Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity

Abstract

We present a pedagogical introduction to SU(2) recoupling theory, focusing on those aspects of the topic which are useful for practical calculations in loop quantum gravity. In particular, we give a self-contained presentation of the powerful graphical formalism, which is an indispensable tool for performing computations in the spin network basis of loop quantum gravity. The use of the graphical techniques in loop quantum gravity is illustrated by several detailed example calculations. Plenty of exercises are included for the benefit of the ambitious student.

66 pages. Based on material from the author's PhD thesis. v2: a few minor corrections

References in corpus (38)

• Background Independent Quantum Gravity: A Status Report
• Discreteness of area and volume in quantum gravity
• Quantum Theory of Gravity I: Area Operators
• Quantum Spin Dynamics (QSD)
• The Spin Foam Approach to Quantum Gravity
• A new spinfoam vertex for quantum gravity
• Fundamental Structure of Loop Quantum Gravity
• Twisted geometries: A geometric parametrisation of SU(2) phase space
• Gauge Field Theory Coherent States (GCS) : II. Peakedness Properties
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Gauge Field Theory Coherent States (GCS) : IV. Infinite Tensor Product and Thermodynamical Limit
• From twistors to twisted geometries
• Spinor Representation for Loop Quantum Gravity
• Coherent spin-networks
• The Fine Structure of SU(2) Intertwiners from U(N) Representations
• Quantum geometry from phase space reduction
• Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators
• U(N) tools for Loop Quantum Gravity: The Return of the Spinor
• Holomorphic Factorization for a Quantum Tetrahedron
• Operators for quantized directions
• Gauge-invariant coherent states for Loop Quantum Gravity I: Abelian gauge groups
• Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint
• New scalar constraint operator for loop quantum gravity
• Hamiltonian operator for loop quantum gravity coupled to a scalar field
• A curvature operator for LQG
• A Discrete and Coherent Basis of Intertwiners
• An elementary introduction to loop quantum gravity
• Asymptotics of the Wigner 9j symbol
• Matrix Elements of Lorentzian Hamiltonian Constraint in LQG
• Loop expansion and the bosonic representation of loop quantum gravity
• Pushing Further the Asymptotics of the 6j-symbol
• Semiclassical Analysis of the Wigner $9J$-Symbol with Small and Large Angular Momenta
• Asymptotics of Wigner 3nj-symbols with Small and Large Angular Momenta: an Elementary Method
• Quantum Theory of Geometry II: Volume operators
• Coherent 3j-symbol representation for the loop quantum gravity intertwiner space
• Graphical method in loop quantum gravity: I. Derivation of the closed formula for the matrix element of the volume operator
• Dynamics in canonical models of loop quantum gravity
• Graphical method in loop quantum gravity: II. The Hamiltonian constraint and inverse volume operators

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