Regge Quantum Gravity Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University July 2014 Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Regge quantum gravity ◮ Path integral quantization of GR with matter based on Regge calculus such that a triangulation T ( M ) of the spacetime manifold M is taken as the fundamenal structure. ◮ If N is the number of cells of T ( M ), then for N ≫ 1, T ( M ) looks like the smooth manifold M . ◮ It is not necessary to define the smooth limit N → ∞ . Insted, we need the large- N asymptotics of the observables. ◮ Semiclassical limit will be described by the effective action Γ( L ), which is computed by using the effective action equation from QFT, in the limit L ǫ ≫ l P . ◮ Inspired by the spin-cube models, which are generalizations of the spin-foam models. Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Regge state-sum models ◮ The fundamental DOF are the edge-lengths L ǫ ≥ 0, and E � � iS Rc ( L ) / l 2 � � Z = dL ǫ µ ( L ) exp , P D E ǫ =1 where F � S Rc = − A ∆ ( L ) θ ∆ ( L ) + Λ c V 4 ( L ) , ∆=1 is the Regge action with a cosmological constant. D E ⊂ R E + such that the triangle inequalities hold. ◮ We choose the following PI measure µ ( L ) = e − V 4 ( L ) / L 4 0 , where L 0 is a free parameter. ◮ We also introduce a classical CC length scale L c such that Λ c = ± 1 / L 2 c . Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Regge state-sum models ◮ We will use the effective action in order to determine the quantum corrections. ◮ EA is different from the Wilsonian approach to quantization which is used in QRC and CDT. ◮ In WQ approach E � � igS R ( L ) / l 2 0 + i λ V 4 ( L ) / l 4 � � Z ( g , λ ) = dL ǫ µ ( L ) exp , 0 D E ǫ =1 and one looks for points ( g ∗ , λ ∗ ) where Z ′′ diverges. ◮ In the vicinity of a critical point the correlation length diverges ⇔ transition from the discrete to a continuum theory. ◮ The semiclassical limit in WQ corresponds to the strong-coupling region | g | ≥ 1 ⇒ can be studied only numerically. Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Effective action equation ◮ Let φ : M → R and S ( φ ) = M d 4 x L ( φ, ∂φ ) a QFT � flat-spacetime action. The effective action Γ( φ ) is determined by the integro-differential equation � i � � S ( φ + h ) − i � δ Γ � e i Γ( φ ) / � = d 4 x D h exp δφ ( x ) h ( x ) . � M ◮ A solution Γ( φ ) ∈ C , so that a Wick rotation is used to obtain Γ( φ ) ∈ R : solve the Euclidean equation � � � − 1 � S E ( φ + h ) + 1 � d 4 x δ Γ E e − Γ E ( φ ) / � = D h exp δφ ( x ) h ( x ) , � M and then put x 0 = − it in Γ E ( φ ). ◮ Wick rotation is equivalent to Γ( φ ) → Re Γ( φ ) + Im Γ( φ ) Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Regge effective action ◮ In the case of a Regge state-sum model � e i Γ( L ) / l 2 d E x µ ( L + x ) e iS Rc ( L + x ) / l 2 P − i � E ǫ ( L ) x ǫ / l 2 ǫ =1 Γ ′ P = P , D E ( L ) P = G N � and D E ( L ) is a subset of R E obtained by where l 2 translating D E by a vector − L . ◮ D E ( L ) ⊆ [ − L 1 , ∞ ) × · · · × [ − L E , ∞ ) . ◮ Semiclassical solution Γ( L ) = S Rc ( L ) + l 2 P Γ 1 ( L ) + l 4 P Γ 2 ( L ) + · · · , where L ǫ ≫ l P and | Γ n ( L ) | ≫ l 2 P | Γ n +1 ( L ) | . Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Perturbative solution ◮ Let L ǫ → ∞ , then D E ( L ) → R E and � P − i � E e i Γ( L ) / l 2 R E d E x µ ( L + x ) e iS Rc ( L + x ) / l 2 ǫ ( L ) x ǫ / l 2 ǫ =1 Γ ′ P ≈ P . ◮ The reason is D E ( L ) ≈ [ − L 1 , ∞ ) × · · · × [ − L E , ∞ ) so that � ∞ w 2 P − wx = √ π l P exp − 1 � dx e − zx 2 / l 2 2 log z + l 2 P 4 z − L e − z ¯ L 2 / l 2 P � � P / z ¯ 1 + O ( l 2 L 2 ) � + l P 2 √ π z ¯ , L where ¯ L = L + l 2 w 2 z and Re z = − (log µ ) ′′ . The non-analytic P terms in � are absent since L →∞ e − z ¯ L 2 / l 2 P = 0 , lim for exponentially damped measures. Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Perturbative solution ◮ For D E ( L ) = R E and µ ( L ) = const . the perturbative solution is given by the EA diagrams Γ 1 = i Γ 2 = � S 2 3 G 3 � + � S 4 G 2 � , 2 Tr log S ′′ Rc , Γ 3 = � S 4 3 G 6 � + � S 2 3 S 4 G 5 � + � S 3 S 5 G 4 � + � S 2 4 G 4 � + � S 6 G 3 � , ... Rc ) − 1 is the propagator and S n = iS ( n ) where G = i ( S ′′ Rc / n ! for n > 2, are the vertex weights. ◮ When µ ( L ) � = const . , the perturbative solution is given by Γ( L ) = ¯ P ¯ P ¯ S Rc ( L ) + l 2 Γ 1 ( L ) + l 4 Γ 2 ( L ) + · · · , where ¯ S Rc = S Rc − il 2 P log µ , while ¯ Γ n is given by the sum of n -loop EA diagrams with ¯ G propagators and ¯ S n vertex weights. Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Perturbative solution ◮ Therefore Γ 1 = − i log µ + i 2 Tr log S ′′ Rc Γ 2 = � S 2 3 G 3 � + � S 4 G 2 � + Res [ l − 4 P Tr log ¯ G ] , P Tr log ¯ Γ 3 = � S 4 3 G 6 � + · · · + � S 6 G 3 � + Res [ l − 6 G ] + Res [ l − 6 P � ¯ S 2 3 ¯ G 3 � ] + Res [ l − 6 P � ¯ S 4 ¯ G 2 � ] , · · · ◮ Since log µ ( L ) = O � ( L / L 0 ) 4 � and for � L ǫ > L c , L 0 > l P L c we get the following large- L asymptotics Γ 1 ( L ) = O ( L 4 / L 4 0 ) + log O ( L 2 / L 2 c ) + log θ ( L ) + O ( L 2 c / L 2 ) and ( L 2 c / L 4 ) n � + L − 2 n ( L 2 c / L 2 ) � � � Γ n +1 ( L ) = O 0 c O , where L 0 c = L 2 0 / L c . Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
QG cosmological constant ◮ For L ǫ ≫ l P and L 0 ≫ √ l P L c the series � l 2 n P Γ n ( L ) n is semiclassical. ◮ Let Γ → Γ / G N so that S eff = ( Re Γ ± Im Γ) / G N ◮ One-loop CC l 2 l 2 S eff = S Rc P P Rc + O ( l 4 Tr log S ′′ ± V 4 ± P ) ⇒ G N L 4 G N 2 G N 0 Λ = Λ c ± l 2 P = Λ c + Λ qg . 2 L 4 0 ◮ The one-loop cosmological constant is exact because there are no O ( L 4 ) terms beyond the one-loop order. Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
QG cosmological constant ◮ This is a consequence of the large- L asymptotics log ¯ Rc ( L ) = log O ( L 2 / ¯ c ) + log θ ( L ) + O (¯ L 2 L 2 c / L 2 ) S ′′ ¯ (¯ L 2 c / L 4 ) n � � Γ n +1 ( L ) = O , � − 1 / 2 . where ¯ L 2 c = L 2 1 + il 2 P ( L 2 c / L 4 � 0 ) c ◮ Note that � l P � 4 P | Λ qg | = 1 l 2 ≪ 1 , 2 L 0 because L 0 ≫ l P is required for the semiclassical approximation. ◮ If Λ c = 0, the observed value of Λ is obtained for L 0 ≈ 10 − 5 m so that l P Λ ≈ 10 − 122 . Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Smooth-spacetime limit ◮ Smooth spacetime is described by T ( M ) with E ≫ 1 ⇒ S R ( L ) ≈ 1 � d 4 x � | g | R ( g ) , 2 M � d 4 x � Λ V 4 ( L ) ≈ Λ | g | = Λ V M , M � a ( L K ) R 2 + b ( L K ) R µν R µν � d 4 x � � Tr log S ′′ R ( L ) ≈ | g | , M where L K is defined by L ǫ ≥ L K ≫ l P . ◮ L K defines a QFT momentum UV cutoff � / L K . LHC experiments ⇒ L K < 10 − 19 m ⇔ � K > 1 TeV. Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Coupling of matter ◮ Scalar field on M S m ( g , φ ) = 1 � | g | [ g µν ∂ µ φ ∂ ν φ − U ( φ )] . d 4 x � 2 M ◮ On T ( M ) we get S m = 1 l − 1 � � g kl � V σ ( L ) σ ( L ) φ ′ k φ ′ V ∗ π ( L ) U ( φ π ) , 2 2 σ π k , l where φ ′ k = ( φ k − φ 0 ) / L 0 k . ◮ The total classical action S ( L , φ ) = 1 S Rc ( L ) + S m ( L , φ ) . G N Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Coupling of matter ◮ The EA equation � � R V d V χ exp � e i Γ( L ,φ ) / l 2 i ¯ S Rm ( L + x , φ + χ ) / l 2 P = d E x P D E ( L ) ∂ Γ ∂ Γ � � x ǫ / l 2 � χ π / l 2 − i P − i , P ∂ L ǫ ∂φ π ǫ π where ¯ S Rm = ¯ S Rc + G N S m ( L , φ ). ◮ Perturbative solution Γ( L , φ ) = S ( L , φ ) + l 2 P Γ 1 ( L , φ ) + l 4 P Γ 2 ( L , φ ) + · · · is semiclassical for L ǫ ≫ l P , L 0 ≫ l P and |√ G N φ | ≪ 1. This can be checked in E = 1 toy model S ( L , φ ) = ( L 2 + L 4 / L 2 c ) θ ( L ) + L 2 θ ( L ) φ 2 (1 + ω 2 L 2 + λφ 2 L 2 ) . Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
Coupling of matter ◮ Γ( L , φ ) = Γ g ( L ) + Γ m ( L , φ ) ◮ Γ m ( L , φ ) = V 4 ( L ) U eff ( φ ) for constant φ and U eff (0) = 0. ◮ Γ g ( L ) = Γ pg ( L ) + Γ mg ( L ) and Γ mg ( L ) ≈ Λ m V M + Ω m ( R , K ) in the smooth-manifold approximation and K = 1 / L K . ◮ Ω m = Ω 1 l 2 P + O ( l 4 P ) and � Ω 1 ( R , K ) = a 1 K 2 d 4 x � | g | R + M � � � d 4 x � a 2 R 2 + a 3 R µν R µν + a 4 R µνρσ R µνρσ + a 5 ∇ 2 R log( K /ω ) | g | M + O ( L 2 K / L 2 ) . Aleksandar Mikovi´ c Lusofona University and GFM - Lisbon University Regge Quantum Gravity
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