Black Holes within Asymptotic Safety Frank Saueressig Research - - PowerPoint PPT Presentation

Black Holes within Asymptotic Safety

Frank Saueressig

Research Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen

• B. Koch and F

. Saueressig, Class. Quant. Grav. 31 (2014) 015006

• B. Koch and F

. Saueressig, Int. J. Mod. Phys. A29 (2014) 8, 1430011 FFP14, Marseille, July 16, 2014

– p. 1/28

Outline

• Why quantum gravity?
• Asymptotic Safety in a nutshell
• Black holes within Asymptotic Safety
• Summary

– p. 2/28

Motivations for Quantum Gravity

1. internal consistency

Rµν − 1

2 gµν R + Λ gµν

• classical

= 8 π GN Tµν

• quantum

– p. 3/28

Motivations for Quantum Gravity

1. internal consistency

Rµν − 1

2 gµν R + Λ gµν

• classical

= 8 π GN Tµν

• quantum

2. singularities in solutions of Einstein equations

• black hole singularities
• Big Bang singularity

– p. 3/28

Motivations for Quantum Gravity

1. internal consistency

Rµν − 1

2 gµν R + Λ gµν

• classical

= 8 π GN Tµν

• quantum

2. singularities in solutions of Einstein equations

• black hole singularities
• Big Bang singularity

3. cosmological observations:

• small positive cosmological constant
• initial conditions for structure formation

– p. 3/28

Motivations for Quantum Gravity

1. internal consistency

Rµν − 1

2 gµν R + Λ gµν

• classical

= 8 π GN Tµν

• quantum

2. singularities in solutions of Einstein equations

• black hole singularities
• Big Bang singularity

3. cosmological observations:

• small positive cosmological constant
• initial conditions for structure formation

General Relativity is incomplete Quantum Gravity may give better answers to these puzzles

– p. 3/28

The quantum gravity landscape

a) Treat gravity as effective field theory:

[J. Donoghue, gr-qc/9405057]

• compute corrections in E2/M2

Pl ≪ 1 (independent of UV-completion)

• breaks down at E2 ≈ M2

Pl

– p. 4/28

The quantum gravity landscape

a) Treat gravity as effective field theory:

[J. Donoghue, gr-qc/9405057]

• compute corrections in E2/M2

Pl ≪ 1 (independent of UV-completion)

• breaks down at E2 ≈ M2

Pl

b) UV-completion requires new physics:

• string theory:
• requires: supersymmetry, extra dimensions
• loop quantum gravity:
• keeps Einstein-Hilbert action as “fundamental”
• new quantization scheme

– p. 4/28

The quantum gravity landscape

a) Treat gravity as effective field theory:

[J. Donoghue, gr-qc/9405057]

• compute corrections in E2/M2

Pl ≪ 1 (independent of UV-completion)

• breaks down at E2 ≈ M2

Pl

b) UV-completion requires new physics:

• string theory:
• requires: supersymmetry, extra dimensions
• loop quantum gravity:
• keeps Einstein-Hilbert action as “fundamental”
• new quantization scheme

c) Gravity makes sense as Quantum Field Theory:

• UV-completion beyond perturbation theory:

Asymptotic Safety

• UV-completion by relaxing symmetries:

Hoˇ rava-Lifshitz

– p. 4/28

The quantum gravity landscape

a) Treat gravity as effective field theory:

[J. Donoghue, gr-qc/9405057]

• compute corrections in E2/M2

Pl ≪ 1 (independent of UV-completion)

• breaks down at E2 ≈ M2

Pl

b) UV-completion requires new physics:

• string theory:
• requires: supersymmetry, extra dimensions
• loop quantum gravity:
• keeps Einstein-Hilbert action as “fundamental”
• new quantization scheme

c) Gravity makes sense as Quantum Field Theory:

• UV-completion beyond perturbation theory:

Asymptotic Safety

• UV-completion by relaxing symmetries:

Hoˇ rava-Lifshitz

– p. 4/28

UV-completion of gravity within QFT

Central ingredient: fixed point of renormalization group flow

β-functions vanish at fixed point {g∗

i }:

• RG flow can “end” at a fixed point keeping limk→∞ gk = g∗ finite!
• trajectory has no unphysical UV divergences
• well-defined continuum limit
• 2 classes of RG trajectories:
• relevant

=

end at FP in UV

• irrelevant

=

go somewhere else...

• predictive power:
• number of relevant directions

= free parameters (determine experimentally)

[scholarpedia ’13]

– p. 5/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN

– p. 6/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN

– p. 6/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling

– p. 6/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling

– p. 6/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
• fundamental theory: interacting
• non-perturbatively renormalizable field theories

– p. 6/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
• fundamental theory: interacting
• non-perturbatively renormalizable field theories

– p. 6/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
• fundamental theory: interacting
• non-perturbatively renormalizable field theories
• anisotropic Gaussian Fixed Point (aGFP)
• fundamental theory: Hoˇ

rava-Lifshitz gravity

• Lorentz-violating renormalizable field theory

– p. 6/28

Proposals for UV fixed points

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
• fundamental theory: interacting
• non-perturbatively renormalizable field theories
• anisotropic Gaussian Fixed Point (aGFP)
• fundamental theory: Hoˇ

rava-Lifshitz gravity

• Lorentz-violating renormalizable field theory

– p. 6/28

Quantum gravity as quantum field theory

Requirements: a) ultraviolet fixed point

• controls the UV-behavior of the RG-trajectory
• ensures the absence of UV-divergences

– p. 7/28

Quantum gravity as quantum field theory

Requirements: a) ultraviolet fixed point

• controls the UV-behavior of the RG-trajectory
• ensures the absence of UV-divergences

b) finite-dimensional UV-critical surface SUV

• fixing the position of a RG-trajectory in SUV

⇐ ⇒ experimental determination of relevant parameters

• guarantees predictive power

– p. 7/28

Quantum gravity as quantum field theory

Requirements: a) ultraviolet fixed point

• controls the UV-behavior of the RG-trajectory
• ensures the absence of UV-divergences

b) finite-dimensional UV-critical surface SUV

• fixing the position of a RG-trajectory in SUV

⇐ ⇒ experimental determination of relevant parameters

• guarantees predictive power

c) classical limit:

• RG-trajectories have part where GR is good approximation
• recover gravitational physics captured by General Relativity:

(perihelion shift, gravitational lensing, nucleo-synthesis, . . .)

– p. 7/28

Quantum gravity as quantum field theory: Asymptotic Safety

Requirements: a) non-Gaussian fixed point (NGFP)

• controls the UV-behavior of the RG-trajectory
• ensures the absence of UV-divergences

b) finite-dimensional UV-critical surface SUV

• fixing the position of a RG-trajectory in SUV

⇐ ⇒ experimental determination of relevant parameters

• guarantees predictive power

c) classical limit:

• RG-trajectories have part where GR is good approximation
• recover gravitational physics captured by General Relativity:

(perihelion shift, gravitational lensing, nucleo-synthesis, . . .) Quantum Einstein Gravity (QEG)

– p. 8/28

Asymptotic Safety in a nutshell

– p. 9/28

Effective average action Γk for gravity

• C. Wetterich, Phys. Lett. B301 (1993) 90
• M. Reuter, Phys. Rev. D 57 (1998) 971

central idea: integrate out quantum fluctuations shell-by-shell in momentum-space

– p. 10/28

Effective average action Γk for gravity

• C. Wetterich, Phys. Lett. B301 (1993) 90
• M. Reuter, Phys. Rev. D 57 (1998) 971

central idea: integrate out quantum fluctuations shell-by-shell in momentum-space

• scale-dependence governed by functional renormalization group equation

k∂kΓk[h, ¯ g] = 1

2STr

• Γ(2)

k

+ Rk

−1

k∂kRk

• vertices of Γk incorporate quantum-corrections with p2 k2

– p. 10/28

Approximate solutions of the flow equation

approximate Γk by scale-dependent Einstein-Hilbert action:

Γk ≈ 1 16πG(k)

• d4x√g [−R + 2Λ(k)] + Sgf + Sgh
• two running couplings: G(k), Λ(k)

– p. 11/28

Approximate solutions of the flow equation

approximate Γk by scale-dependent Einstein-Hilbert action:

Γk ≈ 1 16πG(k)

• d4x√g [−R + 2Λ(k)] + Sgf + Sgh
• two running couplings: G(k), Λ(k)

explicit β-functions for dimensionless couplings gk := k2G(k) , λk := Λ(k)k−2

• Particular choice of Rk (Litim cutoff)

k∂kgk = (ηN + 2)gk , k∂kλk = − (2 − ηN) λk − gk

2π

• 5

1 1−2 λk − 4 − 5 6 1 1−2λk ηN

• anomalous dimension of Newton’s constant:

ηN = gB1 1 − gB2 B1 =

1 3π

• 5

1 1−2λ − 9 1 (1−2λ)2 − 7

• , B2 = −

1 12π

• 5

1 1−2λ + 6 1 (1−2λ)2

• – p. 11/28

Einstein-Hilbert-truncation: the phase diagram

• M. Reuter and F. Saueressig, Phys. Rev. D 65 (2002) 065016 [hep-th/0110054]

−0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −0.75 −0.5 −0.25 0.25 0.5 0.75 1 λ

g

Type IIIa Type Ia Type IIa Type Ib Type IIIb

– p. 12/28

Connecting the quantum and classical regimes

• M. Reuter, H. Weyer, JCAP 0412 (2004) 001, hep-th/0410119

identify RG trajectory realized in Nature by measurement of GN, Λ ? P

1

P

2

0.5 g λ <<1

−70

10 T

• NGFP:

quantum regime (G(k) = k−2g∗, Λ(k) = k2λ∗)

• T:

flow passes extremely close to GFP

• P1 → P2:

classical regime (G(k) = const, Λ(k) = const)

• λ 1/2:

IR fixed point?

– p. 13/28

Charting the RG-flow of the R2-truncation

• O. Lauscher, M. Reuter, Phys. Rev. D66 (2002) 025026, hep-th/0205062
• S. Rechenberger, F.S., Phys. Rev. D86 (2012) 024018, arXiv:1206.0657

Extending Einstein-Hilbert truncation with higher-derivative couplings

Γgrav

k

[g] =

• d4x√g
• 1

16πGk (−R + 2Λk) + 1 bk R2

• 0.5

0.25 0.25 0.5

Λ

0.2 0.4 0.6 0.8 1

g

200 400

b

B A

– p. 14/28

Charting the theory space spanned by Γgrav

k

[g]

. . .

R8 . . . R7 . . . R6 . . . R5 . . . R4 . . . R3 Cµν ρσCρσκλCκλµν R R

+ 7 more

R2 CµνρσCµνρσ RµνRµν R

✎☞ ✍✌ ✒ ✑ ✛ ✚✙ ✏ ✑ ✓

Einstein-Hilbert truncation polynomial f(R)-truncation

R2 + C2-truncation

– p. 15/28

key results: Asymptotic Safety

pure gravity:

• evidence for Asymptotic Safety

⇒

non-Gaussian fixed point provides UV completion of gravity

• low number of relevant parameter:

⇒

dimensionality of UV-critical surface ≃ 3

[ R. Percacci and A. Codello, arXiv:0705.1769] [ P .F. Machado and F. Saueressig, arXiv:0712.0445] [ D. Benedetti, P .F. Machado and F. Saueressig, arXiv:0901.2984]

– p. 16/28

key results: Asymptotic Safety

pure gravity:

• evidence for Asymptotic Safety

⇒

non-Gaussian fixed point provides UV completion of gravity

• low number of relevant parameter:

⇒

dimensionality of UV-critical surface ≃ 3

[ R. Percacci and A. Codello, arXiv:0705.1769] [ P .F. Machado and F. Saueressig, arXiv:0712.0445] [ D. Benedetti, P .F. Machado and F. Saueressig, arXiv:0901.2984]

gravity coupled to matter:

• gravity + scalars: asymptotic safety survives 1-loop counterterm

[ D. Benedetti, P .F. Machado and F. Saueressig, arXiv:0902.4630]

• non-Gaussian fixed point compatible with standard-model matter

[ R. Percacci and D. Perini, hep-th/0207033] [ P . Dona, A. Eichhorn and R. Percacci, arXiv:1311.2898]

• prediction of the Higgs mass mH ≃ 126 GeV

[ M. Shaposhnikov and C. Wetterich, arXiv:0912.0208]

– p. 16/28

Black holes in Asymptotic Safety

– p. 17/28

Classical black hole solutions with cosmological constant

Einstein’s equations in vacuum

Rµν − 1

2 gµν R + Λ gµν = 0

black holes: spherical symmetric, static solutions

ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2

2

f(r) = 1 − 2GM r − 1 3 Λr2

– p. 18/28

Classical black hole solutions with cosmological constant

Einstein’s equations in vacuum

Rµν − 1

2 gµν R + Λ gµν = 0

black holes: spherical symmetric, static solutions

ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2

2

f(r) = 1 − 2GM r − 1 3 Λr2

horizons

• Λ ≤ 0

: black hole horizon rbh

• Λ > 0, M < (3G

√ Λ)−1

: black hole + cosmological horizon rbh < rcosmo

Λ > 0, M ≥ (3G √ Λ)−1

: naked singularity horizon temperature:

T = 1 4π ∂f(r) ∂r

• r=rhorizon

– p. 18/28

Quantum physics from average action Γk

Γk provides effective description of physics at scale k

capture quantum effects by “RG-improvement” scheme:

• transition: classical SEH → average action Γk[g]
• ne-parameter family of effective actions valid at different scales
• k-dependence captures quantum corrections

– p. 19/28

Quantum physics from average action Γk

Γk provides effective description of physics at scale k

capture quantum effects by “RG-improvement” scheme:

• transition: classical SEH → average action Γk[g]
• ne-parameter family of effective actions valid at different scales
• k-dependence captures quantum corrections

extracting physics information from Γk:

• single-scale problem may allow for “cutoff-identification”:
• based on physical intuition:

express RG-scale k through physical cutoff ξ

⇒

modification of classical system by quantum effects

– p. 19/28

Practical RG-improvement schemes

given: physically motivated cutoff-identification k = k(ξ) 1. improved classical solutions

• solve classical equations of motion
• solutions: replace GN −

→ G(k(ξ))

– p. 20/28

Practical RG-improvement schemes

given: physically motivated cutoff-identification k = k(ξ) 1. improved classical solutions

• solve classical equations of motion
• solutions: replace GN −

→ G(k(ξ))

2. improved classical equations of motion

• compute equations of motion from classical action
• equations of motion: replace GN −

→ G(k(ξ))

• solve RG-improved equations of motion

– p. 20/28

Practical RG-improvement schemes

given: physically motivated cutoff-identification k = k(ξ) 1. improved classical solutions

• solve classical equations of motion
• solutions: replace GN −

→ G(k(ξ))

2. improved classical equations of motion

• compute equations of motion from classical action
• equations of motion: replace GN −

→ G(k(ξ))

• solve RG-improved equations of motion

3. improved average action

• Γk: replace GN −

→ G(k(ξ)) k2 ∝ R − → Einstein-Hilbert action → f(R)-gravity theory

• compute modified equations of motion
• solve modified equations of motion

– p. 20/28

Practical RG-improvement schemes

given: physically motivated cutoff-identification k = k(ξ) 1. improved classical solutions

• solve classical equations of motion
• solutions: replace GN −

→ G(k(ξ))

2. improved classical equations of motion

• compute equations of motion from classical action
• equations of motion: replace GN −

→ G(k(ξ))

• solve RG-improved equations of motion

3. improved average action

• Γk: replace GN −

→ G(k(ξ)) k2 ∝ R − → Einstein-Hilbert action → f(R)-gravity theory

• compute modified equations of motion
• solve modified equations of motion

– p. 20/28

Cutoff identification for black holes

[A. Bonanno, M. Reuter, gr-qc/9811026] [A. Bonanno, M. Reuter, hep-th/0002196] [K. Falls, D. F. Litim, A. Raghuraman, arXiv:1002.0260]

requirements for cutoff-identification k = k(physical scale)

• invariance under coordinate transformations
• respect symmetries of solution
• “reasonable” asymptotic behavior

– p. 21/28

Cutoff identification for black holes

[A. Bonanno, M. Reuter, gr-qc/9811026] [A. Bonanno, M. Reuter, hep-th/0002196] [K. Falls, D. F. Litim, A. Raghuraman, arXiv:1002.0260]

requirements for cutoff-identification k = k(physical scale)

• invariance under coordinate transformations
• respect symmetries of solution
• “reasonable” asymptotic behavior

proposal

k(P) = ξ d(P) , d(P) =

• Cr
• |ds2|
• results compatible with improved e.o.m and action scheme

short distance behavior

k(r) = 3ξ 2 √ 2GM r−3/2 (1 + O(r))

• full function k(r) can be found numerically

– p. 21/28

High-energy behavior of RG-improved Schwarzschild black holes

• classical line element

f(r) = 1 − 2 G0 M r

– p. 22/28

High-energy behavior of RG-improved Schwarzschild black holes

• classical line element

f(r) = 1 − 2 G0 M r

• RG-improvement: couplings become scale-dependent

f(r) = 1 − 2 G(k) M r

– p. 22/28

High-energy behavior of RG-improved Schwarzschild black holes

• classical line element

f(r) = 1 − 2 G0 M r

• RG-improvement: couplings become scale-dependent

f(r) = 1 − 2 G(k) M r

• substitute UV-scaling: G(k) = g∗ k−2

f∗(r) = 1 − 2 g∗ k−2 M r

– p. 22/28

High-energy behavior of RG-improved Schwarzschild black holes

• classical line element

f(r) = 1 − 2 G0 M r

• RG-improvement: couplings become scale-dependent

f(r) = 1 − 2 G(k) M r

• substitute UV-scaling: G(k) = g∗ k−2

f∗(r) = 1 − 2 g∗ k−2 M r

• substitute the cutoff-identification k2 ∝ r−3:

f∗(r) = 1 − 1 3

• 4g∗

3G0ξ2

• r2

RG improvement resolves black hole singularity

– p. 22/28

Asymptotically Safe black holes and Planck stars

• S. Hayward, gr-qc/0506126
• C. Rovelli and F. Vidotto, arXiv:1401.6562

Loop quantum gravity: modifications of f(r) due to quantum gravitational repulsion:

f(r) = 1 − 2mr2 r3 + 2α2m

• α: constant determined from fundamental theory

Asymptotics of solution:

f(r) =

  

1 − α−2r2 , r ≪ 2α2m 1 − 2m

r

+ . . . r ≫ 2α2m

• quantum gravitational repulsion resolves black hole singularity
• asymptotics agree with classical Schwarzschild solution

– p. 23/28

Asymptotically Safe black holes and Planck stars

• S. Hayward, gr-qc/0506126
• C. Rovelli and F. Vidotto, arXiv:1401.6562

Loop quantum gravity: modifications of f(r) due to quantum gravitational repulsion:

f(r) = 1 − 2mr2 r3 + 2α2m

• α: constant determined from fundamental theory

Asymptotics of solution:

f(r) =

  

1 − α−2r2 , r ≪ 2α2m 1 − 2m

r

+ . . . r ≫ 2α2m

• quantum gravitational repulsion resolves black hole singularity
• asymptotics agree with classical Schwarzschild solution

Same behavior has RG improved black hole!

– p. 23/28

RG-improved black holes including a cosmological constant

• classical line element

f(r) = 1 − 2 G0 M r − 1 3 Λ0 r2

– p. 24/28

RG-improved black holes including a cosmological constant

• classical line element

f(r) = 1 − 2 G0 M r − 1 3 Λ0 r2

• RG-improvement: couplings become scale-dependent

f(r) = 1 − 2 G(k) M r − 1 3 Λ(k) r2

– p. 24/28

RG-improved black holes including a cosmological constant

• classical line element

f(r) = 1 − 2 G0 M r − 1 3 Λ0 r2

• RG-improvement: couplings become scale-dependent

f(r) = 1 − 2 G(k) M r − 1 3 Λ(k) r2

• substitute UV-scaling: G(k) = g∗ k−2, Λ(k) = λ∗ k2

f∗(r) = 1 − 2 g∗ k−2 M r − 1 3 λ∗ k2 r2

– p. 24/28

RG-improved black holes including a cosmological constant

• classical line element

f(r) = 1 − 2 G0 M r − 1 3 Λ0 r2

• RG-improvement: couplings become scale-dependent

f(r) = 1 − 2 G(k) M r − 1 3 Λ(k) r2

• substitute UV-scaling: G(k) = g∗ k−2, Λ(k) = λ∗ k2

f∗(r) = 1 − 2 g∗ k−2 M r − 1 3 λ∗ k2 r2

• substitute the cutoff-identification k2 ∝ r−3:

f∗(r) = 1 − 2 M r

3

4 G0λ∗ξ2

• − 1

3

• 4g∗

3G0ξ2

• r2

Microscopic black hole is classical Schwarzschild de Sitter solution

– p. 24/28

Temperature of RG-improved Schwarzschild black holes

2 4 6 8 10 m 0.01 0.02 0.03 0.04 0.05 T

classical Schwarzschild black hole RG-improved without cosmological constant

[A. Bonanno, M. Reuter, hep-th/0002196]

RG-improved including Λk with Λ0 = 0

• Λk crucially influences structure of light black holes

Inclusion of Λk prevents remnant formation

– p. 25/28

Temperature of asymptotic (Anti-) de Sitter black holes

2 4 6 8 10 m 0.005 0.010 0.015 0.020 0.025 0.030 T

AdS black hole Λ0 = −0.001 Schwarzschild black hole Λ0 = 0 dS black hole Λ0 = 0.001

• black holes evaporate completely
• non-Gaussian fixed point controls universal short-distance properties

– p. 26/28

Summary

– p. 27/28

Asymptotic Safety Program

Gravitational RG flows:

• strong evidence for a non-Gaussian fixed point:
• predictive: finite number of relevant parameters
• connected to classical gravity

– p. 28/28

Asymptotic Safety Program

Gravitational RG flows:

• strong evidence for a non-Gaussian fixed point:
• predictive: finite number of relevant parameters
• connected to classical gravity

Asymptotically Safe black holes:

• RG improved Schwarzschild black holes
• black hole singularity replaced by de Sitter patch
• formation of black hole remnants
• RG improved black holes including cosmological constant
• microscopic structure: Schwarzschild-de Sitter black hole
• no formation of black hole remnants
• quantum singularity related to dynamical dimensional reduction?

– p. 28/28