The group field theory approach to quantum gravity
Daniele Oriti
TL;DR
The paper surveys group field theories (GFTs) as a higher-dimensional generalisation of matrix models for non-perturbative quantum gravity, presenting GFTs as quantum field theories of spacetime that allow dynamical topology and geometry. It lays out the general formalism (kinematics and dynamics) in which a field on a group manifold encodes simplicial geometry, and shows how perturbative Feynman diagrams yield spin foam amplitudes dual to triangulations, thereby unifying and extending loop quantum gravity, spin foams, and BF-type discretisations. It then reviews concrete models in 2, 3, and 4 dimensions, including pure gravity BF-type theories and Barrett–Crane type constructions, and discusses matter couplings that embed particle degrees of freedom into the spin foam framework, along with connections to Regge calculus, dynamical triangulations, and causal sets. The outlook highlights the potential for renormalisation, continuum limits, and emergent spacetime via condensation of GFT quanta, while stressing the need to integrate and test these ideas across related quantum gravity approaches.
Abstract
We give a very concise review of the group field theory formalism for non-perturbative quantum gravity, a higher dimensional generalisation of matrix models. We motivate it as a simplicial and local realisation of the idea of 3rd quantization of the gravitational field, equivalently of a quantum field theory of simplicial geometry, in which also the topology of space is fully dynamical. We highlight the basic structure of the formalism, and discuss briefly various models that are being studied, some recent results and the many open issues that future research should face. Finally, we point out the connections with other approaches to quantum gravity, such as loop quantum gravity, quantum Regge calculus and dynamical triangulations, and causal sets.