Quantum gravity with torsion and non-metricity

Transcript

1 Quantum gravity with torsion and non-metricity Carlo Pagani work in collaboration with R. Percacci arxiv: Asymptotic Safety seminar,. Carlo Pagani 1

2 Outline Motivations. Definition of torsion and non-metricity. Ansatz. RG flow of torsion and non-metricity due to the gravitons. Adding matter (fermions). Carlo Pagani 2

3 Motivation: exploring theory spaces of gravity Asymptotic Safety proposal. Quantum Gravity can be fully described as a quantum field theory. A non-trivial UV fixed-point with finitely many relevant directions is searched for. Some further choices are available: - symmetries; - field content. - diffeomorphism (e.g.: metric theories of gravity); - foliation preserving diffeomorphism (Horava-Lifshitz); -... Carlo Pagani 3

4 Field content: - Metric formulation of gravity: the metric carries the degree of freedom. g µν Many aspects have been studied: - higher curvatures; - functional truncations; - matter fields; - bimetric aspects. - Tetrad formulation of gravity (Harst and Reuter 12; Donà and Percacci 12). Carlo Pagani

5 - Tetrad+spin connection formulation of gravity (Daum and Reuter 13; Harst and Reuter 1). - Dilaton gravity and non-integrable Weyl theory (Percacci 12; Codello, d Odorico, Pagani, Percacci 12; Henz, Pawlowski, Rodigast, Wetterich 13; Pagani, Percacci 13). General case: A generic connection has both torsion and non-metricity (defined in the next slide). However we did not see any physical effect so far. Maybe the quantum properties of these objects can explain why it is so. Carlo Pagani 5

6 Generic connection: torsion and non-metricity In the GL (n) formalism the metric is written as g µν = θ a µθ b νγ ab. (1) A generic connection - torsion: A µ a b in the tangent bundle has both: T µ a ν = µ θ a ν ν θ a µ + A µ a b θ b ν A ν a b θ b µ (2) - non-metricity: Q µab = µ γ ab A µ c a γ cb A µ c b γ ac. (3) Carlo Pagani 6

7 Going to the metric formalism we can write: where A α β γ = Γ α β γ + φ α β γ φ α β γ = α α β γ + β α β γ. () (5) Here α αβγ is symmetric in (α, γ) and β αβγ is antisymmetric in (β,γ). The relation with torsion and non-metricity is given by the following formulas: T α β γ = A α β γ A γ β α, Q µαβ = A µ g αβ (6) and T αβγ = β αβγ β γβα, Q αβγ = α αβγ + α αγβ. (7) Carlo Pagani 7

8 The curvatures of and are related as follows: A Γ F µν α β = R µν α β + µ φ α ν β νφ α µ β + φ α µ γφ γ ν β φ α ν γφ γ µ β. (8) and (up to surface terms) F µν µν = R + β α αββ βγ γ +α αβγ β αβγ + α α αββ βγ γ + β αβγ β βαγ α αβγ α αγβ + α α αβα βγ γ α α βαβ βγ γ. (9) This can be rewritten as: F µν µν = R + 1 T αβγt αβγ T αβγt αγβ + Tα αβ T γ β γ (10) + 1 Q αβγq αβγ 1 2 Q αβγq βαγ 1 Q γ αγ Q αβγ T αβγ + Q αβ β T γ αγ Qα αβ T γ β γ. Q αβ β Q αβ α Q γ βγ Carlo Pagani 8

9 Ansatz We consider the most general combination of terms appearing at first order in the curvature: S = S H + S 2 (11) where S H (g) = κ d x g (2Λ R), κ = 1 16πG (12) and S 2 = d d x g g 1 β λµν β λµν + g 2 β λµν β µλν + g 3 β λ λµβ ν ν µ (13) +g α λµν α λµν + g 5 α λµν α λνµ + g 6 α λ λµα µν n + g 7 α λ λµα νµ n + g 8 α µ λµ α ν λν +g 9 α λµν β λµν + g 10 α λ λµβ ν ν µ + g 11 α λ µλβ ν ν µ + ε terms. Carlo Pagani 9

10 The parity odd terms built via the epsilon tensor are the following: ε terms = ε αβγδ g12 β ρ αβ β γδρ + g 13 β αβρ β γδ ρ + g 1 β ραβ β ρ γδ +g 15 β ρ ρ α β βγδ + g 16 α ραβ αρ γδ + g 17 α ραβ β γδ ρ +g 18 α ραβ β γδ ρ + g 19 α ρα ρβ βγδ + g 20 α ρ α ρ β βγδ. (1) The number of invariants can been reduced using the following identity: ε αβ[γ [η δ δ] θ] = ε γδ[α [η δ β] θ]. (15) A basis of linearly independent monomials is: ε terms = ε αβγδ g13 β αβρ β γδ ρ + g 1 β ραβ β ρ γδ + g 16 α ραβ α γδ ρ +g 17 α ραβ β γδ ρ + g 19 α ρα ρβ βγδ + g 20 α ρ α ρ β βγδ. (16) Carlo Pagani 10

11 RG flow of torsion and non-metricity We computed the 1-loop beta functions of the couplings {g i } via the flow equation for the EAA: t Γ k = 1 2 Tr Γ (2) k t R k + R k (17) α αβγ and β αβγ do not have a kinetic term. Only gravitons enter in the loop. We expand the flow equation for small α αβγ and β αβγ : t Γ k = 1 2 Tr S (2) H t R k + R k 12 Tr S (2) H 1 + R k S (2) 2 S (2) H 1 + R k t R k +. (18) Carlo Pagani 11

12 For the Cosmological and the Newton constants one obtains beta functions of the following form: d Λ dt d G dt = 2 Λ+ 1 2 A G B G Λ = (d 2) G B G 2. (19) In the de-donder gauge the r.h.s. of the flow equation for torsion and non-metricity couplings has the following form: where t Γ k 1 κ Q d/2 Q n (f) = 1 Γ[n] K µν ρσ = 1 2 t R k (P k 2Λ) 2 0 Tr dzz n 1 f(z) K 1 S (2) 2 δ µ ρ δν σ +δν ρ δµ σ gµν g ρσ. (20) (21) (22) Carlo Pagani 12

13 Computing the traces and using the relations among the various monomials one finds a set of beta functions of the following form: t g i = (d 2) g i + κ j c ij g j (23) where κ = 16π G 1 2 (π) d/2 2 (d/2)!. 1 2 Λ (2) A convenient form is found by performing a linear transformation to the set of couplings {g i }. This defines a new set of couplings {h i }. For instance These new couplings satisfy: h = g +g 8 3. t hj =( (d 2) + κλ j ) h j. (25) (26) Carlo Pagani 13

14 The explicit form of the coefficients {λ j } is: λ 1 = d2 7d 12 (d + 1)(d ) (d + 1)(d ), λ 2 =, λ 3 =, λ = d2 7d 16, (d + 1)(d ) (d + 1)(d ) (d + 1)(d ) λ 5 =, λ 6 =, λ 7 =, λ 8 = d2 7d, λ 9 = d2 7d 16 (d + 1)(d ) (d + 1)(d ), λ 10 =, λ 11 =, λ 12 = (d + 2), λ 13 = (d 1), λ 1 = d, λ 15 = d, λ 16 = d + 3 2, λ 17 =0. The explicit solution of the RG flow is: G[k] = G BG 0k 2 (27) and h j [k] =h j BG 0k 2 λ j /Bπ. (28) Carlo Pagani 1

15 Our system of beta functions has the following fixed points: - Gaussian fixed point for plus a Gaussian fixed point for the hi couplings. Λ, G - UV non-gaussian fixed point for plus a Gaussian fixed point for the couplings hi Λ, G which can be seen from: t hj =( (d 2) + κλ j ) h j. (29) - Gaussian fixed point for plus a Gaussian fixed point for the 1/ h i couplings. Λ, G Carlo Pagani 15

16 The Holst subsector A particular combination of our ansatz corresponds to the Holst action: S Holst = κ d x ε abcd 2Λ θ a θ b θ c θ d F ab θ c θ d + 1γ F ab θ a θ b The correspondence with our ansatz is given by: We can consider the curvatures of the connections Γ + β or Γ + α + β. g 2 = g 3 = κ ; g 13 = κ/γ ; g i =0, for i = {2, 3, 13}. (30) The above relation is broken by the flow equation. The same goes for the connection Γ + α + β. Carlo Pagani 16

17 Adding matter: fermions Scalar and vector field actions which do not need a gravitational connection. We consider the interaction of (minimally coupled) fermions with torsion (they do not couple to non-metricity): S 1/2 = i 2 dx det(θ) ψγ a θ a µ ˆ µ ψ ˆ µ ψγ a θ a µ ψ (31) After integration by parts we get the following Dirac operator: The square of the Dirac operator has the form: D = γ a θ µ a ˆ µ T ρ µ ρ ψγ a θ µ a (32) = ˆ 2 + B ρ ˆ ρ + X. (33) Carlo Pagani 17

18 In our case: B ρ = 1 [γµ,γ ν ] T ρ µ ν T ρα αi X = 1 ˆF µν µν I 1 2 ˆ µ T µα αi 1 [γµ,γ ν ] ˆ µ T ν α α 1 T µα αt µ β β I. (3) The traces can be computed using the heat kernel for torsionful connection. In particular we have: b 2 ˆ = 1 gsi R (πs) d/2 6 X µt µα α 1 T µα αt β µ β 1 2 T α β βb α µb µ 1 B µb µ. (35) The contribution to the flow equation due to a fermion is: t Γ k = 1 2 tr t R k ( ) Q d/2 1 +R k ( ) = 1 (π) d/2 k d 2 d 2 1! 2[ d 2 ] B 2 ( ) 1 β αβγβ αβγ 1 2 β αβγβ αγβ. (36) Carlo Pagani 18

19 The beta functions are modified as follows: t g 1 = (d 2) g 1 + κ 1 ((d 7)d 12) g 1 t g 2 = (d 2) g 2 + κ 1 (d )(d + 1) g (π) d/2 d 2 1! 2[ d 2 ] (π) d/2 d 2 1! 2[ d 2 ] 1 (37) New fixed points appear: Λ G g1 g 2 g 3 FP FP Λ G 1/ g1 1/ g 2 1/ g 3 FP FP FP FP FP FP FP FP Carlo Pagani 19

20 Summary - We considered the most general combination of gravitational terms with torsion and non-metricity up to the first order in the curvatures and computed the one-loop beta functions of the couplings. - We confirm the Asymptotic Safety fixed point in this approximation. - The Holst action is not preserved under renormalization. - If fermions are added new fixed points appear in the torsion sector. Thank you!! Carlo Pagani 20

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