Covariant Loop Gravity

Transcription

1 Covariant Loop Gravity carlo rovelli I. objective II. III. IV. history and ideas math definition of the theory V. quantum space VI. extracting physics

2 Covariant Loop Gravity carlo rovelli I. objective II. III. IV. history and ideas math definition of the theory V. quantum space VI. extracting physics Lectures (``Zakopane Lectures in Loop Gravity ) CR: arxiv: Overall view of field (``LQG, the first 25 years ) CR:. arxiv: Main Barrett et al: arxiv:

3 Covariant Loop Gravity carlo rovelli I. objective II. history and ideas III. math IV. definition of the theory V. quantum space VI. extracting physics cfr: Ashtekar lecture (LQG and LQC) Bojowald: LQC Geller: relation hamiltonian-spinfoam Vidotto: spinfoam cosmology Wetterich: discrete path integral Martin-Benito: effective SC Bonzom: Lessons from topological BF Lectures (``Zakopane Lectures in Loop Gravity ) CR: arxiv: Overall view of field (``LQG, the first 25 years ) CR:. arxiv: Main Barrett et al: arxiv:

4 I. loop quantum gravity Aim: i. Study if there is a quantum theory with GR as classical limit (Lorentzian, 4d, coupled to ordinary matter) Understand how to extract physics from this theory

5 I. loop quantum gravity Aim: i. Study if there is a quantum theory with GR as classical limit (Lorentzian, 4d, coupled to ordinary matter) Understand how to extract physics from this theory Results: i. Definition of the dynamics Z C = X Y j f i e d d jf Y Tr[ e f i e ] v Theorem 1: asymptotic limit Theorem II: finiteness Boundary formalism n-point functions spinfoam cosmology quantum spacetime

6 II. history of the main ideas 1957, Misner Wheeler Z = Z Dg e is EH Curvature 1961, Regge Regge calculus truncation of GR on a manifold with d-2 defects 1971 Penrose Spin-geometry spin network a spin network Loop Quantum gravity quantum geometry Spinfoams 2008 Covariant dynamics of LQG (EPRL) 2010 Asymptotic a spinfoam

7 II. summing over geometries 1957, Misner Wheeler Z = Z Dg e is EH Formal manipulation of conventional perturbative expansions Limit of a discretization: cfr Lattice QCD. ( vertex expansion )

8 II. discretizing GR: Regge 2d Curvature Curvature Conical singularity g R Regge geometry : 3d Curvature Flat except on hinges. 4d Curvature Regge results: g R approximates g R determined by lengths Ll Action: S R (g R )= X h h(l l ) Vol h (L l ) Lattice distance drops out! g

9 II. 3d Triads g ab! e i a g ab = e i a e i b e = e a dx a 2 R 3 Spin connection! =! a dx a 2 so(3) Z!(e) : de +! ^ e =0 GR action S[e,!] = e ^ F [!] Regge discretization Connection: Flat so(3) connection modulo gauge on M-D1 Triad: e l = Z l e 2 R 3 L l = e l Canonical variables Connection: Flat 2d so(3) connection modulo gauge on M-D0 Curvature Phase space: =T (SU(2) P )/Gauge Canonical quantization H = L 2 [SU(2) P ]/Gauge e l! Left invariant vector field: r Discreteness of length L 2 l! Casimir L l = j l (j l + 1) = j l + 1 2

10 II. important Do not confuse: - Regge discretization Truncation of the continuum theory - Discreteness of length L l = j l Quantum effect

11 II. the Ponzano-Regge magic 1968 Z = Z Dg e is EH[g] - Regge discretization Z Dg! Z dl l - Ponzano Regge ansatz L l = j l Z dl l! X j l Z = X Y Y - Define d jl {6j} d j =2j +1 j l l v {6j} =Tr[ e i e ] - Theorem (PR, Roberts) {6j} 1 12 V e is Regge+ 4 + e is Regge 4

12 II. moral from 3d i. The Misner-Wheeler Feynman-integral over geometries can be realized by a strikingly simple algebraic expression based on SU(2) representation theory. Z = X j l Y d jl Y Tr[ e i e ] l v It is UV finite SU(2)! SU(2) q i (It is also IR finite Turaev-Viro = cosmological constant) Length is quantized L l = j l + 1 2

13 III. math l t l Graph: ={N,L}, L N N s l n Graph (nodes, links) Two-complex: C = {V,E,F}, E V V, F P (V ) f e v 2-complex C (vertices, edges, faces) Two-complex as the 2-skeleton of a cellular decomposition (any dimension):

14 III. math SU(2) unitary representations: j; mi 2H j, 2j 2 N, m = j,..., j, v m 2H j Intertwiner space: K j1,...,j n =Inv[H j1... H jn ] 3 i m 1...m n i {6j} = i abc i ade i bdf i cef =Tr[ e i e ] j 1 j 2 j 3 j 6 j 4 i i j 5 i Spin foam: Spins on faces Intertwiners on edges j 3 j 6 j 5 Amplitude at vertex j 2 j 1

15 III. math SU(2) unitary representations: SL(2,C) unitary representations: j; mi 2H j k, ; j, mi 2H k, = M j=k,1 H j k,, 2k 2 N, 2 R SU(2) SL(2,C) map: f : H j!h j, j j; mi 7! j, j; j, mi = j, k = j 0 = j f Main property: ~K + ~ L =0 weakly on the image of Boost generator Rotation generator Extend to intertwiner space: f : K j1...j n! K SL(2,C) (j 1, j 1 )...(j n, j n )

16 IV. 4d theory Tetrads g ab! e i a g ab = e i a e i b e = e a dx a 2 R (1,3) Spin connection! =! a dx a 2 sl(2,c)!(e) : de +! ^ e =0 Z S[e,!] = e ^ e ^ F [!] GR action GR Holst action S[e,!] = Z e ^ e ^ F [!]+ 1 Z e ^ e ^ F [!] Canonical variables!, B =(e ^ e) + 1 (e ^ e) B! (K = nb, L = nb ) Gauge n i e i =0 n i =(1, 0, 0, 0) ~K + ~ L =0 Linear simplicity constraint

17 IV. 4d theory Tetrads g ab! e i a g ab = e i a e i b e = e a dx a 2 R (1,3) Spin connection! =! a dx a 2 sl(2,c)!(e) : de +! ^ e =0 Z S[e,!] = e ^ e ^ F [!] GR action Regge discretization Curvature Connection: Flat so(3) connection modulo gauge on M-D2 On faces: f = Z f e ^ e 2 sl(2,c) A f = f Canonical variables Connection: Flat 3d sl(2,c) connection modulo gauge on M-D1 Curvature Phase space: =T (SL(2,C) L )/Gauge Canonical quantization Linear simplicity constraint H = L 2 [SL(2,C) L ]/Gauge Restrict Discreteness of area A 2 l! Casimir A f = p j l (j l + 1) B l! Left invariant vector field: ~K + ~ L =0 H!L 2 [SU(2) L ]/Gauge

18 IV. spinfoam magic 2010 C Two-complex (dual to a cellular decomposition) j f v i e Define d j =2j +1 d jf A(j f,i e ) A(j l,i e )=Tr[ e (f i e )] Z = X j f,i e Y f Y v Theorem : [Barrett, Pereira, Hellmann, Gomes, Dowdall, Fairbairn 2010] A(j f,i e ) N e is Regge + e is Regge [Freidel Conrady 2008, Bianchi, Satz 2006, Magliaro Perini, 2011] Z C! Z Dg e is EH[g]

19 IV. spinfoam magic 2008 C Two-complex (dual to a cellular decomposition) j f v i e Define d j =2j +1 d jf A(j f,i e ) A(j l,i e )=Tr[ e (f i e )] Z = X j f,i e Y f Y v Theorem : A(j f,i e ) N e is Regge + e is Regge Not to take this striking result as a sign we are on the right track would be a bit like believing that God put fossils into the rocks in order to mislead Darwin about the evolution of life. Stefano Auchino

20 Covariant Loop Gravity carlo rovelli I. objective II. III. IV. history and ideas math definition of the theory V. quantum space VI. extracting physics

21 Covariant Loop Gravity carlo rovelli I. objective II. III. IV. history and ideas math definition of the theory V. quantum space VI. extracting physics Lectures (``Zakopane Lectures in Loop Gravity ) CR: arxiv: Overall view of field (``LQG, the first 25 years ) CR:. arxiv: Main Barrett et al: arxiv:

22 I. loop quantum gravity Aim: i. Study if there is a quantum theory with GR as classical limit (Lorentzian, 4d, coupled to ordinary matter) Understand how to extract physics from this theory

23 I. loop quantum gravity Aim: i. Study if there is a quantum theory with GR as classical limit (Lorentzian, 4d, coupled to ordinary matter) Understand how to extract physics from this theory Results: i. Definition of the dynamics Z C = X Y j f i e d d jf Y Tr[ e f i e ] v Theorem 1: asymptotic limit Theorem II: finiteness Boundary formalism n-point functions spinfoam cosmology quantum spacetime

24 III. math SU(2) unitary representations: SL(2,C) unitary representations: j; mi 2H j k, ; j, mi 2H k, = M j=k,1 H j k,, 2k 2 N, 2 R SU(2) SL(2,C) map: f : H j!h j, j j; mi 7! j, j; j, mi = j, k = j 0 = j Main property: ~K + ~ L =0 weakly on the image of f Boost generator Rotation generator Extend to intertwiner space: f : K j1...j n! K SL(2,C) (j 1, j 1 )...(j n, j n )

25 IV. spinfoam magic 2010 C Two-complex (dual to a cellular decomposition) j f v i e Define d d j =2j +1 jf A(j f,i e ) A(j l,i e )=Tr[ e (f i e )] Z C = X j f,i e Y f Y v Theorem : A(j f,i e ) N e is Regge + e is Regge Z C! Z Dg e is EH[g]

26 IV. limits Infinite dof limit Recovery of all degrees of freedom C Very different from QCD: no lattice spacing a, no critical parameter. Semiclassical limit High quantum numbers Large distance limit j!1 Fix the truncation, disregard Planck scale effects Z C semiclassical limit j!1 S GR,C (q) truncation C Regge truncation of GR on C. Z S GR (q) Regime where small - C it is good: 1 p R L Planck Recovering the continuum limit is not taking a short distance scale cut off to zero.

27 II. Regge 2d Curvature Curvature Conical singularity 3d Curvature Regge results: g R approximates g R determined by lengths Ll Action: S R (g R )= X h g h(l l ) Vol h (L l ) 4d Curvature Lattice distance drops out!

28 V. How to extract physics? the boundary formalism Suppose this is defined : Z = Z Dg e is EH Is physics in these quantities? : W (x 1,...,x n )=Z 1 Z Dg g(x 1 )...g(x n ) e is EH No, because of the gauge invariance of the theory. Observability is tricky already in classical General relativity!

29 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2

30 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x

31 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2

32 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2 x 1 (t) =const, x 1 (t) =const. No observable consequence

33 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2 x 1 (t) =const, x 1 (t) =const. rod L(t) =const No observable consequence No observable consequence

34 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2 x 1 (t) =const, x 1 (t) =const. rod L(t) =const No observable consequence No observable consequence Observable relative motion

35 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2 rod x 1 (t) =const, x 1 (t) =const. L(t) =const No observable consequence No observable consequence Observable relative motion

36 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2 rod x 1 (t) =const, x 1 (t) =const. L(t) =const No observable consequence No observable consequence Observable relative motion

37 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2 rod x 1 (t) =const, x 1 (t) =const. L(t) =const No observable consequence No observable consequence Observable relative motion

38 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2 rod x 1 (t) =const, x 1 (t) =const. L(t) =const No observable consequence No observable consequence Observable relative motion

39 V. observables ds 2 = dt 2 (1 + a cos(!(t z))dx 2 (1 a cos(!(t z))dy 2 dz 2 z x m 1 m 2 rod x 1 (t) =const, x 1 (t) =const. L(t) =const No observable consequence No observable consequence i. Observability is tricky in gravitational physics Locality Relative locality Observable relative motion

40 V. boundary formalism: Hamilton function Hamilton function S(q, t, q 0,t 0 )= Z t 0 t dt L(q(t), q(t)) Hamilton s boundary logic : p(q, t, q 0,t 0 )= S(q, t, q0,t 0 ) q (q, q 0 ) t,t 0! (p, p 0 ) t,t 0 Notice also E(q, t, q 0,t 0 )= S(q, t, q 0,t 0 ) t (q, t) on equal footing (q, t, q 0,t 0 ) (p, E, p 0,E 0 ) } q i } p i Parametrized systems q(t)! q( ), t( ) S(q i,q 0 i)= Z 0 d L(q, t, q, ṫ) Dynamics is the relative evolution of a set of variables, not the evolution of these variables in time. Hamilton dynamics captures this relational dynamics.

41 V. boundary formalism: QM and QFT Quantum theory W (q, t, q 0,t 0 )= q e ih(t0 t) q 0 = q, t q 0,t 0 e i ~ S(q,t,q0,t 0 ) = Z q,t,q 0,t 0 Dx(t) e i ~ S[x(t)] Evolution operator Hamilton function! Field theory W [ b, ]= Z b, D e is[ ] ' 0 ' Boundary conditions ' General covariant field theory W [ b, ]=W [ b] For the gravitational theory: ' b gives the geometry of the boundary

42 V. boundary formalism: classical limit and n-point functions Semiclassical limit W [ b]! e i ~G S GR[ b] + correction in ~G Hamilton function of GR Field propagator Particle propagator: 0 ( x 0,t 0 ) (x, t) 0 = 0 ( x 0 ) e ih(t0 t) (x) 0 = Z d d 0 W [,t, 0,t 0 ] ( x) 0 ( x 0 ) 0[ ] 0[ 0 ] } } } = W ( x) 0 ( x 0 ) 0 Field propagator Field insertion Vacuum boundary state

43 V. boundary formalism: Quantum Gravity Two-complex with boundary C =@C i e s l l t l n j f v Graph (nodes, links) Quantum gravity transition amplitudes Z(j l,i e )= X j f,i n Y f d jf Y v A(j f,i e ) d j =2j +1 A(j l,i e )=Tr[ e (f i e )] Z C (j l,i e ) = L 2 [SU(2) L ]

44 II. 3d quantum geometry Curvature Connection: Flat 3d su(2) connection modulo gauge on M-D1 Phase space: =T (SU(2) L )/Gauge State space H = L 2 [SU(2) L ]/Gauge Derivative operator: G ll l l ~L l = {L i l},i=1, 2, 3 where L i (h) d (he t i ) dt t=0 The gauge invariant operator: G ll 0 = L ~ l ~L l 0 satisfies Is precisely the Penrose metric operator on the graph X ~L l =0 l2n X G ll 0 =0 l2n It satisfies 1971 Penrose spin-geometry, and 1897 Minkowski : semiclassical states have a geometrical interpretation as polyhedra. l G ll l A l Polyhedron

45 II. states (3d quantum geometry) area A 2 l = G ll volume V 2 n = 2 9 L l 1 (L l2 L l3 ) Area and volume (A l,v n ) form a complete set of commuting observables basis,j l,v n j l Nodes: discrete quanta of volume ( quanta of space ) with quantum number v n. Links: discrete quanta of area, with quantum number j l. v n Geometry is quantized: (i) (ii) (iii) eigenvalues are discrete the operators do not commute a generic state is a quantum superposition coherent states theory (based on Perelomov 1986 SU(2) coherent state techniques) States in H = L 2 [SU(2) L /SU(2) N ] describe quantum geometries: not quantum states in spacetime but rather quantum states of spacetime Area eigenvalues A =8 ~G p j l (j l + 1)

46 V. boundary formalism: Quantum Gravity Two-complex with boundary C =@C i e s l l t l n j f v Graph (nodes, links) Quantum gravity transition amplitudes Z C (j l,i e )= X j f,i n Y f d jf Y v A(j f,i e ) d j =2j +1 A(j l,i e )=Tr[ e (f i e )] Z C (j l,i e ) = L 2 [SU(2) L ] Z C (h l ) = L 2 [SU(2) L ] Finite in the q-deformed model

47 IV. limits LQG transition amplitudes. Hamilton function of C a Regge truncation of GR on. Z C (h l ) semiclassical limit j!1 S GR,C (q) truncation C Z(h l ) S GR (q) LQG transition amplitudes. Hamilton function of GR. Regime where small - C it is good: 1 p R L Planck Recovering the continuum limit is not taking a short distance scale cut off to zero.

48 III. boundary formalism (i) cosmology. Transition amplitude Hamilton function a 0, ȧ 0 Classical Hamilton function S(a, a 0 )= 2 3 r 3 (a0 3 a 3 ) a, ȧ W (a, a 0 )! e i ~ S(a,a0 ) Z aȧ aȧ 0 (ii) n-point functions. The background enters in the choice of a background boundary state Z G la l b G lc l d 0 Z 0 0 g ab (x)g cd (y) 0 In principle this technique allows generic n-point functions to be computed, and compared with Effective Quantum GR, and corrections to be computed.

49 IV. results r r (i) Cosmology. Starting from Z C (h l ), it is possible to compute the transition amplitude between homogeneous isotropic geometries u r r W (z i,z f )= Z P t (H, G)= X j SO(4) 4 dg i 1 G. i 2 dgf 1 G. f 2 (2j+1)e 2t~j(j+1) tr Y l i P t (H l (z i ),G i 1G i 1 2 ) Y l f P t (H l (z f ),G f 1 Gf 1 2 ) h D (j) (H)Y D (j+,j i ) (G)Y. e i ~ p (a03 a 3) = e i ~ S(a,a0 ) Vidotto s talk Result: The expanding Friedmann dynamics and the DeSitter Hamilton function are recovered [Bianchi Vidotto Krajewski CR 2010]

50 IV. results (ii) Gravitational waves. Starting from Z C (h l ), it is possible to compute the two point function of the metric on a background. The background enters in the choice of a background boundary state This can be computed at first order in the expansion in the number of vertices. W En E a n b Em E c m j d ab, a(n) = Z 5Y dg a + dg a A na i A nb i A nc i A nd i e P ab,± 2j± ab log n ab (g ± a ) 1 g ± b n ba a=1 A na i = j ± na n an (g ± a ) 1 g ± n i n na n an (g ± a ) 1 g ± n n na h µ (x)h (y) = 1 2 x y 2 ( µ + µ µ ). Result: The free graviton propagator is recovered in the Lorentzian theory [Bianchi Magliaro Perini 2009, Ding 2011, Zhang 2011,]

51 IV. results (ii) Scattering. z Result: The Regge n-point function is recovered in the large j limit (euclidean theory) [Zhang, CR 2011]

52 IV. an overview i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite v. Includes a positive cosmological constant (quantum group). Finite. vi. v vi Lorentz covariant Quantum space (Planck scale discreteness) clear picture of quantum geometry Transition amplitudes background independent amplitudes Unification nothing to say

53 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

54 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

55 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

56 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

57 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

58 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

59 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

60 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

61 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

62 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

63 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

64 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

65 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

66 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

67 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

68 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

69 IV. a comparison i. 4 dimensions, Lorentzian quantum GR: fundamental formulation clear, fundamental degrees of freedom clear i Classical limit: 4d GR not a, but strong indications Couples with Standard Model (fermions, YM) compatible with observed world Ultraviolet finite Loops v. Includes a positive cosmological constant (quantum group). Finite. i. fundamental formulation not clear, fundamental degrees of freedom not clear i Strings Classical limit clear: 10d GR not yet clearly compatible with observed world Ultraviolet finite strong indications. not a v. Problems with positive cosmological constant vi. Lorentz covariant v Lorentz covariant v Quantum space (Planck scale discreteness) clear picture of quantum geometry vi Quantum space (Planck scale discreteness) unclear picture of quantum geometry vi Transition amplitudes background independent amplitudes Unification nothing to say Transition amplitudes background independent transition amplitudes not clear (except if the world is AdS, which it is not) Unification beautiful picture

70 IV. truth in advertising Loops main open problems i. Coupling with fermions and YM not yet studied. Higher corrections not yet studied. i Does the cosine term in the action disturbs the classical limit? Does the limit Z C! Z (vertex expansion) converges in any useful sense? v. The absence of IR divergences in the q-deformed theory means that there may be cosmological constants size radiative corrections. Do these interfere with (iii)? vi. Are gauge degrees of freedom sufficiently suppressed at a finite order expansion? v Radiative corrections and scaling.