We start from the GFT formulation of 4d gravity. Starting from the classical continuum action which is the basis for most model building in LQG and spin foam models, we have the Holst-Palatini action:

Classically equivalent to this is the Plebanski-Holst action derived from topological BF theory and simplicity constraints.

Looking at the classical discrete phase space for a single tetrahedron and the classical tetrahedron in 4d.

Another construction is the EPRL model which is a 4d model for Riemannian Plebanski-Holst gravity in noncommutative bivector variables. Starting from GFT for 4d BF theory with B_{i} ∈ * so(4)* we get:

Where φ(B_{1},..,B_{4};N) the non-commutative bivector, flux representation of tetrahedron wavefunction of the classical GFT field, including normal, satisfies the gauge covariance closure condition and the connection h gives parallel transport across frames:

At the level of Feynman amplitudes, this gives usual BF simplicial path integral – the spin representation of the usual Ooguri spin foam model with SO(4) intertwiners and 15j-symbols

** 4d case (riemannian): quantum tetrahedron**

The simplicity constraints:

We can impose this simplicity constraint as a NC delta function in bivector/flux representation:

The geometricity operator is related to the simplicity and covariance or closure constraint since they commute:

The action for Quantum Gravity model:

Looking at the combinatorics of field arguments in vertex the gluing of 5 tetrahedra across common triangles, to form 4-simplex.

This Feynman diagram is equivalent to stranded graph or a 4d simplicial complex:

We can also give a spin foam formulation of the same amplitudes

The Non-commutative bivector representation of Feynman amplitudes gives the simplicial path integral for discrete Plebanski-Holst gravity:

with non-trivial measure on the connection resulting from parallel transport of simplicity constraints across frames:

Again we can also give a spin foam formulation of the same amplitudes

**Related articles**

- Hamiltonian dynamics of a quantum of space (quantumtetrahedron.wordpress.com)

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