Numerical indications on the semiclassical limit of the flipped vertex by Magliaro, Perini and Rovelli

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

In this paper the authors take  the propagation kernel Wt(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 Wt(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(jnm,in) is a function of ten spin variables jnm where n,m = 1,…,5 and five intertwiner variables in:

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.


  • Anm be the area of the triangle(nm) that separates the tetrahedra n and m.
  • αn(mp,qr) be the angle between the triangles (mp) and (qr) in the tetrahedron n.
  • Θnm 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 jnm are the quantum numbers of the areas Anm . The five intertwiners in are the quantum numbers associated to the angles αn(mp,qr) . They are the eigenvalues of the quantity:


In general relativity, the Einstein equations can be seen as constraints on boundary variables Anm ,  αn(mp,qr)   and Θnm. 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 Anm = j0, all equal angles in = i0, and Θnm = 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:


normal is the normalization factor. The constants σ and θ are fixed by the requirement that the state is peaked on the value in = i0  so all angles of the tetrahedron are equally peaked on in = i0 :


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 flippedequ10with the coherent tetrahedron stateflippedequ7



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:



We compared the two functions ψ(i) -coherent tetrahedron and φ(i)  -evolved state for the cases jn = 2 and jn = 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.

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