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Quantum Gravity as a quantum field theory of simplicial geometry

Daniele Oriti

TL;DR

This work surveys group field theories as a non-perturbative, background-independent framework for quantum gravity in which spacetime and topology are dynamical. By promoting simplicial building blocks to field quanta on group manifolds, GFTs encode histories as spin foams and triangulations, unifying loop quantum gravity, spin foams, dynamical triangulations, and matrix-model ideas within a single algebraic formalism. The Boulatov model for 3D Riemannian gravity provides a concrete realization where the perturbative expansion yields the Ponzano–Regge amplitudes and a topological invariant, illustrating how a third-quantized, simplicial gravity theory can be defined and computed. The paper also highlights the potential of GFTs to address topology change, define a rigorous canonical inner product, and serve as a versatile framework for exploring non-perturbative quantum spacetime, while acknowledging numerous open theoretical challenges ahead.

Abstract

This is an introduction to the group field theory approach to quantum gravity, with emphasis on motivations and basic formalism, more than on recent results; we elaborate on the various ingredients, both conceptual and formal, of the approach, giving some examples, and we discuss some perspectives of future developments.

Quantum Gravity as a quantum field theory of simplicial geometry

TL;DR

This work surveys group field theories as a non-perturbative, background-independent framework for quantum gravity in which spacetime and topology are dynamical. By promoting simplicial building blocks to field quanta on group manifolds, GFTs encode histories as spin foams and triangulations, unifying loop quantum gravity, spin foams, dynamical triangulations, and matrix-model ideas within a single algebraic formalism. The Boulatov model for 3D Riemannian gravity provides a concrete realization where the perturbative expansion yields the Ponzano–Regge amplitudes and a topological invariant, illustrating how a third-quantized, simplicial gravity theory can be defined and computed. The paper also highlights the potential of GFTs to address topology change, define a rigorous canonical inner product, and serve as a versatile framework for exploring non-perturbative quantum spacetime, while acknowledging numerous open theoretical challenges ahead.

Abstract

This is an introduction to the group field theory approach to quantum gravity, with emphasis on motivations and basic formalism, more than on recent results; we elaborate on the various ingredients, both conceptual and formal, of the approach, giving some examples, and we discuss some perspectives of future developments.

Paper Structure

This paper contains 16 sections, 19 equations, 7 figures.

Table of Contents

  1. Introduction: ingredients and motivations for the group field theory approach
  2. Why path integrals? The continuum sum-over-histories approach
  3. Why topology change? Continuum 3rd quantization of gravity
  4. Why going discrete? Matrix models and simplicial quantum gravity
  5. Why groups and representations? Loop quantum gravity and spin foams
  6. Group field theory: what is it? The basic GFT formalism
  7. A discrete superspace
  8. The field and its symmetries
  9. The space of states or a third quantized simplicial space
  10. Quantum histories or a third quantized simplicial spacetime
  11. The third quantized simplicial gravity action
  12. The partition function and its perturbative expansion
  13. GFT definition of the canonical inner product
  14. Summary: GFT as a general framework for quantum gravity
  15. An example: 3d Riemannian Quantum Gravity
  16. ...and 1 more sections

Figures (7)

  • Figure 1: Propagator and vertex
  • Figure 2: Dual picture
  • Figure 3: A spin foam
  • Figure 4: A spin network
  • Figure :
  • ...and 2 more figures