A Primer of Group Theory for Loop Quantum Gravity and Spin-foams

Abstract

Calculations in Loop Quantum Gravity (LQG) and spin-foam theory rely heavily on the group theory of SU(2) and SL(2,C). Even though many monographs are devoted to this material, the various tools required (representation theory, harmonic analysis, recoupling theory, ...) are often scattered across different books, each with its own conventions and notations. This was the initial motivation for the present document, which serves three main purposes: a concise introduction for students to the essential mathematical tools of LQG, a convenient compendium of formulae for researchers, and a translation hub between the conventions of the main references. These notes are aimed both at physicists, who want their tools to be mathematically well-grounded, and at mathematicians, curious to see how some of their familiar abstract structures reveal the beauty of quantum gravity.

References in corpus (16)

• The Spin Foam Approach to Quantum Gravity
• LQG vertex with finite Immirzi parameter
• Lorentzian spin foam amplitudes: graphical calculus and asymptotics
• Gauge Field Theory Coherent States (GCS) : I. General Properties
• Gauge Field Theory Coherent States (GCS) : II. Peakedness Properties
• A "general boundary" formulation for quantum mechanics and quantum gravity
• Gauge Field Theory Coherent States (GCS) : III. Ehrenfest Theorems
• Gauge Field Theory Coherent States (GCS) : IV. Infinite Tensor Product and Thermodynamical Limit
• From twistors to twisted geometries
• General boundary quantum field theory: Foundations and probability interpretation
• Numerical methods for EPRL spin foam transition amplitudes and Lorentzian recoupling theory
• Boosting Wigner's nj-symbols
• Twistor Networks and Covariant Twisted Geometries
• Infrared divergences in the EPRL-FK Spin Foam model
• Twisted geometries, twistors and conformal transformations
• 2-vertex Lorentzian Spin Foam Amplitudes for Dipole Transitions

Cited by in corpus (18)

• A Short Review of Loop Quantum Gravity
• Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude
• Canonical transformations and squeezing formalism in cosmology
• How-To compute EPRL spin foam amplitudes
• Emergent Cosmology from Quantum Gravity in the Lorentzian Barrett-Crane Tensorial Group Field Theory Model
• The Complete Barrett-Crane Model and its Causal Structure
• Quantum gravity states, entanglement graphs and second-quantized tensor networks
• Asymptotic analysis of spin-foams with time-like faces in a new parameterisation
• Phase transitions in TGFT: a Landau-Ginzburg analysis of Lorentzian quantum geometric models
• Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors
• Holographic entanglement in spin network states: a focused review
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Causal structure in spin-foams
• All you need is spin: SU(2) equivariant variational quantum circuits based on spin networks
• Spherically-symmetric geometries in a matter reference frame as quantum gravity condensates
• Phase transitions in TGFT: Landau-Ginzburg analysis of the causally complete Lorentzian Barrett-Crane model
• Quantum Geometry II : The Mathematics of Loop Quantum Gravity Three dimensional quantum gravity
• Categorical Quantum Volume Operator