so(4) Plebanski Action and Relativistic Spin Foam Model

L.Freidel; R. De Pietri

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so(4) Plebanski Action and Relativistic Spin Foam Model

R. De Pietri, L.Freidel

General Relativity and Quantum Cosmology · Submitted 26 Apr 1998

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We argue that the $so(4)$ Plebanski model is the continuum analog of the relativistic spin foam model.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.81 · compares with prior results 0.84 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

In this note we study the correspondence between the ``relativistic spin foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski action. We argue that the $so(4)$ Plebanski model is the continuum analog of the relativistic spin foam model. We prove that the Plebanski action possess four phases, one of which is gravity and outline the discrepancy between this model and the model of Euclidean gravity. We also show that the Plebanski model possess another natural dicretisation and can be associate with another, new, spin foam model that appear to be the $so(4)$ counterpart of the spin foam model describing the self dual formulation of gravity.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this note we study the correspondence between the ``relativistic spin foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski action.	contribution 0.86	0.71	0.08	unclear 0.57
s1	We argue that the $so(4)$ Plebanski model is the continuum analog of the relativistic spin foam model.	interpretation 0.63	0.71	0.72	asserted 0.78
s2	We prove that the Plebanski action possess four phases, one of which is gravity and outline the discrepancy between this model and the model of Euclidean gravity.	result 0.94	0.93	0.23	asserted 1.00
s3	We also show that the Plebanski model possess another natural dicretisation and can be associate with another, new, spin foam model that appear to be the $so(4)$ counterpart of the spin foam model describing the self dual formulation of gravity.	result 0.64	0.92	0.14	hedged possibility 0.59

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.95.

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The paper

Original abstract

In this note we study the correspondence between the ``relativistic spin foam'' model introduced by Barrett, Crane and Baez and the so(4) Plebanski action. We argue that the $so(4)$ Plebanski model is the continuum analog of the relativistic spin foam model. We prove that the Plebanski action possess four phases, one of which is gravity and outline the discrepancy between this model and the model of Euclidean gravity. We also show that the Plebanski model possess another natural dicretisation and can be associate with another, new, spin foam model that appear to be the $so(4)$ counterpart of the spin foam model describing the self dual formulation of gravity.

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References

Forward citations

Cited by 15 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Flipped spinfoam vertex and loop gravity
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A new spinfoam vertex derived from quantized Regge calculus reproduces the SO(3) loop quantum gravity boundary states, linking the covariant and canonical formulations.
Renormalization group flows in area-metric gravity
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The first renormalization group analysis of area-metric gravity shows shape-mismatching masses grow toward the infrared, parity is not emergent, and the Immirzi parameter flow has fixed points at γ=0 and γ=∞.
Edge modes of gravity. Part III. Corner simplicity constraints
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The corner area element equals the Poincaré spin Casimir, and the continuum simplicity constraints split into two second class and one first class component, leaving the full Lorentz algebra as Dirac observables.
A New Spin Foam Model for 4d Gravity
gr-qc 2007-08 conditional novelty 7.0 of 10

A new spin foam vertex is constructed that solders neighbouring tetrahedra, is argued to quantize the gravitational rather than the topological sector of Plebanski gravity, and admits a quantized Immirzi parameter.
Edge modes of gravity -- II: Corner metric and Lorentz charges
hep-th 2020-07 conditional novelty 6.0 of 10

The corner metric of tetrad gravity satisfies an ultra-local sl(2,R) algebra whose Casimir is the corner area element, and its space-like nature places that Casimir in the unitary discrete series, giving a quantized a...

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Effective Friedmann equations derived from group field theory condensates are non-singular: a conserved charge keeps the volume from reaching zero, producing a bounce in the Planck regime.
Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
hep-th 2011-11 conditional novelty 6.0 of 10

A new spin foam model for 4d gravity with the Immirzi parameter is constructed from group field theory, with amplitudes given as non-commutative simplicial path integrals and as Wigner 15j-symbol state sums.
On the semiclassical limit of 4d spin foam models
gr-qc 2008-09 conditional novelty 6.0 of 10

In the large-spin limit the FK-gamma spin foam amplitude on an arbitrary triangulation is exponentially suppressed unless the face spins come from a discrete geometry, in which case it reduces to exp(i times the Regge...
Area-angle variables for general relativity
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Replacing edge lengths with triangle areas and 3d dihedral angles, plus local closure and gluing constraints, reproduces Regge calculus on a 4-simplex and removes the non-locality that blocked area-only formulations.
Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model
gr-qc 2011-08 conditional novelty 5.0 of 10

Adding a dynamical normal vector to each tetrahedron in group field theory lets the simplicity constraints be imposed covariantly, yielding a Barrett-Crane-type spin foam model whose edge amplitudes differ from earlie...
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Weakening the simplicity constraints to hold only in expectation values, using coherent states, reproduces the Engle-Pereira-Rovelli spinfoam model and gives its boundary states a clear quantum-tetrahedron geometry.
Group Field Theory: An overview
hep-th 2005-05 conditional novelty 4.0 of 10

Group field theory's tree-level two-point function is proposed as a triangulation-independent physical inner product for loop quantum gravity, with the full two-point function giving a sum over spacetime topologies.
Les Houches lectures on Spinfoam Path Integrals
gr-qc 2026-07 accept novelty 2.0 of 10

A pedagogical review of spinfoam path integrals, from 1d quantum mechanics and 2d BF theory through Ponzano-Regge/Turaev-Viro to the 4d EPRL model, accurate but with no new results.
Group Field Theory and Loop Quantum Gravity
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Group field theory reformulates loop quantum gravity as a quantum field theory of spin network vertices, giving spin foam models a generating functional and a route to continuum and cosmological limits.
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Group field theory gives a discrete third quantization of 3d Riemannian gravity whose Feynman amplitudes equal Ponzano-Regge spin foam and simplicial BF path integrals, while 4d extensions remain tentative.

How the literature uses it

54 arXiv papers cite this in 82 sentences

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Papers citing it · 55 sentences without a confident label

Cites it as background 20 papers

In Plebanski (or BF, as it was subsequently called) formulation of gravitational theory [ref] , [ref] , [ref] , [ref] one starts by the $SO(4)$ (or $SO(3,1)$ in the Lorentzian case) BF theory ∫B^IJ F(A)_IJ where $F(A)$ is the curvature of an $SO(4)$ connection and $B$ is a Lie-algebra valued two-form field.

A New Spin Foam Model for 4d Gravity · BF formulation of gravity and path integral quantization · 0.99
The developments in spin foam models of the last few years [ref] are based on a new way to implement the simplicity constraints which reduce BF theory to general relativity [ref] .

Holomorphic Lorentzian Simplicity Constraints · Introduction · 0.99
The traditional way to enforce the restriction ([ref]), the one matching the original classical Plebanski formulation of gravity, was to add quadratic simplicity constraints[“Simplicity” because a two-form that can be written as a wedge product of one-forms is called simple.] to the action [ref] , ϵ_ABCDΣ^AB_abΣ^CD_cd=Vϵ_abcd,

Classical general relativity as BF-Plebanski theory with linear constraints · Introduction · 0.92

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