Loop gravity from a spinor action Frontiers of Fundamental Physics 14, Marseille Wolfgang Martin Wieland Institute for Gravitation and the Cosmos, Penn State 15 July 2014
Outline of the talk I present an action for discretized gravity with spinors as the fundamental configuration variables. The theory has a Hamiltonian and local gauge symmetries. Generic solutions represent twisted geometries, and have curvature – there is a deficit angle around triangles. Table of contents 1 New action for simplicial gravity in first-order spin-variables 2 Hamiltonian formulation, twisted geometries and curvature 3 Conclusion References: *WMW, New action for simplicial gravity in four dimensions, (2014), arXiv:1407.0025 . *WMW, One-dimensional action for simplicial gravity in three dimensions, accepted for publication in Phys. Rev. D (2014), arXiv:1402.6708 . *WMW, Hamiltonian spinfoam gravity, Class. Quant. Grav. 31 (2014), arXiv:1301.5859 . *E Freidel and S Speziale, From twistors to twisted geometries, Phys. Rev. D 82 (2010), arXiv:1001.2748 . *E Livine and J Tambornino, Spinor Representation for Loop Quantum Gravity, J. Math. Phys. 53 (2012), arXiv:1105.3385 . *M Dupuis, L Freidel, E Livine and S Speziale, Holomorphic Lorentzian Simplicity Constraints, J. Math. Phys. 53 (2012), arXiv:1107.5274 . *M Dupuis, S Speziale and J Tambornino, Spinors and Twistors in Loop Gravity and Spin Foams, PoS QGQGS2011 (2011), arXiv:1201.2120 . *S Speziale and WMW, Twistorial structure of loop-gravity transition amplitudes, Phys. Rev. D 86 (2012), arXiv:1207.6348 . arXiv:1204.0539 . *E Livine, M Martín-Benito, Group theoretical Quantization of I sotropic Loop Cosmology, Phys. Rev. D 85 (2012), *EF Borja, L Freidel, I Garay, and E Livine, U(N) tools for loop quantum gravity: the return of the spinor, Class. Quantum Grav. 28 (2011), arXiv:1010.5451 . 2 / 23
Motivation LQG boundary states twisted geometries ∪ ? spinfoam amplitudes Regge geometries general relativity twisted Regge calculus quantization continuum limit Tension between L QG kinematics and dynamics Kinematics: The LQG boundary states represent twisted geometries: Every tetrahedron has a unique volume, and every triangle has a unique area, yet there are no unique edge lengths. Dynamics: Spinfoam gravity provides us with the transition amplitudes between generic boundary states. A conceptual tension: We always try to fi nd just Regge gravity in the semi-classical limit. Yet, our kinematical framework is more general: Twisted geometries are less restrictive than Regge discretizations. Key question: Can we formulate the dynamics of discretized gravity in terms of twisted geometries? 3 / 23
New action for simplicial gravity in first-order spin-variables
Plebański principle The BF action is topological, and determines the symplectic structure of the theory: � � � � � ∗ Σ αβ − β − 1 Σ αβ ∧ F αβ [ A ] ≡ Π αβ ∧ F αβ . S BF [Σ , A ] = (1) 2 ℓ P2 M M General relativity follows from the simplicity constraints added to the action: Σ αβ ∧ Σ µν ∝ ǫ αβµν . (2) � With the solutions: ± e α ∧ e β , Σ αβ = (3) ± ∗ ( e α ∧ e β ) . Notation: α, β, γ . . . are internal Lorentz indices. Σ α β is an so (1 , 3) -valued two-form. A α β is an SO (1 , 3) connection, with F α β = d A α β + A α µ ∧ A µ β denoting its curvature. e α is the tetrad, diagonalizing the four-dimensional metric g = e α ⊗ e α . 2 = 8 π � /Gc 3 , and β is the Barbero–Immirzi parameter. ℓ P 5 / 23
Discretized BF theory with spinors on a lattice We can write the discretized BF action as a sum over the two-dimensional simplicial faces f 1 , f 2 , . . . : � S BF [ Z f 1 ,Z f 2 , . . . ; Z f 1 , Z f 2 , . . . ; ζ f 1 , ζ f 2 , . . . ; Λ e 1 , Λ e 2 , . . . ] = S f � � f : faces � � �� � � (4) π f A Dω A f A π f A ω A f A = f − π A d ω f + ζ f f − π A ω + cc . f � � � � ∂f f : faces Notation: A, B, C, . . . are spinor indices, and cc . denotes complex conjugation. π A ′ , ω A ) . f : ∂f → T ≃ C 4 , Z = (¯ Each face f carries two twistors: Z f , Z � A − π A ω A . ζ f : ∂f → C is a Lagrange multiplier imposing the constraint ∆ f = π A ω � � e � Dπ A = ˙ π A + [Λ e ] A B π B . D is the covariant differential, ˙ e an edge’s tangent vector: ˙ 6 / 23
Key ideas of the proof, 1/2 Step 1: Discretize the action: � � � � � Π αβ ∧ F αβ ≈ F αβ ≡ S BF [Σ , A ] = Π αβ S f . M τ f f f : faces f : faces Step 2: Define the smeared fl ux: � � � µν . Π αβ d x d y [ h γ ( t,x,y ) ] α µ [ h γ ( t,x,y ) ] β f ( t ) = Π p ( x,y ) ( ∂ x , ∂ y ) ν τ f Step 3: Employ the non-Abelian Stoke ’ s theorem: � D d z h − 1 γ t ( z ) F γ t ( z ) ( ∂ z , ∂ t ) h γ t ( z ) = h − 1 d th γ t (1) , γ t (1) γ t to eventually fi nd the one-dimensional action: � � � D h − 1 αβ Π αβ S f = − d t f ( t ) . d th γ t (1) γ t (1) ∂f 7 / 23
Key ideas of the proof, 2/2 Step 4: Introduce spinors to diagonalize both holonomies and fl uxes: f ( t ) = 1 ǫ A ′ B ′ ω ( A Π αβ f ( t ) π B ) 2¯ f ( t ) + cc ., � f ( t ) π f A f ( t ) ω f A B ( t ) − π B ( t ) � � A � � A B = ω h γ t B = Pexp − A � � � . � E f ( t ) E f ( t ) γ t � We also need the area-matching constraint: f A f − π f A ω A ∆ f := π f ≡ E f ( t ) − E f ( t ) . A ω � � � Putting the pieces together yields the face action: S f [ Z, Z , A, ζ ] = � � � � π A D A d d tω A − π A − ζ∆ = d t d tω + cc . (6) � � ∂f 8 / 23
Linear simplicity constraints Instead of discretizing the quadratic simplicity constraints Σ αβ ∧ Σ µν ∝ ǫ αβµν , (7) we will use the linear simplicity constraints: For every tetrahedron T e (dual to an edge e ) there exist an internal future-oriented four-normal n α e such that the fluxes through its four bounding triangles τ f (dual to a face f : e ⊂ ∂f ) annihilate n α e : � Σ αβ n β e = 0 . (8) τ f The spinorial parametrization turns the simplicity constraints into the following complex conditions: i ! β + i π f A ω A V f = f + cc . = 0 , (9a) W ef = n AA ′ ! π f ω f A ¯ = 0 . (9b) e A ′ 9 / 23
Adding the simplicity constraints The simplicity constraints reduce the SO (1 , 3) spin connection A α β to the SU (2) n Asthekar–Barbero connection: A α = n µ � 1 � αρ A ν ρ + βA α 2 ǫ µν . µ (10) We introduce Lagrange multipliers λ ∈ R and z ∈ C and get the following constrained action for each face in the discretization: � � � A − π A ω A � π A D ω A − π A − ζ S face [ Z, Z | ζ, z, λ |A , n ] = A d ω + π A ω � � � � � ∂f � � � − λ i β + i π A ω A + cc . − z n AA ′ π A ¯ ω A ′ + cc ., (11) 2 where D π A = d π A + A α τ A Bα π B is the SU (2) n covariant differential. Problem: There is no term in the action that would determine the t -dependence of the normal n α e along the edges e ( t ) . We now have to make a proposal. 10 / 23
Four-dimensional closure constraint Any proposal for the dynamics of the time normals must respect the closure constraint at the vertices (four-simplices): We define the volume-weighted four-normal: p e α = n e α Vol( e ) . (12) At every four simplex we have the closure constraint: � � p e p e α = α . (13) outgoing edges e incoming edges e at v at v Notation: α ω f 1 A π f 1 Vol( e ) ∝ 2 9 n α ǫ αβµν L 1 β L 2 µ L 3 ν , with e.g.: L 1 α = − τ AB B + cc . 11 / 23
The proposal for the dynamics of the time-normals Any proposal for the dynamics of the time-normals - must respect the four-dimensional closure constraint, and - be consistent with all symmetries of the action. The following action fulfills these requirements: � � �� � p α d X α − N p α p α + Vol 2 ( e ) S edge [ X, p | N, Vol( e )] = . (14) 2 e We just need an additional boundary term at the vertices: � � � Y α v − X α v ev S vertex [ Y v , { X ev } e ∋ v , { v ev } e ∋ v ] = α . ev (15) e : e ∋ v Where N is a Lagrange multiplier imposing the mass-shell condition: � � C := 1 p α p α + Vol 2 ( e ) ! = 0 . (16) 2 12 / 23
Putting the pieces together – defining the action Adding the face, edge and vertex contributions gives us a proposal for an action for discretized gravity in first-order variables: � � � � � � ζ f , z f , λ f � A ∂f , n ∂f S spin-Regge = S face Z f , Z + f � f : faces � � � � � N e , Vol( e ) + S edges X e , p e + e : edges � � � + Y v , { X ev } e ∋ v , { v ev } e ∋ v S vertex . (17) v : vertices Notation: f are the twistors Z f : ∂f → T ≃ C 4 parametrizing the SL (2 , C ) Z f and Z � holonomy- fl ux variables. ζ f , λ f and z f are Lagrange multipliers imposing the area-matching constraint and simplicity constraints respectively. A is the SU (2) n Ashtekar–Barbero connection along the edges of the discretization. n denotes the time normal of the elementary tetrahedra. p e is the volume-weighted time-normal, of the tetrahedron dual to the edge e . Vol( e ) denotes the corresponding three-volume. N is a Lagrange multiplier imposing the mass-shell condition C = 0 . 13 / 23
Hamiltonian formulation, twisted geometries and curvature
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