This week is I’ve reviewing an old but interesting paper on the flipped vertex. I’m working on replicating and extesnding the calculations in this paper and will post about them next week.

This is related to the posts

- A new spinfoam vertex for quantum gravity
- The spinfoam framework for quantum gravity
- Quantum geometry from phase space reduction

In this paper the authors take the propagation kernel W_{t}(x,y) of a one-dimensional nonrelativistic quantum system defined by a hamiltonian operator H:

Then they consider a semiclassical wave packet centered on its initial values and compute its evolution under the kernel W_{t}(x,y):

and see whether or not the initial state evolves into a semiclassical wave packet centered on the correct final values.

The flipped vertex W(j_{nm},in) is a function of ten spin variables j_{nm} where n,m = 1,…,5 and five intertwiner variables i_{n}:

The process described by one vertex can be seen as the dynamics of a single cell in a Regge triangulation of general relativity. This gives a simple and direct geometrical interpretation to the dynamical variables entering the vertex amplitude and a simple formulation of

the dynamical equations. The boundary of a Regge cell is formed by five tetrahedra joined along all their faces, forming a closed space with the topology of a 3-sphere.

Let

- A
_{nm}be the area of the triangle(nm) that separates the tetrahedra n and m. - α
_{n}^{(mp,qr)}be the angle between the triangles (mp) and (qr) in the tetrahedron n. - Θ
_{nm}be the angle between the normals to the tetrahedra n and m.

These quantities determine entirely the intrinsic and extrinsic classical geometry of the boundary surface.

The ten spins j_{nm }are the quantum numbers of the areas A_{nm} . The five intertwiners i_{n} are the quantum numbers associated to the angles α_{n}^{(mp,qr)} . They are the eigenvalues of the quantity:

In general relativity, the Einstein equations can be seen as constraints on boundary variables A_{nm} , α_{n}^{(mp,qr)} and Θ_{nm}. These can be viewed as the ensemble of the initial, boundary and final data for a process happening inside the boundary 3-sphere.

In general finding a solution to these constraints is complicated but one is easy – that corresponding to a flat space and to the boundary

of a regular 4-simplex. This is given by all equal areas A_{nm} = j_{0}, all equal angles i_{n} = i_{0}, and Θ_{nm} = Θ, where elementary geometry gives:

A boundary wave packet centered on these values must be correctly propagated by the vertex amplitude, if the vertex amplitude is to give the Einstein equations in the classical limit.

The simplest wave packet is a diagonal gaussian wave packet:

is the normalization factor. The constants *σ* and *θ *are fixed by the requirement that the state is peaked on the value i_{n} = i_{0} so all angles of the tetrahedron are equally peaked on i_{n} = i_{0} :

The state considered is formed by a gaussian state on the spins, with phases given by the extrinsic curvature and by a coherent tetrahedron state:

for each tetrahedron.

We can also write the wave packet

as an initial state times a final state:

We can then test the classical limit of the vertex amplitude by computing the evolution of the four incoming tetrahedra generated by the vertex amplitude and comparing *φ*(i) with* ψ*(i).:

So compare the evolved state with the coherent tetrahedron state

where

If the function φ(i) turns out to be close to the coherent tetrahedron state *ψ*(i), we can say that the flipped vertex amplitude appears to evolve four coherent tetrahedra into one coherent tetrahedron, consistently with the at solution of the classical Einstein equations.

The flipped vertex in the present case is:

** Results**

We compared the two functions *ψ(i)* -coherent tetrahedron and φ*(i)* -evolved state for the cases j_{n} = 2 and j_{n} = 4. The numerical results are shown below:

The agreement between the evolved state and the coherent tetrahedron state is quite good. Besides the overall shape of the state, there is a concordance of the mean values and the widths of the wave packet.

**Related articles**