Hilbert space formalisms for group field theory

Abstract

Group field theory is a background-independent approach to quantum gravity whose starting point is the definition of a quantum field theory on an auxiliary group manifold (not interpreted as spacetime, but rather as the finite-dimensional configuration space of a single "atom" of geometry). Group field theory models can be seen as an extension of matrix and tensor models by additional data, and are traditionally defined through a functional integral whose perturbative expansion generates a sum over discrete geometries. More recently, some efforts have been directed towards formulations of group field theory based on a Hilbert space and operators, in particular in applications to cosmology. This is an attempt to review some of these formulations and their main ideas, to disentangle these constructions as much as possible from applications and phenomenology, and to put them into a wider context of quantum gravity research.

22 pages, 1 figure. Review that makes a few technical points not easily found in the literature; v2: added some additional comments and extended some discussions; v3: minimal change in presentation, accepted version

References in corpus (40)

• Loop Quantum Cosmology: A Status Report
• The Spin Foam Approach to Quantum Gravity
• Nonperturbative Quantum Gravity
• Colored Tensor Models - a Review
• Group Field Theory: An overview
• Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space
• Colored Group Field Theory
• Spacetime as a Feynman diagram: the connection formulation
• Time and a physical Hamiltonian for quantum gravity
• The Trinity of Relational Quantum Dynamics
• Cosmology from Group Field Theory Formalism for Quantum Gravity
• Homogeneous cosmologies as group field theory condensates
• Gravity quantized
• Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
• The Ponzano-Regge model
• Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates
• Diffeomorphisms in group field theories
• Bouncing cosmologies from quantum gravity condensates
• Ten questions on Group Field Theory (and their tentative answers)
• EPRL/FK Group Field Theory
• Switching internal times and a new perspective on the 'wave function of the universe'
• Flowing in Group Field Theory Space: a Review
• Radiative corrections in the Boulatov-Ooguri tensor model: The 2-point function
• Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint
• The Ostrogradsky instability can be overcome by quantum physics
• Group field theory condensate cosmology: An appetizer
• Group field theory for quantum gravity minimally coupled to a scalar field
• A relational Hamiltonian for group field theory
• Effective relational cosmological dynamics from Quantum Gravity
• Fermions in three-dimensional spinfoam quantum gravity
• Generalised effective cosmology from group field theory
• Suppression of non-manifold-like sets in the causal set path integral
• Mean-field phase transitions in TGFT quantum gravity
• A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity
• Thermal representations in group field theory: squeezed vacua and quantum gravity condensates
• Frontiers of Quantum Gravity: shared challenges, converging directions
• Toward matter dynamics in spin foam quantum gravity
• Generalised Gaussian states in group field theory and $\mathfrak{su}(1,1)$ quantum cosmology
• Relational dynamics and Page-Wootters formalism in group field theory
• Stationary cosmology in group field theory