This week I have been studying a nice paper about Polyhedra in spacetime.

The paper considers convex spacelike polyhedra oriented in Minkowski space. These are classical analogues of spinfoam intertwiners. There is a parametrization of these shapes using null face normals. This construction is dimension-independent and in 3+1d, it provides the spacetime picture behind the property of the loop quantum gravity intertwiner space in spinor form that the closure constraint is always satisfied after some SL(2,C) rotation.These variables can be incorporated in a 4-simplex action that reproduces the large-spin behaviour of the Barrett–Crane vertex amplitude.

In loop quantum gravity and in spinfoam models, convex polyhedra are fundamental objects. Specifically, the intertwiners between rotation-group representations that feature in these theories can be viewed as the quantum versions of convex polyhedra. This makes the parametrization of such shapes a subject of interest for LQG.

In kinematical LQG, one deals with the SU(2) intertwiners, which correspond to 3d polyhedra in a local 3d Euclidean frame. These polyhedra are naturally parametrized in terms of area-normal vectors: each face i is associated with a vector x_{i}, such that its norm

equals the face area A_{i}, and its direction is orthogonal to the face. The area normals must satisfy a ‘closure constraint’:

Minkowski’s reconstruction theorem guarantees a one-to-one correspondence between space-spanning sets of vectors x_{i} that satisfy (1) and convex polyhedra with a spatial orientation. In

LQG, the vectors x_{i} correspond to the SU(2) fluxes. The closure condition then encodes the Gauss constraint, which also generates spatial rotations of the polyhedron.

In the EPRL/FK spinfoam, the SU(2) intertwiners get lifted into SL(2,C) and are acted on by SL(2,C) ,Lorentz, rotations. Geometrically, this endows the polyhedra with an orientation in the local 3+1d Minkowski frame of a spinfoam vertex. The polyhedron’s

orientation is now correlated with those of the other polyhedra surrounding the vertex, so that together they define a generalized 4-polytope. In analogy with the spatial case, a polyhedron with spacetime orientation can be parametrized by a set of area-normal

simple bivectors B_{i}. In addition to closure, these bivectors must also satisfy a cross-simplicity

constraint:

In this paper, the author presents a different parametrization of convex spacelike polyhedra with spacetime orientation. Instead of bivectors B_{i}, they associate null vectors i to the polyhedron’s

faces. This parametrization does not require any constraints between the variables on different faces. It is unusual in that both the area and the full orientation of each face are functions of the data on all the faces. This construction, like the area-vector and

area-bivector constructions above, is dimension-independent. So we can parametrize d-dimensional convex spacelike polytopes with (d − 1)-dimensional faces, oriented in a (d + 1)-dimensional Minkowski spacetime. These variables can be to construct an action principle for a Lorentzian 4-simplex. The action principle reproduces the large spin behaviour of the Barrett–Crane spinfoam vertex. In particular, it recovers the Regge action for the classical simplicial gravity, up to a possible sign and the existence of additional,degenerate solutions.

In d = 2, 3 spatial dimensions, the parametrization is contained in the spinor-based description of the LQG intertwiners. There, the face normals are constructed as squares of spinors. It was observed that the closure constraint in these variables can always be satisfied by acting on the spinors with an SL(2,C) boost. The simple spacetime picture presented in this paper is new. Hopefully, it will contribute to the geometric interpretation of the modern spinor and twistor variables in LQG.

**The parametrization**

Consider a set of N null vectors *l*_{i}^{μ} in the (d + 1)-dimensional Minkowski space R^{d,1}, where i = 1, 2, . . . ,N and d ≥2. Assume the following conditions on the null vectors *l*_{i}^{μ}.

- The
*l*_{i}^{μ}span the Minkowski space and N ≥ d + 1. - The
*l*_{i}^{μ}are either all future-pointing or all past-pointing.

The central observation in this paper is that such sets of null vectors are in one-to-one correspondence with convex d-dimensional spacelike polytopes oriented in R^{d,1}.

**Constructing the polytrope**

Consider a set {*l*_{i}^{μ}} ,take the sum of the *l*_{i}^{μ }normalized

to unit length:

The unit vector n^{μ} is timelike, with the same time orientation as the *l*_{i}^{μ}. Now take n^{μ} to be the unit normal to the spacelike polytope. To construct the polytope in the spacelike hyperplane ∑ orthogonal to n^{μ }define the projections of the null vectors *l*_{i}^{μ}into this hyperplane:

The spacelike vectors *s*_{i}^{μ }^{ }automatically sum up to zero. Also, since the *l*_{i}^{μ }span the spacetime, the *s*_{i}^{μ }must span the hyperplane ∑ . By the Minkowski reconstruction theorem, it follows that the *s*_{i}^{μ }are the (d − 1)-area normals of a unique convex d-dimensional polytope in . In this way, the null vectors *l*_{i} define a d-polytope oriented in spacetime.

**Basic features of the parametrization.**

The vectors are *l*_{i}^{μ} associated to the polytope’s (d −1)-dimensional faces and are null normals to these faces. The orientation of a spacelike (d − 1)-plane in R^{d,1 }is in one-to-one correspondence with the directions of its two null normals. So each *l*_{i}^{μ }carries partial information about the orientation of the i^{th} face. The second null normal to the face is a function of all the *l*_{i}^{μ}. It can be expressed as:

where n^{μ} is given by

Similarly, the area A_{i} of each face is a function of the

null normals *l*_{i}^{μ} to all the faces:

The total area of the faces has the simple expression:

**A (d+1)-simplex action**

To construct a (d + 1)-simplex action that reproduces in the d = 3 case the large-spin behaviour of the Barrett–Crane spinfoam vertex.

At the level of degree-of-freedom counting, the shape of a (d +1)-simplex is determined by the (d + 1)(d + 2)/2 areas A_{ab} of its (d − 1)-faces. These areas are directly analogous to the spins that appear in the Barrett–Crane spinfoam. Let us fix a set of values for A_{ab} and consider the action:

Then restrict to the variations where:

The stationary points of the action have the following properties. For each a, the vectors *l*_{ab}^{μ }define a d-simplex with unit normal n_{a}^{μ}

and (d − 1)-face areas A_{ab}.

*A (d − 1)-face in a (d + 1)-simplex, shared by two d-simplices a and b. The diagram depicts the 1+1d plane orthogonal to the face. The dashed lines are the two null rays in this normal plane*.

The d-simplices automatically agree on the areas of their shared (d −1)-faces. The two d-simplices agree not only on the area of their shared (d − 1)-face, but also on the orientation of its (d − 1)-plane in spacetime. In other words, they agree on the face’s area-normal bivector:

The area bivectors defined automatically satisfy closure and cross-simplicity:

We conclude that the stationary points are in one-to-one correspondence with the bivector geometries of the Barrett-Crane model with an action of the form:

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