Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals - PowerPoint PPT Presentation

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals Etera Livine ENS Lyon - Laboratoire de Physique July 2014 in Marseille @ FFP ’14 Frontiers of Fundamental Physics Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators A Large Project: Probe the Geometry of Loop Quantum Gravity Goal of the programme: Understand the space-time geometry of the LQG quantum states and parameterize their deformations (quantum diffeomorphisms?) in order to describe the dynamics of this QG theory LQG spin network states define quantum geometry, but. . . Interpretation as an effective discrete classical geometry? Emergence of smooth classical geometry at large scale? Symmetries encode the whole physics of GR and maps between observers. We assume the same will be true for quantum gravity, but. . . How to go from “discrete diffeomorphisms” at Planck scale to classical diffeomorphisms of smooth manifolds? Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals 1 Loop Quantum Gravity in a Nutshell 2 Twisted Geometries and Coherent States 3 Spinfoam Amplitudes as Spinor Integrals 4 The Hamiltonian Constraints as Differential Operators Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators A 1-slide Loop Quantization Programme for General Relativity Loop Quantum Gravity = Canonical Quantization of GR 1 GR as a gauge theory with gauge group SU (2) in terms of Ashtekar-Barbero connection A and conjugate triad field E . Metric reconstructed a posteriori from A and E . 2 Canonical quantization with quantum states of geometry evolving in time, the spin network states . 3 Length, area, volume, holonomy, raised to quantum operators. Spin networks describe quantum excitations of geometry. 4 Fully constrained theory with Hamiltonian constraints: Gauss constraints imposing SU (2) gauge symmetry Scalar & vector constraints imposing diffeomorphism invariance 5 Transition amplitudes given by spinfoam path integral over random discrete histories of spin networks Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Whats’s to be done in Loop Quantum Gravity? The hard tasks: 1 Implement the Hamiltonian constraints as quantum operators (without anomaly) 2 Develop a mathematically consistent perturbative scheme to identify physical states solving the constraints and compute their scalar product 3 Identify suitable physical states corresponding to actual physical configurations (spherical symmetry, two-body problem, cosmology, ..) Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Loop Quantum Gravity Concretely. . . Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Spin Networks and Geometric Operators Choice of wave-functions: cylindrical functionals of connection A ψ ( A ) = ψ Γ ( { U e [ A ] } e ∈ Γ ) Choose oriented graph Γ. 1 ψ Γ ( { h s ( e ) U e h − 1 = t ( e ) } e ∈ Γ ) , Build holonomies U e of connection A 2 ∀ h v ∈ SU (2) along edges e . Consider gauge-invariant functions under 3 SU (2) transformations at vertices v . Spin Network Basis of L 2 ( SU (2) E / SU (2) V ) on fixed graph Γ SU (2) irreps on edges : spin j e ∈ N / 2 1 Intertwiners (singlet states) at vertices, 2 I v ∈ Inv SU (2) V j e 1 ⊗ .. ⊗ V j en Spin network ψ j e , I v e D j e ( U e ) � ( U e ) = Tr � 3 v I v Γ Gluing of SU (2) elements by Clebsh-Gordan coefficients Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Spin Networks and Geometric Operators Geometric Operators built out of triad E raised to operator ˆ E Area ˆ A of S intersecting edge carrying spin j e : 1 A S ∼ j e l 2 Planck in Planck units Volume excitations at spin network vertices: 2 ˆ V diagonalized by spin network basis and value depend on intertwiners I v inside region LQG Hilbert Space allowing superposition of different graphs ֒ → Have to describe how graphs share parts and can be refined ֒ → Done cleanly through projective limit by Ashtekar & Lewandowski ֒ → Compatible with diffeomorphism invariance A discrete picture of geometry. . . ? Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Spinfoam Amplitudes and Evolving States of Geometry Spinfoams as histories of evolving spin networks: Graphs evolving into Spinfoam models define transition 2-complexes amplitudes between spin network states, as product of vertex ampli- tudes Spinfoam vertices are Space-time events Models define probability Framework allow more general non- amplitude for each spin- trivial space-time topologies. foam vertex. Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators First List of Questions What discrete (classical) geometry behind spin networks? related to Regge triangulations? Can we build coherent spin network states of geometry? Can spinfoam transition amplitudes be written as discrete path integrals of 4d geometry? Do they admit some semi-classical interpretation as transition amplitudes for quantum general relativity, e.g. at least in the framework of cosmology? What are the symmetries and invariance of the spinfoam amplitudes? and what are their relation to deformations and diffeomorphisms of the geometry? A second list of questions will come later. . . Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals 1 Loop Quantum Gravity in a Nutshell 2 Twisted Geometries and Coherent States 3 Spinfoam Amplitudes as Spinor Integrals 4 The Hamiltonian Constraints as Differential Operators Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators The Holonomy-Flux Poisson algebra on a fixed graph Phase space of loop gravity in terms of standard holonomy-flux variables on a fixed oriented graph: Holonomy along edge e g e ∈ SU (2) Flux for edge e around vertex v X v e ∈ su (2) ∼ R 3 These are discretization of connection-triad fields A and E � g e = P exp( i e A ) � X v p ∈S g v → p E g − 1 S = v → p Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals

Loop Quantum Gravity in a Nutshell Twisted Geometries and Coherent States Spinfoam Amplitudes as Spinor Integrals The Hamiltonian Constraints as Differential Operators The Holonomy-Flux Poisson algebra on a fixed graph Phase space of loop gravity in terms of standard holonomy-flux variables on a fixed oriented graph: Holonomy along edge e g e ∈ SU (2) Flux for edge e around vertex v X v e ∈ su (2) ∼ R 3 T ∗ SU (2) Poisson bracket : With constraints: Parallel transport of vectors by { g e , g e ′ } = 0 3d rotations along edges: { X a e , X b e ′ } = i δ ee ′ ǫ abc X c e X t ( e ) = − g e ⊲ X s ( e ) e e { � X e , g e } = � σ g e Closure constraint around vertices: � e ∋ v X v e = 0 generating local SU (2)-inv Etera Livine ENS Lyon - Laboratoire de Physique Loop Quantum Gravity: From Spin Networks to Spinor Path Integrals