Download Dilaton Quantum Gravity - A Functional

Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Dilaton Quantum Gravity
A Functional Renormalization Group Approach
Tobias Henz
with Jan Martin Pawlowski & Christof Wetterich
Institute for Theoretical Physics, University of Heidelberg
TR33 Winter School, December 2012
Summary and Outlook
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Outline
1
Introduction: Asymptotic Safety & Quantum Einstein Gravity
Nonrenormalizibility and Asymptotic Safety
Beyond Perturbation Theory: Functional Renormalization
Challenges & Open Questions
2
Scalar Tensor Theories and Dilatation Symmetry
Dynamical Constants
Results
3
Summary and Outlook
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Renormalizibility and Quantum Gravity
Einstein-Hilbert action
1
√
dd x g (2Λ − R[gµν ])
Γ [gµν ] =
16πGN
mass dimensions of the couplings
Z
dim(Λ) = 2
dim(GN ) = 2 − d
=⇒ Perturbatively not renormalizable if d > 2.
Summary and Outlook
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Alternative Approaches
Introduction of radically new concepts
(Super) String Theory, Loop Quantum Gravity, ...
What if we give up perturbation theory instead?
The Asymptotic Safety Scenario
Summary and Outlook
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Alternative Approaches
Introduction of radically new concepts
(Super) String Theory, Loop Quantum Gravity, ...
What if we give up perturbation theory instead?
The Asymptotic Safety Scenario
Summary and Outlook
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Asymptotic Safety
Weinberg, 1979
A theory is said to be asymptotically safe if the essential coupling
parameters approach a fixed point as the momentum scale of their
renormalization point goes to infinity.
working translation
Quantum gravity is considered asymptotically safe if the
UV-critical surface is finite dimensional and if the dimensionless
coupling constants cease to increase if the momentum scale k goes
to infinity, but approach an ultraviolet fixed point instead.
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Theory Space and the Functional Renormalization Group
Wetterich Equation
"
#
1
1
∂ t Γk = STr (2)
∂ t Rk ,
2
Γk + Rk
t = log(k/k0 )
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Theory Space and the Functional Renormalization Group
Wetterich Equation
"
#
1
1
∂ t Γk = STr (2)
∂ t Rk ,
2
Γk + Rk
t = log(k/k0 )
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
The Flow Diagram of Quantum Einstein Gravity
Reuter, 1998
Coupling Constants approach a nontrivial UV Fixed Point
=⇒ Prospect of Gravity being Asymptotically Safe
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
The Flow Diagram of Quantum Einstein Gravity
"
0.015
10
gN
A
2
Ib
1
0.2
0.1
C
0
0.45
0.5
0.5
Ia
C
B
D
0
!"
!2
!0.5
0 0.1 0.2
0.3
0.4
0.45
Christiansen, Litim, Pawlowski, Rodigast, 2012
Stable Infrared Scenarios
=⇒ Prospect of UV and IR consistent theory
λ
0.5
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Asymptotically Safe Quantum Gravity
FRG Technicalities
Coupling to SM
Regulator Dependence
Yang Mills Theory
Background Dependence
Background Dependence
Truncation Stability
Asymptotic Freedom
Infrared Limit
Hierarchy Problem
Trajectory UV → IR
MSM ≈ 102 GeV
GR as limiting case?
MPlanck ≈ 1019 GeV
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Dynamical Constants I: Scalar-Tensor Theories
1
Γk [gµν ] =
16πGN,k
⇓
Z
Γk [gµν , χ] =
√
Z
√
dd x g (2Λk − R[gµν ])
⇓
⇓
1
d x g Vk [χ] − Fk [χ]R[gµν ] + gµν ∂ µ χ∂ ν χ
2
d
Assumptions:
O(R) truncation
no wave function renormalization
curved background spacetime & optimized cut-offs
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Dynamical Constants I: Scalar-Tensor Theories
1
Γk [gµν ] =
16πGN,k
⇓
Z
Γk [gµν , χ] =
√
Z
√
dd x g (2Λk − R[gµν ])
⇓
⇓
1
d x g Vk [χ] − Fk [χ]R[gµν ] + gµν ∂ µ χ∂ ν χ
2
d
Assumptions:
O(R) truncation
no wave function renormalization
curved background spacetime & optimized cut-offs
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Dynamical Constants II: Dilatation Symmetry
Dilatations ↔ Conformal Transformations
gµν (x ) 7→ Ω(x )gµν (x )
with Ω = const.
Dilatation ↔ global resetting of the physical scale
Dilatation Symmetry ↔ Physical Scale is introduced only by
expectation value of the scalar field
Arising Goldstone Boson: Dilaton
Dilatation Symmetric Actions
Γ is invariant under dilatations
m
all couplings have scaling dimension 0.
(d = 4 : F ∝ χ2 , V ∝ χ4 )
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Limiting Cases
dimensionless field χ̃ (d = 4 : χ̃ = k −2 χ)
Ultraviolet: χ̃ → 0
χ→0
Infrared: χ̃ → ∞
χ→∞
k→0
k→∞
V , F power series in
Percacci, 2009
χ̃2
V , F power series in χ̃−2
current project, 2012
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
The Infrared Limit I: Expansions
Dilatation Symmetric Couplings have vanishing β-functions
Closed set of flow equations to each order in χ̃−2
Fixed point value only in constant order nontrivial
⇒ stable Einstein-Hilbert infrared limit
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
The Infrared Limit II: Dilatation Symmetry
Dilatation Symmetric Infrared Scenario
V = 0 and F = ξχ2
m
√
1
Γk→0 =
d x g
gµν ∂ µ χ∂ ν χ − ξχ2 R
2
Flow diverges at conformal coupling parameters
Z
d
=⇒ Weyl Anomaly is realized
Introduction: Asymptotic Safety
Current Research: Dilatation Symmetry
Summary and Outlook
Multifaceted evidence for asymptotic safety scenario
Dilatation symmetric infrared limit
Stability surveys
Continuation past asymptotic cases
Thank you very much!
Summary and Outlook