e⁺₁

0 1

1 2e₁⁻

3 Path integration:

DcⁱDp^c_i → DP_χδ(Φ_α) → D(ω₁,e⁻₁,e₁⁺)→ D(X,X⁺,X⁻) DetM trivial δ(EOM(X,X⁺X⁻)) ( ˆX,Xˆ⁺,Xˆ⁻)

(DetM)⁻¹

∂₀X =j₁+X⁺

∂0X⁺=j₂− i

√

2F(X)(χ^∗₁←→

∂0χ1) (∂0+U(X)X⁺)X⁻=j₃−V(X) + i

√

2F(X)(χ^∗₀←→

∂0χ0) +H(X)g(χχ)

Quantizing Gravity Three steps to Quantized Gravity

Three steps to Quantized Gravity

1 Ghosts(cⁱ,p^c_j)for theG_i, BRST chargeΩ =cⁱG_i+¹₂cⁱc^jC_ij^kp_k^c terminates at Yang-Mills level

2 Fix EF gauge(ω₀,e⁻₀,e⁺₀) = (0,1,0):gµν =e⁺₁

0 1

1 2e₁⁻

3 Path integration:

DcⁱDp^c_i → DP_χδ(Φ_α) → D(ω₁,e⁻₁,e₁⁺)→ D(X,X⁺,X⁻) DetM trivial δ(EOM(X,X⁺X⁻)) ( ˆX,Xˆ⁺,Xˆ⁻)

(DetM)⁻¹

∂₀X =j₁+X⁺

∂0X⁺=j₂− i

√

2F(X)(χ^∗₁←→

∂0χ1) (∂0+U(X)X⁺)X⁻=j₃−V(X) + i

√

2F(X)(χ^∗₀←→

∂0χ0) +H(X)g(χχ)

Quantizing Gravity Three steps to Quantized Gravity

Three steps to Quantized Gravity

1 Ghosts(cⁱ,p^c_j)for theG_i, BRST chargeΩ =cⁱG_i+¹₂cⁱc^jC_ij^kp_k^c terminates at Yang-Mills level

2 Fix EF gauge(ω₀,e⁻₀,e⁺₀) = (0,1,0):gµν =e⁺₁

0 1

1 2e₁⁻

3 Path integration:

DcⁱDp^c_i → DP_χδ(Φ_α) → D(ω₁,e⁻₁,e₁⁺)→ D(X,X⁺,X⁻) DetM trivial δ(EOM(X,X⁺X⁻)) ( ˆX,Xˆ⁺,Xˆ⁻)

(DetM)⁻¹

∂₀X =j₁+X⁺

∂0X⁺=j₂− i

√

2F(X)(χ^∗₁←→

∂0χ1) (∂0+U(X)X⁺)X⁻=j₃−V(X) + i

√

2F(X)(χ^∗₀←→

∂0χ0) +H(X)g(χχ)

Quantizing Gravity The effective action

The effective action

L_eff = JⁱXˆ_i+ i

√2F( ˆX)(χ^∗₁←→

∂₁χ₁) +e^{Q( ˆX)}

j₃−V( ˆX) +H( ˆX)g(χχ) + i

√2F( ˆX)(χ^∗₀←→

∂₀χ₀)

Nonlocal: ∂₀⁻¹

Includes all backreactions of matter fields on geometry Procedure is background independent

Integration over(ω₀,e⁻₁,e⁺₁)is not restricted

So far: No quantum corrections from matter included, and matter is still off-shell.

Quantizing Gravity The effective action

The effective action

L_eff = JⁱXˆ_i+ i

√2F( ˆX)(χ^∗₁←→

∂₁χ₁) +e^{Q( ˆX)}

j₃−V( ˆX) +H( ˆX)g(χχ) + i

√2F( ˆX)(χ^∗₀←→

∂₀χ₀)

Nonlocal: ∂₀⁻¹

Includes all backreactions of matter fields on geometry Procedure is background independent

Integration over(ω₀,e⁻₁,e⁺₁)is not restricted

So far: No quantum corrections from matter included, and matter is still off-shell.

Quantizing Gravity The effective action

The effective action

L_eff = JⁱXˆ_i+ i

√2F( ˆX)(χ^∗₁←→

∂₁χ₁) +e^{Q( ˆX)}

j₃−V( ˆX) +H( ˆX)g(χχ) + i

√2F( ˆX)(χ^∗₀←→

∂₀χ₀)

Nonlocal: ∂₀⁻¹

Includes all backreactions of matter fields on geometry Procedure is background independent

Integration over(ω₀,e⁻₁,e⁺₁)is not restricted

So far: No quantum corrections from matter included, and matter is still off-shell.

Quantizing Gravity The effective action

The effective action

L_eff = JⁱXˆ_i+ i

√2F( ˆX)(χ^∗₁←→

∂₁χ₁) +e^{Q( ˆX)}

j₃−V( ˆX) +H( ˆX)g(χχ) + i

√2F( ˆX)(χ^∗₀←→

∂₀χ₀)

Nonlocal: ∂₀⁻¹

Includes all backreactions of matter fields on geometry Procedure is background independent

Integration over(ω₀,e⁻₁,e⁺₁)is not restricted

So far: No quantum corrections from matter included, and matter is still off-shell.

Quantizing Gravity The effective action

The effective action

L_eff = JⁱXˆ_i+ i

√2F( ˆX)(χ^∗₁←→

∂₁χ₁) +e^{Q( ˆX)}

j₃−V( ˆX) +H( ˆX)g(χχ) + i

√2F( ˆX)(χ^∗₀←→

∂₀χ₀)

Nonlocal: ∂₀⁻¹

Includes all backreactions of matter fields on geometry Procedure is background independent

Integration over(ω₀,e⁻₁,e⁺₁)is not restricted

So far: No quantum corrections from matter included, and matter is still off-shell.

Matter perturbation theory

Outline

1 First Order Gravity with Matter First Order Gravity

Fermions

Non-linear gauge theory

2 Quantizing Gravity

Three steps to Quantized Gravity The effective action

3 Matter perturbation theory 1-Loop effects

4-Point Vertices & Virtual Black Holes

4 Conclusions & Outlook

Matter perturbation theory 1-Loop effects

1-Loop effects

Expansion: L_eff =L⁽⁰⁾+L⁽²⁾+L_int Kinetic term (m=0,F(X)=1):

L⁽²⁾= i

2E_a^µ[j,J](χγ^a←→

∂µχ)

→effective Background Conformal anomaly:

T_µ^µ= R 24π 1-loop in the matter fields:

WPoly.=−logDetD/ = 1 96π

Z

d²x√

−gR1

∆R

Matter perturbation theory 1-Loop effects

1-Loop effects

Expansion: L_eff =L⁽⁰⁾+L⁽²⁾+L_int Kinetic term (m=0,F(X)=1):

L⁽²⁾= i

2E_a^µ[j,J](χγ^a←→

∂µχ)

→effective Background Conformal anomaly:

T_µ^µ= R 24π 1-loop in the matter fields:

WPoly.=−logDetD/ = 1 96π

Z

d²x√

−gR1

∆R

Matter perturbation theory 1-Loop effects

1-Loop effects

Expansion: L_eff =L⁽⁰⁾+L⁽²⁾+L_int Kinetic term (m=0,F(X)=1):

L⁽²⁾= i

2E_a^µ[j,J](χγ^a←→

∂µχ)

→effective Background Conformal anomaly:

T_µ^µ= R 24π 1-loop in the matter fields:

WPoly.=−logDetD/ = 1 96π

Z

d²x√

−gR1

∆R

Matter perturbation theory 1-Loop effects

1-Loop effects

Expansion: L_eff =L⁽⁰⁾+L⁽²⁾+L_int Kinetic term (m=0,F(X)=1):

L⁽²⁾= i

2E_a^µ[j,J](χγ^a←→

∂µχ)

→effective Background Conformal anomaly:

T_µ^µ= R 24π 1-loop in the matter fields:

WPoly.=−logDetD/ = 1 96π

Z

d²x√

−gR1

∆R

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

For free massless fermions:

AB CD

1) χ^∗₁←→

∂0χ1 χ^∗₁←→

∂0χ1

2) χ^∗₁←→

∂₀χ₁ χ^∗₁←→

∂₁χ₁ 3) χ^∗₁←→

∂₀χ₁ χ^∗₀←→

∂₀χ₀ Nonlocal of formR

d²yR

d²xΘ(y⁰−x⁰)δ(x¹−y¹)V(x,y).

Vanish forx⁰=y⁰.

(1) & (2) same as for a real scalar

(1) & (2) conformally invariant; (3) not conformally invariant, but external legsχ₀also have conformal weight−2

No gravitational interaction ofχ₀with itself in this gauge.

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

For free massless fermions:

AB CD

1) χ^∗₁←→

∂0χ1 χ^∗₁←→

∂0χ1

2) χ^∗₁←→

∂₀χ₁ χ^∗₁←→

∂₁χ₁ 3) χ^∗₁←→

∂₀χ₁ χ^∗₀←→

∂₀χ₀ Nonlocal of formR

d²yR

d²xΘ(y⁰−x⁰)δ(x¹−y¹)V(x,y).

Vanish forx⁰=y⁰.

(1) & (2) same as for a real scalar

(1) & (2) conformally invariant; (3) not conformally invariant, but external legsχ₀also have conformal weight−2

No gravitational interaction ofχ₀with itself in this gauge.

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

For free massless fermions:

AB CD

1) χ^∗₁←→

∂0χ1 χ^∗₁←→

∂0χ1

2) χ^∗₁←→

∂₀χ₁ χ^∗₁←→

∂₁χ₁ 3) χ^∗₁←→

∂₀χ₁ χ^∗₀←→

∂₀χ₀ Nonlocal of formR

d²yR

d²xΘ(y⁰−x⁰)δ(x¹−y¹)V(x,y).

Vanish forx⁰=y⁰.

(1) & (2) same as for a real scalar

(1) & (2) conformally invariant; (3) not conformally invariant, but external legsχ₀also have conformal weight−2

No gravitational interaction ofχ₀with itself in this gauge.

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

For free massless fermions:

AB CD

1) χ^∗₁←→

∂0χ1 χ^∗₁←→

∂0χ1

2) χ^∗₁←→

∂₀χ₁ χ^∗₁←→

∂₁χ₁ 3) χ^∗₁←→

∂₀χ₁ χ^∗₀←→

∂₀χ₀ Nonlocal of formR

d²yR

d²xΘ(y⁰−x⁰)δ(x¹−y¹)V(x,y).

Vanish forx⁰=y⁰.

(1) & (2) same as for a real scalar

(1) & (2) conformally invariant; (3) not conformally invariant, but external legsχ₀also have conformal weight−2

No gravitational interaction ofχ₀with itself in this gauge.

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

For free massless fermions:

AB CD

1) χ^∗₁←→

∂0χ1 χ^∗₁←→

∂0χ1

2) χ^∗₁←→

∂₀χ₁ χ^∗₁←→

∂₁χ₁ 3) χ^∗₁←→

∂₀χ₁ χ^∗₀←→

∂₀χ₀ Nonlocal of formR

d²yR

d²xΘ(y⁰−x⁰)δ(x¹−y¹)V(x,y).

Vanish forx⁰=y⁰.

(1) & (2) same as for a real scalar

(1) & (2) conformally invariant; (3) not conformally invariant, but external legsχ₀also have conformal weight−2

No gravitational interaction ofχ₀with itself in this gauge.

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

For free massless fermions:

AB CD

1) χ^∗₁←→

∂0χ1 χ^∗₁←→

∂0χ1

2) χ^∗₁←→

∂₀χ₁ χ^∗₁←→

∂₁χ₁ 3) χ^∗₁←→

∂₀χ₁ χ^∗₀←→

∂₀χ₀ Nonlocal of formR

d²yR

d²xΘ(y⁰−x⁰)δ(x¹−y¹)V(x,y).

Vanish forx⁰=y⁰.

(1) & (2) same as for a real scalar

(1) & (2) conformally invariant; (3) not conformally invariant, but external legsχ₀also have conformal weight−2

No gravitational interaction ofχ₀with itself in this gauge.

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

For free massless fermions:

AB CD

1) χ^∗₁←→

∂0χ1 χ^∗₁←→

∂0χ1

2) χ^∗₁←→

∂₀χ₁ χ^∗₁←→

∂₁χ₁ 3) χ^∗₁←→

∂₀χ₁ χ^∗₀←→

∂₀χ₀ Nonlocal of formR

d²yR

d²xΘ(y⁰−x⁰)δ(x¹−y¹)V(x,y).

Vanish forx⁰=y⁰.

(1) & (2) same as for a real scalar

(1) & (2) conformally invariant; (3) not conformally invariant, but external legsχ₀also have conformal weight−2

No gravitational interaction ofχ₀with itself in this gauge.

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

Virtual Black Hole

VBH: Intermediary effective geometry in scattering processes encoding the vertices and the classical background

e.g. for Spherical Reduced Gravity:

(ds)²=2 drdu+

1−θ(r_y −r)δ(u−u_y) 2m

r +ar

(du)²

Important: Quantum

Triviality in the geometric sector on matter tree level Generic to all

2D Dilaton Gravity models, and scalars and fermions.

Many Worlds?

Warning: It is virtual!

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

Virtual Black Hole

VBH: Intermediary effective geometry in scattering processes encoding the vertices and the classical background

e.g. for Spherical Reduced Gravity:

(ds)²=2 drdu+

1−θ(r_y −r)δ(u−u_y) 2m

r +ar

(du)²

Important: Quantum

Triviality in the geometric sector on matter tree level Generic to all

2D Dilaton Gravity models, and scalars and fermions.

Many Worlds?

Warning: It is virtual!

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

Virtual Black Hole

VBH: Intermediary effective geometry in scattering processes encoding the vertices and the classical background

e.g. for Spherical Reduced Gravity:

(ds)²=2 drdu+

1−θ(r_y −r)δ(u−u_y) 2m

r +ar

(du)²

Important: Quantum

Triviality in the geometric sector on matter tree level Generic to all

2D Dilaton Gravity models, and scalars and fermions.

Many Worlds?

Warning: It is virtual!

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

Virtual Black Hole

VBH: Intermediary effective geometry in scattering processes encoding the vertices and the classical background

e.g. for Spherical Reduced Gravity:

(ds)²=2 drdu+

1−θ(r_y −r)δ(u−u_y) 2m

r +ar

(du)²

Important: Quantum

Triviality in the geometric sector on matter tree level Generic to all

2D Dilaton Gravity models, and scalars and fermions.

Many Worlds?

Warning: It is virtual!

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

Virtual Black Hole

VBH: Intermediary effective geometry in scattering processes encoding the vertices and the classical background

e.g. for Spherical Reduced Gravity:

(ds)²=2 drdu+

1−θ(r_y −r)δ(u−u_y) 2m

r +ar

(du)²

Important: Quantum

Triviality in the geometric sector on matter tree level Generic to all

2D Dilaton Gravity models, and scalars and fermions.

Many Worlds?

Warning: It is virtual!

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

Virtual Black Hole

VBH: Intermediary effective geometry in scattering processes encoding the vertices and the classical background

e.g. for Spherical Reduced Gravity:

(ds)²=2 drdu+

1−θ(r_y −r)δ(u−u_y) 2m

r +ar

(du)²

Important: Quantum

Triviality in the geometric sector on matter tree level Generic to all

2D Dilaton Gravity models, and scalars and fermions.

Many Worlds?

Warning: It is virtual!

Matter perturbation theory 4-Point Vertices & Virtual Black Holes

Virtual Black Hole

VBH: Intermediary effective geometry in scattering processes encoding the vertices and the classical background

e.g. for Spherical Reduced Gravity:

(ds)²=2 drdu+

1−θ(r_y −r)δ(u−u_y) 2m

r +ar

(du)²

Important: Quantum

Triviality in the geometric sector on matter tree level Generic to all

2D Dilaton Gravity models, and scalars and fermions.

Many Worlds?

Warning: It is virtual!

Conclusions & Outlook

Outline

1 First Order Gravity with Matter First Order Gravity

Fermions

Non-linear gauge theory

2 Quantizing Gravity

Three steps to Quantized Gravity The effective action

3 Matter perturbation theory 1-Loop effects

4-Point Vertices & Virtual Black Holes

4 Conclusions & Outlook

Conclusions & Outlook

Conclusions & Outlook

Conclusions:

Consistent, background independent quantization of geometry, taking into account matter backreactions.

Nontrivial scattering exists. Off-shell geometries resembling Virtual BHs can be found.

Crucial: Eddington-Finkelstein gauge Outlook:

Tree-level S-Matrix, Unitarity, CPT invariance Bosonization in Quantum Gravity?

One-loop effects: Corrections to the specific heat of the Dilaton BH for fermionsScalars: [D. Grumiller, W. Kummer, D. Vassilevich, hep-th/0305036]

Boundaries & Matter (w. D. Grumiller, L. Bergamin, D. Vassilevich)

Conclusions & Outlook

Conclusions & Outlook

Conclusions:

Consistent, background independent quantization of geometry, taking into account matter backreactions.

Nontrivial scattering exists. Off-shell geometries resembling Virtual BHs can be found.

Crucial: Eddington-Finkelstein gauge Outlook:

Tree-level S-Matrix, Unitarity, CPT invariance Bosonization in Quantum Gravity?

One-loop effects: Corrections to the specific heat of the Dilaton BH for fermionsScalars: [D. Grumiller, W. Kummer, D. Vassilevich, hep-th/0305036]

Boundaries & Matter (w. D. Grumiller, L. Bergamin, D. Vassilevich)

Conclusions & Outlook

Conclusions & Outlook

Conclusions:

Consistent, background independent quantization of geometry, taking into account matter backreactions.

Nontrivial scattering exists. Off-shell geometries resembling Virtual BHs can be found.

Crucial: Eddington-Finkelstein gauge Outlook:

Tree-level S-Matrix, Unitarity, CPT invariance Bosonization in Quantum Gravity?

One-loop effects: Corrections to the specific heat of the Dilaton BH for fermionsScalars: [D. Grumiller, W. Kummer, D. Vassilevich, hep-th/0305036]

Boundaries & Matter (w. D. Grumiller, L. Bergamin, D. Vassilevich)

Conclusions & Outlook

Conclusions & Outlook

Conclusions:

Consistent, background independent quantization of geometry, taking into account matter backreactions.

Nontrivial scattering exists. Off-shell geometries resembling Virtual BHs can be found.

Crucial: Eddington-Finkelstein gauge Outlook:

Tree-level S-Matrix, Unitarity, CPT invariance Bosonization in Quantum Gravity?

One-loop effects: Corrections to the specific heat of the Dilaton BH for fermionsScalars: [D. Grumiller, W. Kummer, D. Vassilevich, hep-th/0305036]

Boundaries & Matter (w. D. Grumiller, L. Bergamin, D. Vassilevich)

Conclusions & Outlook

Conclusions & Outlook

Conclusions:

Consistent, background independent quantization of geometry, taking into account matter backreactions.

Nontrivial scattering exists. Off-shell geometries resembling Virtual BHs can be found.

Crucial: Eddington-Finkelstein gauge Outlook:

Tree-level S-Matrix, Unitarity, CPT invariance Bosonization in Quantum Gravity?

One-loop effects: Corrections to the specific heat of the Dilaton BH for fermionsScalars: [D. Grumiller, W. Kummer, D. Vassilevich, hep-th/0305036]

Boundaries & Matter (w. D. Grumiller, L. Bergamin, D. Vassilevich)