Polyhedra in spacetime from null vectors by Neiman

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 xi, such that its norm
equals the face area Ai, and its direction is orthogonal to the face. The area normals must satisfy a ‘closure constraint’:

polyequ1

Minkowski’s reconstruction theorem guarantees a one-to-one correspondence between space-spanning sets of vectors xi that satisfy (1) and convex polyhedra with a spatial orientation. In
LQG, the vectors xi 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 Bi. In addition to closure, these bivectors must also satisfy a cross-simplicity
constraint:

polyequ2

In this paper, the author presents a different parametrization of convex spacelike polyhedra with spacetime orientation. Instead of bivectors Bi, 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 liμ in the (d + 1)-dimensional Minkowski space Rd,1, where i = 1, 2, . . . ,N and d ≥2.  Assume the following conditions on the null vectors liμ.

  • The liμ span the Minkowski space and  N ≥  d + 1.
  • The  liμ  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 Rd,1.

Constructing the polytrope

Consider a set {liμ} ,take the sum of the liμ normalized
to unit length:

polyequ3

The unit vector nμ is timelike, with the same time orientation as the liμ. Now take nμ to be the unit normal to the spacelike polytope. To construct the polytope in the spacelike hyperplane ∑ orthogonal to nμ define the projections of the null vectors liμinto this hyperplane:

polyequ4

The spacelike vectors siμ  automatically sum up to zero. Also, since the liμ span the spacetime, the siμ must span the hyperplane ∑ . By the Minkowski reconstruction theorem, it follows that the siμ are the (d − 1)-area normals of a unique convex d-dimensional polytope in . In this way, the null vectors li define a d-polytope oriented in spacetime.

Basic features of the parametrization.

The vectors  are liμ  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 Rd,1 is in one-to-one correspondence with the directions of its two null normals. So each liμ carries partial information about the orientation of the ith face. The second null normal to the face is a function of all the liμ. It can be expressed as:

polyequ6

where  nμ is given by

polyequ3

Similarly, the area Ai of each face is a function of the
null normals liμ to all the faces:

polyequ7

The total area of the faces has the simple expression:

polyequ8

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 Aab 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 Aab and consider the action:

polyequ9

Then restrict to the variations where:

polyequ10

The stationary points of the action  have the following properties. For each a, the vectors  labμ define a d-simplex with unit normal naμ
and (d − 1)-face areas Aab.

polyfig1

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:

polyequ15a

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

polyequ17

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:

polyequ25

 

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