The ultraviolet behavior of quantum gravity with fakeons

Transcription

1 The ultraviolet behavior of quantum gravity with fakeons Dipartimento di Fisica Enrico Fermi SW12: Hot Topics in Modern Cosmology Cargèse May 18 th, 2018

2 Outline 1 Introduction 2 Lee-Wick quantum eld theory 3 Superrenormalizable quantum gravity 4 Fakeons and higher-derivative quantum gravity 5 Conclusions

3 The problem of renormalizability in QG

4 The problem of renormalizability in QG Einstein-Hilbert action: unitary but nonrenormalizable theory S EH = 1 2κ 2 d 4 x gr, κ 2 = 8πG. Γ (ct) EH = 1 2κ 2 d 4 x g [ R + c 1 R 2 + c 2 R µνr µν + c 3 R ρσ µν R ρσ ] αβ Rαβ µν }{{}

5 The problem of renormalizability in QG Einstein-Hilbert action: unitary but nonrenormalizable theory S EH = 1 2κ 2 d 4 x gr, κ 2 = 8πG. Γ (ct) EH = 1 2κ 2 d 4 x g [ R + c 1 R 2 + c 2 R µνr µν + c 3 R ρσ µν R ρσ ] αβ Rαβ µν }{{} Stelle action: renormalizable but not unitary theory S HD = 1 2κ 2 d 4 x g [ γr + αr µνr µν + βr 2 ] + 2Λ C. Γ (ct) HD = 1 2κ 2 d 4 x g [ a γr + a αr µνr µν + a β R 2 ] + 2a C.

6 Unitarity S S = 1. i(t T ) = T T, S = 1 + it. Cutting equations (Cutkosky, 't Hooft and Veltman) DiscM = 2iImM = j C j, C j = cut diagrams. T fi = (2π) 4 δ (4) (p i p f )M(i f).

7 In general, higher-derivative theories cannot be dened in Minkowski spacetime. U.G. Aglietti and D. Anselmi, Inconsistency of Minkowski higher-derivative theories, Eur. Phys. J. C 77 (2017) 84 and arxiv: [hep-th]. Inconsistencies: nonlocal and non-hermitian divergences.

8 In general, higher-derivative theories cannot be dened in Minkowski spacetime. U.G. Aglietti and D. Anselmi, Inconsistency of Minkowski higher-derivative theories, Eur. Phys. J. C 77 (2017) 84 and arxiv: [hep-th]. Inconsistencies: nonlocal and non-hermitian divergences. SHD = 1 2κ 2 d 4 x [ g R 1 M 4 (DρRµν )(Dρ R µν ) + 1 ] 2M 4 (DρR)(Dρ R). Propagator in De Donder gauge (expansion g µν = η µν + 2κh µν) im 4 h µν (p) h ρσ( p) 0 = 2(p 2 + iɛ) η µρη νσ + η µση νρ η µν η ρσ (p 2 ) 2 + M 4.

9 In general, higher-derivative theories cannot be dened in Minkowski spacetime. U.G. Aglietti and D. Anselmi, Inconsistency of Minkowski higher-derivative theories, Eur. Phys. J. C 77 (2017) 84 and arxiv: [hep-th]. Inconsistencies: nonlocal and non-hermitian divergences. SHD = 1 2κ 2 d 4 x [ g R 1 M 4 (DρRµν )(Dρ R µν ) + 1 ] 2M 4 (DρR)(Dρ R). Propagator in De Donder gauge (expansion g µν = η µν + 2κh µν) im 4 h µν (p) h ρσ( p) 0 = 2(p 2 + iɛ) h µν(p) h ρσ( p) nld 1 = η µρη νσ + η µση νρ η µν η ρσ (p 2 ) 2 + M 4. κ 2 M 8 [(68r + 240π 2 (p 2 i)(ηµρηνσ + ηνρηµσ) + (373r 4i)ηµνηρσ ) 2 1 8p 2 (125ir r + 8i) (p µp ρη νσ + p µp ση νρ + p νp ρη µσ + p νp ση µρ) + 1 4p 2 (255ir2 1522r + 36i) (p µp νη ρσ + p ρp ση µν) 1 2(p 2 ) 2 (185r3 + 75ir r + 24i)p µp νp ρp σ ] ( ) Λ 2 UV ln, r p 2 /M 2. M 2

10 Higher-derivative theories must be dened from Euclidean space. A dierent formulation has been proposed by Lee and Wick. T.D. Lee and G.C. Wick, Negative metric and the unitarity of the S-matrix, Nucl. Phys. B 9 (1969) 209. T.D. Lee and G.C. Wick, Finite theory of quantum electrodynamics, Phys. Rev. D 2 (1970) R.E. Cutkosky, P.V Landsho, D.I. Olive, J.C. Polkinghorne, A non-analytic S matrix, Nucl.Phys. B12 (1969) B. Grinstein, D. O'Connell and M.B. Wise, Causality as an emergent macroscopic phenomenon: The Lee-Wick O(N) model, Phys. Rev. D 79 (2009) and arxiv: [hep-th].

11 Higher-derivative theories must be dened from Euclidean space. A dierent formulation has been proposed by Lee and Wick. T.D. Lee and G.C. Wick, Negative metric and the unitarity of the S-matrix, Nucl. Phys. B 9 (1969) 209. T.D. Lee and G.C. Wick, Finite theory of quantum electrodynamics, Phys. Rev. D 2 (1970) R.E. Cutkosky, P.V Landsho, D.I. Olive, J.C. Polkinghorne, A non-analytic S matrix, Nucl.Phys. B12 (1969) B. Grinstein, D. O'Connell and M.B. Wise, Causality as an emergent macroscopic phenomenon: The Lee-Wick O(N) model, Phys. Rev. D 79 (2009) and arxiv: [hep-th]. We reformulate the models as nonanalytically Wick rotated Euclidean theories. The new formulation solves the previous problems and makes the models both renormalizable and unitary. D. Anselmi and M. Piva, A new formulation of Lee-Wick quantum eld theory, JHEP 1706 (2017) 066 and arxiv: [hep-th]. D. Anselmi and M. Piva, Perturbative unitarity of Lee-Wick quantum eld theory, Phys. Rev. D 96 (2017) and arxiv: [hep-th].

12 Average continuation Pinching

13 Average continuation Pinching Branch cuts Im[p 0 ] 2M 2M Re[p 0 ]

14 Average continuation Pinching Branch cuts Im[p 0 ] 2M 2M Re[p 0 ] J AV (p) = 1 2 [J +(p) + J (p)]. D. Anselmi and M. Piva, A new formulation of Lee-Wick quantum eld theory, JHEP 1706 (2017) 066, and arxiv: [hep-th]. D. Anselmi, Fakeons and Lee-Wick models, JHEP 02 (2018) 141, and arxiv: [hep-th].

15 Superrenormalizable quantum gravity L QG = [ g 2κ 2 2Λ C M 2 + ζr γ M 2 RµνRµν + 1 (γ η)r2 2M 2 1 M 4 (DρRµν)(Dρ R µν ) + 1 2M 4 (1 ξ)(dρr)(dρ R) + 1 ( α1 M 4 R µνr µρ Rρ ν + α 2 RR µνr µν + α 3 R 3 + α 4 RR µνρσr µνρσ + α 5 R µνρσr µρ R νσ + α 6 R µνρσr ρσαβ R µν αβ )].

16 Superrenormalizable quantum gravity L QG = [ g 2κ 2 2Λ C M 2 + ζr γ M 2 RµνRµν + 1 (γ η)r2 2M 2 1 M 4 (DρRµν)(Dρ R µν ) + 1 2M 4 (1 ξ)(dρr)(dρ R) + 1 ( α1 M 4 R µνr µρ Rρ ν + α 2 RR µνr µν + α 3 R 3 + α 4 RR µνρσr µνρσ + α 5 R µνρσr µρ R νσ + α 6 R µνρσr ρσαβ R µν αβ )]. Expansion g µν = η µν + 2κh µν, Propagator in De Donder gauge η µν = diag(1, 1, 1, 1). h µν(p)h ρσ( p) free η=ξ=0 = im 4 2 η µρη νσ + η µση νρ η µνη ρσ, P (1, γ, ζ, 2Λ C ) P (a, b, c, d) = a(p 2 ) 3 + bm 2 (p 2 ) 2 + cm 4 p 2 + dm 6.

17 L QG = [ g 2κ 2 2Λ C M 2 + ζr γ M 2 RµνRµν + 1 (γ η)r2 2M 2 1 M 4 (DρRµν)(Dρ R µν ) + 1 2M 4 (1 ξ)(dρr)(dρ R) + 1 ( α1 M 4 R µνr µρ Rρ ν + α 2RR µνr µν + α 3 R 3 + α 4 RR µνρσr µνρσ + α 5 R µνρσr µρ R νσ + α 6 R µνρσr ρσαβ R µν αβ Counterterms (up to three loops) g Lcount = [2a (4π) 2 C M 4 + a ζ M 2 R a γr µνr µν + 12 ] ε (aγ aη)r2. )]. Unitarity cannot be proved at non vanishing Λ C. Can be set to zero by solving a chain of RG conditions. D. Anselmi, On the quantum eld theory of gravitational interactions, JHEP 1706 (2017) 086 and arxiv: [hep-th].

18 The problem of uniqueness L QG is the simplest representative of a general class

19 The problem of uniqueness L QG is the simplest representative of a general class L QG = g 2κ 2 [ 2Λ C M 2 + ζr 1 M 2 RµνPn( c/m 2 )R µν P n and Q n polynomials of degree n, c µ µ. + 1 ] 2M 2 RQn( c/m 2 )R + V(R, M, α i ).

20 The problem of uniqueness L QG is the simplest representative of a general class L QG = g 2κ 2 [ 2Λ C M 2 + ζr 1 M 2 RµνPn( c/m 2 )R µν P n and Q n polynomials of degree n, c µ µ. + 1 ] 2M 2 RQn( c/m 2 )R + V(R, M, α i ). All these theories are superrenormalizable and of the Lee-Wick type (for suitable choices of the polynomials). n = 1 counterterms up to three loops. n = 2 counterterms up to two loops. n 3 counterterms up to one loop.

21 The problem of uniqueness L QG is the simplest representative of a general class L QG = g 2κ 2 [ 2Λ C M 2 + ζr 1 M 2 RµνPn( c/m 2 )R µν P n and Q n polynomials of degree n, c µ µ. + 1 ] 2M 2 RQn( c/m 2 )R + V(R, M, α i ). All these theories are superrenormalizable and of the Lee-Wick type (for suitable choices of the polynomials). n = 1 counterterms up to three loops. n = 2 counterterms up to two loops. n 3 counterterms up to one loop. Problem of uniquness Superrenormalizable models of quantum gravity are innitely many.

22 New quantization prescription and fakeons D. Anselmi, On the quantum eld theory of gravitational interactions, JHEP 1706 (2017) 086, and arxiv: [hep-th].

23 New quantization prescription and fakeons D. Anselmi, On the quantum eld theory of gravitational interactions, JHEP 1706 (2017) 086, and arxiv: [hep-th]. Modied Euclidean propagator After the Wick rotation L = 1 2 µϕ µ ϕ λ 4! ϕ4. p 2 E (p 2, E = cticious LW scale. E )2 + E4 p 2 lim E 0 [(p 2 ) 2 + E 4. ] AV

24 New quantization prescription and fakeons D. Anselmi, On the quantum eld theory of gravitational interactions, JHEP 1706 (2017) 086, and arxiv: [hep-th]. Modied Euclidean propagator After the Wick rotation L = 1 2 µϕ µ ϕ λ 4! ϕ4. p 2 E (p 2, E = cticious LW scale. E )2 + E4 p 2 lim E 0 [(p 2 ) 2 + E 4. ] AV One-loop bubble diagram (after renormalizing the UV divergence). Feynman prescription i 2(4π) 2 ln p2 iɛ µ 2. New presciption i 4(4π) 2 ln (p2 ) 2 µ 4. We can turn ghosts into fake degrees of freedom.

25 Quantum gravity with fakeons S HD = 1 [ g 2κ 2 2Λ C + ζr + α (R µνr µν 13 ) R2 ξ6 ] R2. The Lagrangian coincides with Stelle theory but we quantize it in a dierent way.

26 Quantum gravity with fakeons S HD = 1 [ g 2κ 2 2Λ C + ζr + α (R µνr µν 13 ) R2 ξ6 ] R2. The Lagrangian coincides with Stelle theory but we quantize it in a dierent way. The propagator in De Donder gauge (Λ C = 0, α = ξ) is h µν(p)h ρσ( p) 0 = i { 1 2p 2( ζ αp 2) Iµνρσ = p 2 + α } i ζ αp 2 2ζ Iµνρσ, I µνρσ (η µρη νσ + η µση νρ η µνη ρσ).

27 Quantum gravity with fakeons S HD = 1 [ g 2κ 2 2Λ C + ζr + α (R µνr µν 13 ) R2 ξ6 ] R2. The Lagrangian coincides with Stelle theory but we quantize it in a dierent way. The propagator in De Donder gauge (Λ C = 0, α = ξ) is h µν(p)h ρσ( p) 0 = i { 1 2p 2( ζ αp 2) Iµνρσ = p 2 + α } i ζ αp 2 2ζ Iµνρσ, I µνρσ (η µρη νσ + η µση νρ η µνη ρσ). With the new prescription it turns into { 1 p 2 + iɛ + α(ζ αp 2 ) [ (ζ α(p 2 + iɛ)) 2 + E 4] AV } i (ηµρηνσ + ηµσηνρ ηµνηρσ). 2ζ

28 Quantum gravity with fakeons S HD = 1 [ g 2κ 2 2Λ C + ζr + α (R µνr µν 13 ) R2 ξ6 ] R2. The Lagrangian coincides with Stelle theory but we quantize it in a dierent way. The propagator in De Donder gauge (Λ C = 0, α = ξ) is h µν(p)h ρσ( p) 0 = i { 1 2p 2( ζ αp 2) Iµνρσ = p 2 + α } i ζ αp 2 2ζ Iµνρσ, I µνρσ (η µρη νσ + η µση νρ η µνη ρσ). With the new prescription it turns into { 1 p 2 + iɛ + α(ζ αp 2 ) [ (ζ α(p 2 + iɛ)) 2 + E 4] AV } i (ηµρηνσ + ηµσηνρ ηµνηρσ). 2ζ

29 Graviton/Fakeon prescription a) Replace p 2 with p 2 + iɛ everywhere in the denominators of the propagators;

30 Graviton/Fakeon prescription a) Replace p 2 with p 2 + iɛ everywhere in the denominators of the propagators; b) turn the massive poles into fakeons by means of the replacement 1 ζ u(p 2 + iɛ) ζ up 2 (ζ u(p 2 + iɛ)) 2, u = α, ξ, + E4

31 Graviton/Fakeon prescription a) Replace p 2 with p 2 + iɛ everywhere in the denominators of the propagators; b) turn the massive poles into fakeons by means of the replacement 1 ζ u(p 2 + iɛ) ζ up 2 (ζ u(p 2 + iɛ)) 2, u = α, ξ, + E4 c) calculate the diagrams in the Euclidean, nonanalytically Wick rotate them, send ɛ 0 rst, E 0 last.

32 Graviton/Fakeon prescription a) Replace p 2 with p 2 + iɛ everywhere in the denominators of the propagators; b) turn the massive poles into fakeons by means of the replacement 1 ζ u(p 2 + iɛ) ζ up 2 (ζ u(p 2 + iɛ)) 2, u = α, ξ, + E4 c) calculate the diagrams in the Euclidean, nonanalytically Wick rotate them, send ɛ 0 rst, E 0 last.

33 Graviton/Fakeon prescription a) Replace p 2 with p 2 + iɛ everywhere in the denominators of the propagators; b) turn the massive poles into fakeons by means of the replacement 1 ζ u(p 2 + iɛ) ζ up 2 (ζ u(p 2 + iɛ)) 2, u = α, ξ, + E4 c) calculate the diagrams in the Euclidean, nonanalytically Wick rotate them, send ɛ 0 rst, E 0 last. h µν(p)h ρσ( p) 0 = h µν(p)h ρσ( p) 0grav + h µν(p)h ρσ( p) 0fake.

34 Absorptive part of graviton self energy at Λ C = 0 D. Anselmi and M. Piva, The ultraviolet behavior of quantum gravity, JHEP 05 (2018) 027, arxiv: [hep-th]. M abs = ImM D abs = ReD, M = amplitude, D = diagram.

35 Absorptive part of graviton self energy at Λ C = 0 D. Anselmi and M. Piva, The ultraviolet behavior of quantum gravity, JHEP 05 (2018) 027, arxiv: [hep-th]. M abs = ImM D abs = ReD, M = amplitude, D = diagram. h µν(p)h ρσ( p) 0 = h µν(p)h ρσ( p) 0grav + h µν(p)h ρσ( p) 0fake. Possible cases in the diagram i) Fake-Fake below the pinching threshold: i real integral purely imaginary;

36 Absorptive part of graviton self energy at Λ C = 0 D. Anselmi and M. Piva, The ultraviolet behavior of quantum gravity, JHEP 05 (2018) 027, arxiv: [hep-th]. M abs = ImM D abs = ReD, M = amplitude, D = diagram. h µν(p)h ρσ( p) 0 = h µν(p)h ρσ( p) 0grav + h µν(p)h ρσ( p) 0fake. Possible cases in the diagram i) Fake-Fake below the pinching threshold: i real integral purely imaginary; ii) Fake-Fake above the pinching threshold: average continuation purely imaginary;

37 Absorptive part of graviton self energy at Λ C = 0 D. Anselmi and M. Piva, The ultraviolet behavior of quantum gravity, JHEP 05 (2018) 027, arxiv: [hep-th]. M abs = ImM D abs = ReD, M = amplitude, D = diagram. h µν(p)h ρσ( p) 0 = h µν(p)h ρσ( p) 0grav + h µν(p)h ρσ( p) 0fake. Possible cases in the diagram i) Fake-Fake below the pinching threshold: i real integral purely imaginary; ii) Fake-Fake above the pinching threshold: average continuation purely imaginary; iii) Grav-Fake: no threshold on the real axis, analytical Wick rotation purely imaginary;

38 Absorptive part of graviton self energy at Λ C = 0 D. Anselmi and M. Piva, The ultraviolet behavior of quantum gravity, JHEP 05 (2018) 027, arxiv: [hep-th]. M abs = ImM D abs = ReD, M = amplitude, D = diagram. h µν(p)h ρσ( p) 0 = h µν(p)h ρσ( p) 0grav + h µν(p)h ρσ( p) 0fake. Possible cases in the diagram i) Fake-Fake below the pinching threshold: i real integral purely imaginary; ii) Fake-Fake above the pinching threshold: average continuation purely imaginary; iii) Grav-Fake: no threshold on the real axis, analytical Wick rotation purely imaginary; iv) Grav-Grav: nontrivial real part

39 Absorptive part of graviton self energy at Λ C = 0 D. Anselmi and M. Piva, The ultraviolet behavior of quantum gravity, JHEP 05 (2018) 027, arxiv: [hep-th]. M abs = ImM D abs = ReD, M = amplitude, D = diagram. h µν(p)h ρσ( p) 0 = h µν(p)h ρσ( p) 0grav + h µν(p)h ρσ( p) 0fake. Possible cases in the diagram i) Fake-Fake below the pinching threshold: i real integral purely imaginary; ii) Fake-Fake above the pinching threshold: average continuation purely imaginary; iii) Grav-Fake: no threshold on the real axis, analytical Wick rotation purely imaginary; iv) Grav-Grav: nontrivial real part

40 Absorptive part of graviton self energy at Λ C = 0 D. Anselmi and M. Piva, The ultraviolet behavior of quantum gravity, JHEP 05 (2018) 027, arxiv: [hep-th]. M abs = ImM D abs = ReD, M = amplitude, D = diagram. h µν(p)h ρσ( p) 0 = h µν(p)h ρσ( p) 0grav + h µν(p)h ρσ( p) 0fake. Possible cases in the diagram i) Fake-Fake below the pinching threshold: i real integral purely imaginary; ii) Fake-Fake above the pinching threshold: average continuation purely imaginary; iii) Grav-Fake: no threshold on the real axis, analytical Wick rotation purely imaginary; iv) Grav-Grav: nontrivial real part The contributions of type iv) can be evaluated by using h µν(p)h ρσ( p) 0 = h µν(p)h ρσ( p) 0grav.

41 The divergent part and the absorptive part can be unambiguously related to each other 1 ε 1 2 ln Λ2 1 Λ2 ln 2 p ln( p2 ) prescr 1 2 ln( p2 iɛ) abs i π 2 θ(p2 ).

42 The divergent part and the absorptive part can be unambiguously related to each other 1 ε 1 2 ln Λ2 1 Λ2 ln 2 p ln( p2 ) prescr 1 2 ln( p2 iɛ) abs i π 2 θ(p2 ). 1 ε 1 2 ln Λ2 1 Λ2 ln 2 p ln( p2 ) prescr 1 4 ln( p2 ) 2 abs 0.

43 The divergent part and the absorptive part can be unambiguously related to each other 1 ε 1 2 ln Λ2 1 Λ2 ln 2 p ln( p2 ) prescr 1 2 ln( p2 iɛ) abs i π 2 θ(p2 ). 1 ε 1 2 ln Λ2 1 Λ2 ln 2 p ln( p2 ) prescr 1 4 ln( p2 ) 2 abs 0. We expand S HD around the Hilbert term in powers of α and ξ. 1 p 2 (ζ up 2 ) = 1 ζp 2 + u ζ 2 + u2 p 2 ζ , u = α, ζ.

44 The divergent part and the absorptive part can be unambiguously related to each other 1 ε 1 2 ln Λ2 1 Λ2 ln 2 p ln( p2 ) prescr 1 2 ln( p2 iɛ) abs i π 2 θ(p2 ). 1 ε 1 2 ln Λ2 1 Λ2 ln 2 p ln( p2 ) prescr 1 4 ln( p2 ) 2 abs 0. We expand S HD around the Hilbert term in powers of α and ξ. 1 p 2 (ζ up 2 ) = 1 ζp 2 + u ζ 2 + u2 p 2 ζ , u = α, ζ. In the bubble diagram, powers higher than the second give massless tadpoles, which are set to zero by the dimensional regularization. The result is exact in α and ξ.

45 In the case of pure gravity the absorptive part can be written as δshd Γ abs = g µν, g µν piecewise local function of h µν. δg µν

46 In the case of pure gravity the absorptive part can be written as δshd Γ abs = g µν, g µν piecewise local function of h µν. δg µν Adding (massless) matter elds of all types the absorptive part is Γ abs = iµ ε 16π [ g c (R µνθ( c)r µν 13 ) ] Rθ( c)r + Nsη2 36 Rθ( c)r δshd δg µν g µν, c = (Ns + 6N f + 12N v), η nonminimal coupling for scalar elds. N s = number of scalars; N v = number of vectors; N f = number of Dirac fermions plus 1/2 the number of Weyl fermions.

47 Conclusions In the end we have a unique, renormalizable and unitary theory of QG in 4 dim.

48 Conclusions In the end we have a unique, renormalizable and unitary theory of QG in 4 dim. Unitarity can be proved only if Λ C = 0 but realistic models have Λ C 0. It is not known how to construct a scattering theory in the presence of Λ C 0. It could be due to our lack of knowledge, or the cosmological constant might be an anomaly of unitarity.

49 Conclusions In the end we have a unique, renormalizable and unitary theory of QG in 4 dim. Unitarity can be proved only if Λ C = 0 but realistic models have Λ C 0. It is not known how to construct a scattering theory in the presence of Λ C 0. It could be due to our lack of knowledge, or the cosmological constant might be an anomaly of unitarity. Higher-derivative theories must be dened from the Euclidean space and then Wick rotate.

50 Conclusions In the end we have a unique, renormalizable and unitary theory of QG in 4 dim. Unitarity can be proved only if Λ C = 0 but realistic models have Λ C 0. It is not known how to construct a scattering theory in the presence of Λ C 0. It could be due to our lack of knowledge, or the cosmological constant might be an anomaly of unitarity. Higher-derivative theories must be dened from the Euclidean space and then Wick rotate. The Wick rotation is not analytic. Nevertheless, a new formulation gives well dened amplitudes, physically dierent from the previous formulations and consistent with unitarity.

51 Conclusions In the end we have a unique, renormalizable and unitary theory of QG in 4 dim. Unitarity can be proved only if Λ C = 0 but realistic models have Λ C 0. It is not known how to construct a scattering theory in the presence of Λ C 0. It could be due to our lack of knowledge, or the cosmological constant might be an anomaly of unitarity. Higher-derivative theories must be dened from the Euclidean space and then Wick rotate. The Wick rotation is not analytic. Nevertheless, a new formulation gives well dened amplitudes, physically dierent from the previous formulations and consistent with unitarity. Reconcile renormalizability and unitarity in quantum gravity.

52 Conclusions In the end we have a unique, renormalizable and unitary theory of QG in 4 dim. Unitarity can be proved only if Λ C = 0 but realistic models have Λ C 0. It is not known how to construct a scattering theory in the presence of Λ C 0. It could be due to our lack of knowledge, or the cosmological constant might be an anomaly of unitarity. Higher-derivative theories must be dened from the Euclidean space and then Wick rotate. The Wick rotation is not analytic. Nevertheless, a new formulation gives well dened amplitudes, physically dierent from the previous formulations and consistent with unitarity. Reconcile renormalizability and unitarity in quantum gravity. It is possible to compute physical quantities in order to discriminate our model from others.

53 Outlook Absorptive part in the presence of massive elds (coming soon).

54 Outlook Absorptive part in the presence of massive elds (coming soon). Include the scalar degree of freedom in the spectrum (coming soon).

55 Outlook Absorptive part in the presence of massive elds (coming soon). Include the scalar degree of freedom in the spectrum (coming soon). Handling the presence of the cosmological constant.

56 Outlook Absorptive part in the presence of massive elds (coming soon). Include the scalar degree of freedom in the spectrum (coming soon). Handling the presence of the cosmological constant. Investigate the new possible phenomenology of fundamental interactions due to the new quantization prescription.

57 Outlook Absorptive part in the presence of massive elds (coming soon). Include the scalar degree of freedom in the spectrum (coming soon). Handling the presence of the cosmological constant. Investigate the new possible phenomenology of fundamental interactions due to the new quantization prescription. A lot of other things...