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:

flippedequ1

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

flippedequ2

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.

Let

  • 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:

flippedequ3

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:

flippedequ4

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:

flippedequ5

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 :

flippedequ6

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:

flippedequ7

for each tetrahedron.

We can also write the wave packet

wave

as an initial state times a final state:

flippedequ8

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).:

flippedequ9

So compare the evolved state flippedequ10with the coherent tetrahedron stateflippedequ7

where

flippedequ11

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:

flippedequ12

 Results

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:

flippedfig1

flippedfig3

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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Tensorial methods and renormalization in Group Field Theories by Sylvain Carrozza

This week I am going to look at a the PhD thesis, Tensorial methods and renormalization in Group Field Theories by  Sylvain Carrozza .

The thesis looks at the two main ways of understanding the construction of GFT models. One way stems from the quantization program for quantum gravity, in the form of loop quantum gravity and spin foam models. Here GFTs are generating functionals for spin foam amplitudes, in the same way as quantum field theories are generating functionals for Feynman amplitudes. They complete the definition of spin foam models by assigning canonical weights to the different foams contributing to a same transition between boundary
states i.e. spin networks. See the post

A second route for GFTs is given by  discrete approaches to quantum gravity. Starting from  matrix models, which allowed us to define random two dimensional surfaces, and so achieve a quantization of two-dimensional quantum gravity. The natural extensions of matrix models are tensor models. From this perspective GFTs appear as enriched tensor models, which allow to the definition of  finer notions of discrete quantum geometries such as the emergence of the continuum.

The thesis then looks at recent aspects of GFTs and tensor models, particularly those following the introduction of colored models. The main results and tools include the combinatorial and topological properties of coloured graphs.

The thesis has two main results, the first set of results concerns the so-called 1/N expansion of topological GFT models. This applies to GFTs with cut-off, given by the parameter N, in which a particular scaling of the coupling constant allows them to reach an asymptotic many-particle regime at large N. The second set of results concerns full-fledge renormalization. Tensorial Group Field Theories (TGFTs), are refined versions of the cut-off models with new non trivial propagators. They have a built-in notion of scale, which generates a well-defined renormalization group flow, and gives rise to dynamical versions of the 1/N expansions.

The thesis looks at a renormalizable TGFT based on the group SU(2) in three dimensions, and incorporating the closure constraint of spin foam models. This TGFT can be considered a field theory realization of the original Boulatov model for three-dimensional
quantum gravity .

Let’s look at the relationship between  the quantum tetrahedron, GFT and the Boulatov model. If the GFT field ϕ is assumed to represent an elementary building block of geometry, then the geometric data should refer to this building block.The Boulatov model generates Ponzano-Regge amplitudes. In its simplicial version, the boundary states of the Ponzano-Regge model are labeled byclosed graphs with three-valent vertices, whose analogue in the field theory formalism are convolutions of the fields ϕ. Following general QFT procedure, we encode the boundary states of the model into functionals of  a single scalar field ϕ. Using the Boulatov model as an example ϕ(g1, g2, g3) is to be interpreted as a flat triangle, and the variables gi label its edges. It is the role of the constraint to introduce an SU(2) flat discrete connection at the level of the amplitudes, encoded in the elementary line holonomies hℓ. The natural interpretation of the variable gi is as the holonomy from a reference point inside the triangle, to the center of the edge i. Thanks to the flatness assumption, this holonomy is independent of the path one chooses to compute gi. The constraint  encodes the freedom in the choice of reference point. From the discrete geometric perspective, the Boulatov model can therefore naturally be called a second quantization of a flat triangle: the GFT field ϕ is the wave-function of a quantized flat triangle, and the path integral provides an interacting theory for such quantum geometric degrees of freedom.

bovlatov

In four dimensions, the correspondence between group and Lie algebra representation can also be put to use. There, the GFT field represents a quantum tetrahedron, and Lie algebra elements correspond to bivectors associated to its boundary triangles. In addition to the closure constraint – equivalent to the Gauss constraint in group space, additional geometricity conditions have to be imposed to guarantee that the bivectors are built from edge vectors of a geometric tetrahedron. These additional constraints are nothing but the simplicity constraints, and non-commutative δ-functions can be used to implement them.

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The 4d Quantum Tetrahedron

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:

4dtetrafig1 Classically equivalent to this is the  Plebanski-Holst action derived from topological BF theory and simplicity constraints.

4dtetrafig2

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

4dtetrafig3

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 Bi ∈ so(4) we get:

4dtetrafig4


Where φ(B1,..,B4;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:

4dtetrafig5At 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:

4dtetrafig6

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

4dtetrafig7

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

4dtetrafig8

The action for Quantum Gravity model:

4dtetrafig9

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:

4dtetrafig10

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:

4dtetrafig11

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

4dtetrafig12

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

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