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Spacetime as a Feynman diagram: the connection formulation
General Relativity and Quantum Cosmology · Submitted 29 Feb 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime.
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The abstract, sentence by sentence
Spin foam models are the path integral counterparts to loop quantized canonical theories. In the last few years several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime. The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set. Models that can accomodate an infinite set of degrees of freedom and that are independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models by summing them over all lattice spacetimes. Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime. We give an explicit formula for the action of the field theory corresponding to any given spin foam model in a wide class which includes several gravity models. Such a field theory was recently found for a particular gravity model [De Pietri et al, hep-th/9907154]. Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology. A first version of our result has appeared in a companion paper [gr-qc\0002083]: here we present a new and more detailed derivation based on the connection formulation of the spin foam models.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Spin foam models are the path integral counterparts to loop quantized canonical theories. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s1 | In the last few years several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime. | background 1.00 | 0.05 | 0.03 | asserted 0.99 |
| s2 | The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set. | background 0.95 | 0.08 | 0.03 | asserted 0.99 |
| s3 | Models that can accomodate an infinite set of degrees of freedom and that are independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models by summing them over all lattice spacetimes. | background 0.61 | 0.11 | 0.11 | asserted 0.91 |
| s4 | Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime. | contribution 0.53 | 0.94 | 0.83 | asserted 1.00 |
| s5 | We give an explicit formula for the action of the field theory corresponding to any given spin foam model in a wide class which includes several gravity models. | contribution 0.67 | 0.95 | 0.25 | asserted 1.00 |
| s6 | Such a field theory was recently found for a particular gravity model [De Pietri et al, hep-th/9907154]. | background 0.99 | 0.04 | 0.03 | asserted 1.00 |
| s7 | Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology. | contribution 0.61 | 0.72 | 0.10 | asserted 1.00 |
| s8 | A first version of our result has appeared in a companion paper [gr-qc\0002083]: here we present a new and more detailed derivation based on the connection formulation of the spin foam models. | background 0.65 | 0.87 | 0.05 | asserted 1.00 |
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The paper
Original abstract
Spin foam models are the path integral counterparts to loop quantized canonical theories. In the last few years several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime. The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set. Models that can accomodate an infinite set of degrees of freedom and that are independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models by summing them over all lattice spacetimes. Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime. We give an explicit formula for the action of the field theory corresponding to any given spin foam model in a wide class which includes several gravity models. Such a field theory was recently found for a particular gravity model [De Pietri et al, hep-th/9907154]. Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology. A first version of our result has appeared in a companion paper [gr-qc\0002083]: here we present a new and more detailed derivation based on the connection formulation of the spin foam models.
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References
How the literature uses it
47 arXiv papers cite this in 59 sentences
Papers citing it · 42 sentences without a confident label
Compares with it 1 paper
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As the pregeometric foundation, the topoverse has several novel features: (i) is unique, whereas other proposals (e.g. [ref] ) consider Feynman sums over all or large classes of 4-manifold topologies; (ii) the topological connectivity of serves as the complete foundation for the real world throughout time, whereas other proposals assume additional structure, such as geometry, time, fields, group representations or causal structure, while generally ignoring the global topological information; and (iii) particles and quantum phenomena emerge naturally from the structure of , as we shall see below.
Unified meta-theory of information, consciousness, time and the classical-quantum universe · Topology as foundation? · 0.96
Cites it as support 3 papers
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This correspondence between spin foam models and GFTs is one to one and fully generic [ref] : for any given spin foam model, there is a GFT model such that the spin foam amplitudes are reproduced as Feynman amplitudes, and any GFT model defines a spin foam model in its perturbative expansion.
Group Field Theory and Loop Quantum Gravity · The quantum dynamics · 0.96
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In fact, one can show that the correspondence between GFTs and spin foam models is generic [ref] : for any given spin foam model defined on a given cellular complex, there is a GFT model whose Feynman amplitudes coincide with it on the Feynman diagram dual to the same complex (and vice versa, any given GFT model defines a corresponding spin foam model, in its perturbative expansion).
Tullio Regge's legacy: Regge calculus and discrete gravity · Group field theories · 0.99
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It can be motivated by results from GFT renormalisation which show that Laplace–Beltrami operators with respect to group arguments are generated by radiative corrections [ref] , even if one started with a pure mass term as is often done in the spin foam literature [ref] .
Effective cosmology from one-body operators in group field theory · Effective GFT quantum constraint · 0.98
Cites it as background 13 papers
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Group field theories (GFT) represent a second quantized framework for both spin networks and simplicial geometry [ref] , field theories on group manifolds (or Lie algebras) producing Feynman amplitudes which can be equivalently expressed as simplicial gravity path integrals [ref] or spin foam models [ref] , in turn covariant formulations of spin networks dynamics [ref] [They can also be seen, from a more statistical field theory perspective, as a higher dimensional generalization of matrix models [ref] and a particular class of tensor models [ref] ].
Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model · Introduction · 1.00
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While GFT is closely related to the covariant spin foam formulation of LQG [ref] and to tensor models [ref] , a canonical framework was also proposed in [ref] .
Relational dynamics and Page-Wootters formalism in group field theory · Introduction · 0.91
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The formalism combines ideas from tensor models [ref] and spin foam models [ref] , yielding a background-independent description of discrete quantum geometries and their collective dynamics.
Collective excitations in quantum gravity condensates · A specific model: GFT condensate cosmology · 0.91
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