Gravity in two dimensions

Transcription

1 Gravity in two dimensions Daniel Grumiller Institute for Theoretical Physics Vienna University of Technology Université catholique de Louvain, Belgium, April 2011

2 Outline Why lower-dimensional gravity? Which 2D theory? Quantum dilaton gravity with matter D. Grumiller Gravity in two dimensions 2/26

3 Outline Why lower-dimensional gravity? Which 2D theory? Quantum dilaton gravity with matter D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 3/26

4 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

5 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

6 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

7 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of black hole entropy D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

8 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of black hole entropy Conceptual insight information loss problem essentially resolved D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

9 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of black hole entropy Conceptual insight information loss problem essentially resolved There is a lot we still do not know about quantum gravity D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

10 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of black hole entropy Conceptual insight information loss problem essentially resolved There is a lot we still do not know about quantum gravity Reasonable alternatives to string theory? D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

11 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of black hole entropy Conceptual insight information loss problem essentially resolved There is a lot we still do not know about quantum gravity Reasonable alternatives to string theory? Cosmological constant problem? Gravity at large distances? D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

12 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of black hole entropy Conceptual insight information loss problem essentially resolved There is a lot we still do not know about quantum gravity Reasonable alternatives to string theory? Cosmological constant problem? Gravity at large distances? Full history of evaporating quantum black hole? D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

13 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of black hole entropy Conceptual insight information loss problem essentially resolved There is a lot we still do not know about quantum gravity Reasonable alternatives to string theory? Cosmological constant problem? Gravity at large distances? Full history of evaporating quantum black hole? Experimental signatures? Data? D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 4/26

14 Gravity in lower dimensions Riemann-tensor D2 (D 2 1) 12 components in D dimensions: 11D: 1210 (1144 Weyl and 66 Ricci) 10D: 825 (770 Weyl and 55 Ricci) 5D: 50 (35 Weyl and 15 Ricci) 4D: 20 (10 Weyl and 10 Ricci) D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 5/26

15 Gravity in lower dimensions Riemann-tensor D2 (D 2 1) 12 components in D dimensions: 11D: 1210 (1144 Weyl and 66 Ricci) 10D: 825 (770 Weyl and 55 Ricci) 5D: 50 (35 Weyl and 15 Ricci) 4D: 20 (10 Weyl and 10 Ricci) 3D: 6 (Ricci) 2D: 1 (Ricci scalar) D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 5/26

16 Gravity in lower dimensions Riemann-tensor D2 (D 2 1) 12 components in D dimensions: 11D: 1210 (1144 Weyl and 66 Ricci) 10D: 825 (770 Weyl and 55 Ricci) 5D: 50 (35 Weyl and 15 Ricci) 4D: 20 (10 Weyl and 10 Ricci) 3D: 6 (Ricci) 2D: 1 (Ricci scalar) 2D: lowest dimension exhibiting black holes (BHs) Simplest gravitational theories with BHs in 2D D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 5/26

17 Gravity in lower dimensions Riemann-tensor D2 (D 2 1) 12 components in D dimensions: 11D: 1210 (1144 Weyl and 66 Ricci) 10D: 825 (770 Weyl and 55 Ricci) 5D: 50 (35 Weyl and 15 Ricci) 4D: 20 (10 Weyl and 10 Ricci) 3D: 6 (Ricci) 2D: 1 (Ricci scalar) 2D: lowest dimension exhibiting black holes (BHs) Simplest gravitational theories with BHs in 2D 3D: lowest dimension exhibiting BHs and gravitons Simplest gravitational theories with BHs and gravitons in 3D D. Grumiller Gravity in two dimensions Why lower-dimensional gravity? 5/26

18 Outline Why lower-dimensional gravity? Which 2D theory? Quantum dilaton gravity with matter D. Grumiller Gravity in two dimensions Which 2D theory? 6/26

19 Attempt 1: Einstein Hilbert in and near two dimensions Let us start with the simplest attempt. Einstein-Hilbert action in 2 dimensions: I EH = 1 d 2 x g R = 1 (1 γ) 16π G 2G D. Grumiller Gravity in two dimensions Which 2D theory? 7/26

20 Attempt 1: Einstein Hilbert in and near two dimensions Let us start with the simplest attempt. Einstein-Hilbert action in 2 dimensions: I EH = 1 d 2 x g R = 1 (1 γ) 16π G 2G Action is topological No equations of motion Formal counting of number of gravitons: -1 [in D dimensions Einstein gravity has D(D 3)/2 graviton polarizations] D. Grumiller Gravity in two dimensions Which 2D theory? 7/26

21 Attempt 1: Einstein Hilbert in and near two dimensions Let us continue with the next simplest attempt. Einstein-Hilbert action in 2+ɛ dimensions: I ɛ 1 EH = d 2+ɛ x g R 16π G(ɛ) D. Grumiller Gravity in two dimensions Which 2D theory? 7/26

22 Attempt 1: Einstein Hilbert in and near two dimensions Let us continue with the next simplest attempt. Einstein-Hilbert action in 2+ɛ dimensions: I ɛ 1 EH = d 2+ɛ x g R 16π G(ɛ) Weinberg: theory is asymptotically safe Mann: limit ɛ 0 should be possible and lead to 2D dilaton gravity DG, Jackiw: limit ɛ 0 (with G(ɛ 0) 0) yields Liouville gravity lim I EH ɛ 1 = d 2 x g [ XR ( X) 2 + λe 2X] ɛ 0 16π G 2 D. Grumiller Gravity in two dimensions Which 2D theory? 7/26

23 Attempt 1: Einstein Hilbert in and near two dimensions Let us continue with the next simplest attempt. Einstein-Hilbert action in 2+ɛ dimensions: I ɛ 1 EH = d 2+ɛ x g R 16π G(ɛ) Weinberg: theory is asymptotically safe Mann: limit ɛ 0 should be possible and lead to 2D dilaton gravity DG, Jackiw: limit ɛ 0 (with G(ɛ 0) 0) yields Liouville gravity lim I EH ɛ 1 = d 2 x g [ XR ( X) 2 + λe 2X] ɛ 0 16π G 2 Result of attempt 1: A specific 2D dilaton gravity model D. Grumiller Gravity in two dimensions Which 2D theory? 7/26

24 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ b a P b describes constant curvature gravity in 2D. Algorithm: D. Grumiller Gravity in two dimensions Which 2D theory? 8/26

25 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ b a P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj D. Grumiller Gravity in two dimensions Which 2D theory? 8/26

26 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X D. Grumiller Gravity in two dimensions Which 2D theory? 8/26

27 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X Construct non-abelian BF theory [ ] I = X A F A = X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b ΛX D. Grumiller Gravity in two dimensions Which 2D theory? 8/26

28 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X Construct non-abelian BF theory [ ] I = X A F A = X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b ΛX Eliminate X a (Torsion constraint) and ω (Levi-Civita connection) D. Grumiller Gravity in two dimensions Which 2D theory? 8/26

29 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X Construct non-abelian BF theory [ ] I = X A F A = X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b ΛX Eliminate X a (Torsion constraint) and ω (Levi-Civita connection) Obtain the second order action 1 I = d 2 x g X [R Λ] 16π G 2 D. Grumiller Gravity in two dimensions Which 2D theory? 8/26

30 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X Construct non-abelian BF theory [ ] I = X A F A = X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b ΛX Eliminate X a (Torsion constraint) and ω (Levi-Civita connection) Obtain the second order action 1 I = d 2 x g X [R Λ] 16π G 2 Result of attempt 2: A specific 2D dilaton gravity model D. Grumiller Gravity in two dimensions Which 2D theory? 8/26

31 Attempt 3: Dimensional reduction For example: spherical reduction from D dimensions Line element adapted to spherical symmetry: ds 2 = g µν (D) }{{} dx µ dx ν = g αβ (x γ ) dx α dx β }{{} φ 2 (x α ) }{{} dω 2 S D 2, full metric 2D metric surface area D. Grumiller Gravity in two dimensions Which 2D theory? 9/26

32 Attempt 3: Dimensional reduction For example: spherical reduction from D dimensions Line element adapted to spherical symmetry: ds 2 = g µν (D) }{{} dx µ dx ν = g αβ (x γ ) dx α dx β }{{} φ 2 (x α ) }{{} dω 2 S D 2, full metric 2D metric surface area Insert into D-dimensional EH action I EH = κ d D x g (D) R (D) : I EH = κ 2π(D 1)/2 Γ( D 1 2 ) d 2 x g φ D 2[ (D 2)(D 3) ( R + ( φ) 2 φ 2 1 ) ] D. Grumiller Gravity in two dimensions Which 2D theory? 9/26

33 Attempt 3: Dimensional reduction For example: spherical reduction from D dimensions Line element adapted to spherical symmetry: ds 2 = g µν (D) }{{} dx µ dx ν = g αβ (x γ ) dx α dx β }{{} φ 2 (x α ) }{{} dω 2 S D 2, full metric 2D metric surface area Insert into D-dimensional EH action I EH = κ d D x g (D) R (D) : I EH = κ 2π(D 1)/2 Γ( D 1 2 ) d 2 x g φ D 2[ (D 2)(D 3) ( R + ( φ) 2 φ 2 1 ) ] Cosmetic redefinition X (λφ) D 2 : 1 I EH = d 2 x [ g XR + D 3 16π G 2 (D 2)X ( X)2 λ 2 X (D 4)/(D 2)] Result of attempt 3: A specific class of 2D dilaton gravity models D. Grumiller Gravity in two dimensions Which 2D theory? 9/26

34 Attempt 4: Poincare gauge theory and higher power curvature theories Basic idea: since EH is trivial consider f(r) theories or/and theories with torsion or/and theories with non-metricity D. Grumiller Gravity in two dimensions Which 2D theory? 10/26

35 Attempt 4: Poincare gauge theory and higher power curvature theories Basic idea: since EH is trivial consider f(r) theories or/and theories with torsion or/and theories with non-metricity Example: Katanaev-Volovich model (Poincare gauge theory) I KV d 2 x g [ αt 2 + βr 2] Kummer, Schwarz: bring into first order form: [ ] I KV X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b (αx a X a + βx 2 ) Use same algorithm as before to convert into second order action: 1 I KV = d 2 x g [XR + α( X) 2 + βx 2] 16π G 2 D. Grumiller Gravity in two dimensions Which 2D theory? 10/26

36 Attempt 4: Poincare gauge theory and higher power curvature theories Basic idea: since EH is trivial consider f(r) theories or/and theories with torsion or/and theories with non-metricity Example: Katanaev-Volovich model (Poincare gauge theory) I KV d 2 x g [ αt 2 + βr 2] Kummer, Schwarz: bring into first order form: [ ] I KV X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b (αx a X a + βx 2 ) Use same algorithm as before to convert into second order action: 1 I KV = d 2 x g [XR + α( X) 2 + βx 2] 16π G 2 Result of attempt 4: A specific 2D dilaton gravity model D. Grumiller Gravity in two dimensions Which 2D theory? 10/26

37 Attempt 5: Strings in two dimensions Conformal invariance of the σ model I σ d 2 ξ h [ g µν h ij i x µ j x ν + α φr +... ] requires vanishing of β-functions β φ 4b 2 4( φ) φ + R +... βµν g R µν + 2 µ ν φ +... Conditions β φ = βµν g = 0 follow from target space action 1 I target = d 2 x g [XR π G 2 X ( X)2 4b 2] where X = e 2φ D. Grumiller Gravity in two dimensions Which 2D theory? 11/26

38 Attempt 5: Strings in two dimensions Conformal invariance of the σ model I σ d 2 ξ h [ g µν h ij i x µ j x ν + α φr +... ] requires vanishing of β-functions β φ 4b 2 4( φ) φ + R +... βµν g R µν + 2 µ ν φ +... Conditions β φ = βµν g = 0 follow from target space action 1 I target = d 2 x g [XR π G 2 X ( X)2 4b 2] where X = e 2φ Result of attempt 5: A specific 2D dilaton gravity model D. Grumiller Gravity in two dimensions Which 2D theory? 11/26

39 Selected List of Models Black holes in (A)dS, asymptotically flat or arbitrary spaces (Wheeler property) Model U(X) V (X) 1. Schwarzschild (1916) 1 λ 2 2X 2. Jackiw-Teitelboim (1984) 0 ΛX 3. Witten Black Hole (1991) 1 2b 2 X X 4. CGHS (1992) 0 2b 2 5. (A)dS 2 ground state (1994) a BX X 6. Rindler ground state (1996) a BX a X 7. Black Hole attractor (2003) 0 BX 1 8. Spherically reduced gravity (N > 3) N 3 λ 2 X (N 4)/(N 2) (N 2)X 9. All above: ab-family (1997) a BX a+b X 10. Liouville gravity a be αx 11. Reissner-Nordström (1916) 1 2X 12. Schwarzschild-(A)dS 1 2X λ 2 + Q2 X λ 2 lx 13. Katanaev-Volovich (1986) α βx 2 Λ Q 14. BTZ/Achucarro-Ortiz (1993) 0 2 J ΛX X 4X KK reduced CS (2003) 0 X(c 2 X2 ) 16. KK red. conf. flat (2006) 1 tanh (X/2) A sinh X D type 0A string Black Hole 1 2b 2 X + b2 q 2 X 8π 18. exact string Black Hole (2005) lengthy lengthy D. Grumiller Gravity in two dimensions Which 2D theory? 12/26

40 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) D. Grumiller Gravity in two dimensions Which 2D theory? 13/26

41 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R D. Grumiller Gravity in two dimensions Which 2D theory? 13/26

42 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) D. Grumiller Gravity in two dimensions Which 2D theory? 13/26

43 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) Self-interaction potential V (X) leads to non-trivial geometries D. Grumiller Gravity in two dimensions Which 2D theory? 13/26

44 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) Self-interaction potential V (X) leads to non-trivial geometries Gibbons Hawking York boundary term guarantees Dirichlet boundary problem for metric D. Grumiller Gravity in two dimensions Which 2D theory? 13/26

45 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) Self-interaction potential V (X) leads to non-trivial geometries Gibbons Hawking York boundary term guarantees Dirichlet boundary problem for metric Hamilton Jacobi counterterm contains superpotential S(X) S(X) 2 = e X U(y) dy X V (y)e y U(z) dz dy and guarantees well-defined variational principle δi = 0 D. Grumiller Gravity in two dimensions Which 2D theory? 13/26

46 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) Self-interaction potential V (X) leads to non-trivial geometries Gibbons Hawking York boundary term guarantees Dirichlet boundary problem for metric Hamilton Jacobi counterterm contains superpotential S(X) S(X) 2 = e X U(y) dy X V (y)e y U(z) dz dy and guarantees well-defined variational principle δi = 0 Interesting option: couple 2D dilaton gravity to matter D. Grumiller Gravity in two dimensions Which 2D theory? 13/26

47 Outline Why lower-dimensional gravity? Which 2D theory? Quantum dilaton gravity with matter D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 14/26

48 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

49 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

50 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom Consider 2D dilaton gravity with a scalar field D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

51 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom Consider 2D dilaton gravity with a scalar field Do the BRST gymnastics D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

52 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom Consider 2D dilaton gravity with a scalar field Do the BRST gymnastics Integrate out geometry exactly! D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

53 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom Consider 2D dilaton gravity with a scalar field Do the BRST gymnastics Integrate out geometry exactly! Obtain non-local non-polynomial matter action D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

54 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom Consider 2D dilaton gravity with a scalar field Do the BRST gymnastics Integrate out geometry exactly! Obtain non-local non-polynomial matter action Treat matter perturbatively D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

55 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom Consider 2D dilaton gravity with a scalar field Do the BRST gymnastics Integrate out geometry exactly! Obtain non-local non-polynomial matter action Treat matter perturbatively Calculate Feynman rules D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

56 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom Consider 2D dilaton gravity with a scalar field Do the BRST gymnastics Integrate out geometry exactly! Obtain non-local non-polynomial matter action Treat matter perturbatively Calculate Feynman rules Reconstruct intermediate geometry (Virtual BHs) D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

57 Exact path integral quantization of 2D dilaton gravity Overview of the quantization procedure Implement the following algorithm: Matter provides propagating degrees of freedom Consider 2D dilaton gravity with a scalar field Do the BRST gymnastics Integrate out geometry exactly! Obtain non-local non-polynomial matter action Treat matter perturbatively Calculate Feynman rules Reconstruct intermediate geometry (Virtual BHs) Calculate S-matrix or corrections to classical quantities (like specific heat) D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 15/26

58 Non-minimally coupled matter Prominent example: Einstein-massless Klein-Gordon model (Choptuik) no matter: integrability, no scattering, no propagating physical modes with matter: no integrability in general, scattering, critical collapse D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 16/26

59 Non-minimally coupled matter Prominent example: Einstein-massless Klein-Gordon model (Choptuik) no matter: integrability, no scattering, no propagating physical modes with matter: no integrability in general, scattering, critical collapse Massless scalar field S: I (m) = d 2 x gf (X)( S) 2 minimal coupling: F = const. non-minimal coupling otherwise spherical reduction: F X D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 16/26

60 Non-perturbative path integral quantization Integrating out geometry exactly constraint analysis { G i (x), G j (x ) } = G k C k ij δ(x x ) BRST charge Ω = c i G i + c i c j C ij k p k (ghosts c i, p k ) gauge fixing fermion to achieve EF gauge D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 17/26

61 Non-perturbative path integral quantization Integrating out geometry exactly constraint analysis { G i (x), G j (x ) } = G k C k ij δ(x x ) BRST charge Ω = c i G i + c i c j C ij k p k (ghosts c i, p k ) gauge fixing fermion to achieve EF gauge integrating ghost sector yields Z[sources] = ( Dfδ f + iδ/δj e + 1 ) Z[f, sources] with ( S = S f) Z[f, sources] = D SD(ω, e a, X, X a ) det F.P. exp i(i g.f. + sources) Can integrate over all fields except matter non-perturbatively! D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 17/26

62 Non-local effective theory Convert local gravity theory with matter into non-local matter theory without gravity Generating functional for Green functions (F = 1): Z[f, sources] = D S exp i (L k + L v + L s )d 2 x L k = 0 S 1 S E1 ( 0S) 2, L v = w ( ˆX), L s = σs + j e + Ê , 1 D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 18/26

63 Non-local effective theory Convert local gravity theory with matter into non-local matter theory without gravity Generating functional for Green functions (F = 1): Z[f, sources] = D S exp i (L k + L v + L s )d 2 x L k = 0 S 1 S E1 ( 0S) 2, L v = w ( ˆX), L s = σs + j e + Ê , 1 S = Sf 1/2, Ê 1 + = eq( ˆX), ˆX = a + bx 0 }{{} X 0 ( 0S) , a = 0, b = 1, }{{} non local + 2 E 1 = w(x) + M, Ê+ 1 = eq(x) + e Q(X) U(X) 2 0 ( 0S) D S exp i ( L k = exp i/96π x y fr x 1 xy R y ) } {{ } Polyakov Red: geometry, Magenta: matter, Blue: boundary conditions D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 18/26

64 Some Feynman diagrams lowest order non-local vertices: 0 S 0 S 0 S 1 S vertex corrections: q V (4) (x,y) a k q V (4) (x,y) b k x y + x y q k q k 0 S 0 S 0 S 0 S propagator corrections: vacuum bubbles: so far: calculated only lowest order vertices and propagator corrections partial resummations possible (similar to Bethe-Salpeter)? non-local loops vanish to this order D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 19/26

65 S-matrix for s-wave gravitational scattering Quantizing the Einstein-massless-Klein-Gordon model ingoing s-waves q = αe, q = (1 α)e interact and scatter into outgoing s-waves k = βe, k = (1 β)e D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 20/26

66 S-matrix for s-wave gravitational scattering Quantizing the Einstein-massless-Klein-Gordon model ingoing s-waves q = αe, q = (1 α)e interact and scatter into outgoing s-waves k = βe, k = (1 β)e T (q, q ; k, k ) T δ(k + k q q )/ kk qq 3/2 with Π = (k + k )(k q)(k q) and T = Π ln Π2 E Π Plot of cross-section p p 2 ln p2 E 2 ( ) 3kk qq 1 r 2 s 2 2 r p s r,p result finite and simple monomial scaling with E forward scattering poles Π = 0 decay of s-waves possible not understood why so simple! (1a) (1b) (intermediate results vastly more complicated) D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 20/26

67 Other selected successes of (quantum) dilaton gravity with/without matter Gravity as non-abelian gauge theory Jackiw, Teitelboim 84 Black holes in string theory Witten 91 Black hole evaporation Callan, Giddings, Harvey, Strominger 92 Gravity as non-linear gauge theory Ikeda, Izawa 93 Dirac quantization Louis-Martinez, Gegenberg, Kunstatter 94 All classical solutions Klösch, Strobl Virtual black holes DG, Kummer, Vassilevich 00 Unitary S-matrix DG, Kummer, Vassilevich 01 Quantum corrected specific heat DG, Kummer, Vassilevich 03 Liouville Field Theory Nakayama 04 Duality DG, Jackiw 06 Holographic renormalization DG, McNees 07 Central charge in AdS 2 Hartman, Strominger 08 AdS 2 holography Castro, DG, Larsen, McNees 08 Model for gravity at large distances DG 10 Quantization of cosmological constant? Govaerts, Zonetti 11 D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 21/26

68 Summary Dilaton gravity in two dimensions is surprisingly rich! D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 22/26

69 Summary Dilaton gravity in two dimensions is surprisingly rich! Dilaton gravity in two dimensions provides valuable lessons for black hole physics and quantum gravity D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 22/26

70 Summary Dilaton gravity in two dimensions is surprisingly rich! Dilaton gravity in two dimensions provides valuable lessons for black hole physics and quantum gravity Dilaton gravity in two dimensions is also capable of providing insights into gravity at large distances D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 22/26

71 Summary Dilaton gravity in two dimensions is surprisingly rich! Dilaton gravity in two dimensions provides valuable lessons for black hole physics and quantum gravity Dilaton gravity in two dimensions is also capable of providing insights into gravity at large distances... there still may be surprises waiting to be discovered! D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 22/26

72 Summary Thank you for your attention! Dilaton gravity in two dimensions is surprisingly rich! Dilaton gravity in two dimensions provides valuable lessons for black hole physics and quantum gravity Dilaton gravity in two dimensions is also capable of providing insights into gravity at large distances... there still may be surprises waiting to be discovered! D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 22/26

73 Some literature J. D. Brown, LOWER DIMENSIONAL GRAVITY, World Scientific Singapore (1988). D. Grumiller, W. Kummer, and D. Vassilevich, Dilaton gravity in two dimensions, Phys. Rept. 369 (2002) , hep-th/ D. Grumiller, R. Meyer, Ramifications of lineland, Turk. J. Phys. 30 (2006) , hep-th/ D. Grumiller, R. McNees, Thermodynamics of black holes in two (and higher) dimensions, JHEP 0704 (2007) 074, hep-th/ D. Grumiller, R. Jackiw, Liouville gravity from Einstein gravity, J. Govaerts, S. Zonetti, Quantized cosmological constant in 1+1 dimensional quantum gravity with coupled scalar matter, Thanks to Bob McNees for providing the LATEX beamerclass! D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 23/26

74 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

75 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

76 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

77 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

78 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

79 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

80 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive δφ EOM: R = 8 AdS L 2 2! D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

81 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive δφ EOM: R = 8 AdS L 2 2! δa EOM: µ F µν = 0 E = constant D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

82 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive δφ EOM: R = 8 AdS L 2 2! δa EOM: µ F µν = 0 E = constant δg EOM: complicated for non-constant dilaton... µ ν e 2φ g µν 2 e 2φ + 4 L 2 e 2φ g µν + L2 2 F µ λ F νλ L2 8 g µν F 2 = 0 D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

83 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive δφ EOM: R = 8 AdS L 2 2! δa EOM: µ F µν = 0 E = constant δg EOM:...but simple for constant dilaton: e 2φ = L4 4 E2 µ ν e 2φ g µν 2 e 2φ + 4 L 2 e 2φ g µν + L2 2 F µ λ F νλ L2 8 g µν F 2 = 0 D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 24/26

84 Some surprising results Hartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM Holographic renormalization leads to boundary mass term (CGLM) I dx γ ma 2 Nevertheless, total action gauge invariant D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 25/26

85 Some surprising results Hartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM Holographic renormalization leads to boundary mass term (CGLM) I dx γ ma 2 Nevertheless, total action gauge invariant Boundary stress tensor transforms anomalously (HS) (δ ξ + δ λ ) T tt = 2T tt t ξ + ξ t T tt c 24π L 3 t ξ where δ ξ + δ λ is combination of diffeo- and gauge trafos that preserve the boundary conditions (similarly: δ λ J t = k 4π L tλ) D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 25/26

86 Some surprising results Hartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM Holographic renormalization leads to boundary mass term (CGLM) I dx γ ma 2 Nevertheless, total action gauge invariant Boundary stress tensor transforms anomalously (HS) (δ ξ + δ λ ) T tt = 2T tt t ξ + ξ t T tt c 24π L 3 t ξ where δ ξ + δ λ is combination of diffeo- and gauge trafos that preserve the boundary conditions (similarly: δ λ J t = k 4π L tλ) Anomalous transformation above leads to central charge (HS, CGLM) c = 24αe 2φ = 3 G 2 = 3 2 ke2 L 2 D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 25/26

87 Some surprising results Hartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM Holographic renormalization leads to boundary mass term (CGLM) I dx γ ma 2 Nevertheless, total action gauge invariant Boundary stress tensor transforms anomalously (HS) (δ ξ + δ λ ) T tt = 2T tt t ξ + ξ t T tt c 24π L 3 t ξ where δ ξ + δ λ is combination of diffeo- and gauge trafos that preserve the boundary conditions (similarly: δ λ J t = k 4π L tλ) Anomalous transformation above leads to central charge (HS, CGLM) c = 24αe 2φ = 3 G 2 = 3 2 ke2 L 2 Positive central charge only for negative coupling constant α (CGLM) α < 0 D. Grumiller Gravity in two dimensions Quantum dilaton gravity with matter 25/26

88 Virtual black holes Reconstruct geometry from matter i + Intermediate geometry (caveat: off-shell!): y I + i 0 I - i - ds 2 = 2 du dr + [1 δ(u u 0 )θ(r 0 r) (2M/r + ar + d)] du 2 }{{} localized Schwarzschild and Rindler terms nontrivial part localized geometry is non-local (depends on r, u, r 0, u }{{} 0 ) D. Grumiller Gravity in two dimensions Quantum dilaton gravityy with matter 26/26