spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. In this paper the authors develop the condensed matter point of view on extracting the physical and geometrical information out of spin network states: they

introduce Ising spin network states, both in 2d on a square lattice and in 3d on a hexagonal lattice, whose correlations map onto the usual Ising model in statistical physics. They construct these states from the basic holonomy operators of loop gravity and derive a set of local Hamiltonian constraints which entirely characterize the states. By studying their phase diagram distance can be reconstructed from the correlations in the various phases.

The line of research pursued in this paper is at the interface between condensed matter and quantum information and quantum gravity on the other: the aim is to understand how the distance can be recovered from correlation and entanglement between sub-systems of the quantum gravity state. The core of the investigation is the correlations and entanglement entropy on spin network states. Correlations and especially entropy are of special importance

for the understanding of black holes dynamics. Understanding the microscopic origin of black holes entropy is one the major test of any attempt to quantify gravity and entanglement between the horizon and its environment degrees of freedom appears crucial

A spin network state is defined on a graph, dressed

with spins on the edge and intertwiners at the vertices.

A spin on an edge e is a half-integer je 2 N=2 giving an

irreducible representation of SU(2) while an intertwiner

at a vertex v is an invariant tensor, or singlet state, between

the representations living on the edges attached

to that vertex. Spins and intertwiners respectively carry

the basic quanta of area and volume. The authors build the spin

network states based on three clear simplifications:

- Use a fixed graph, discarding graph superposition and graph changing dynamics and work with a fixed regular

lattice. - Freeze all the spins on all the graph edges. Fix to smallest possible value, ½, which correspond to the most basic excitation of geometry in loop quantum gravity, thus representing a quantum geometry directly at the Planck scale.
- Restricted to 4-valent vertices, which represent the basic quanta of volume in loop quantum gravity, dual to quantum tetrahedra.

These simplifications provide us with the perfect setting to map spin network states, describing the Planck scale quantum geometry, to qubit-based condensed matter models. Such models have been extensively studied in statistical physics and much is known on their phase diagrams and correlation functions, and we hope to be able to import these results to the context of loop quantum gravity. One of the most useful model is the Ising model whose relevance goes from modeling binary mixture to the magnetism of matter. We thus naturally propose to construct and investigate Ising spin

network states.

The paper reviews the definition of spin network and analyzes the structure of 4-valent intertwiners between spins ½ leading to the

effective two-state systems used to define the Ising spin network states. Different equivalent definitions are given in terms of the high and low temperature expansions of the Ising model. The loop representation of the spin network is then obtained and studied as well as the associated density which gives information about

parallel transport in the classical limit. Section III introduces

It then introduces a set of local Hamiltonian constraints for which

the Ising state is a unique solution and elaborates on their usefulness for understanding the coarse-graining of

spin network and the dynamic of loop quantum gravity.

After this, it discusses the phase diagram of the Ising

states and their continuum limit as well as the distance

from correlation point of view.

**Ising Spin Network State**

Spin network basis states define the basic excitations of the quantum geometry and they are provided with a natural interpretation in terms of discrete geometry with the spins giving the quanta of area and the intertwiners giving the quanta of volume.

These 4-valent vertices will be organized along a regular lattice. The 3d diamond lattice and the 2d square lattice are considered . Looking initially at the 2d square the square lattice: in this setting, the space of 4-valent interwiners between four spins ½ is two dimensional – it can be decomposed into spin 0 and spin 1 states by combining the spins by pairs, as

Different such decompositions exist and are shown as a graphical representation below. There are three such decompositions, depending on which spins are paired together, the s, t and u channels.

The spin 0 and 1 states in the s channel can be explicitly written in terms of the up and down states of the four spins:

Those two states form a basis of the intertwiner space. There transformation matrices between this basis and the two other channels:

Let’s look at the intertwiner basis defined in terms of the square volume operator U of loop quantum gravity. Since the spins, and the area quanta, are fixed, the only freedom left in the spin network states are the volume quanta defined by the intertwiners. This will provide the geometrical interpretation of our spin network states as

excitations of volumes located at each lattice node. For a 4-valent vertex, this operator is defined as:

where J are the spin operators acting on the i link.

The volume itself can then obtained by taking the square root of the absolute value of U . Geometrically, 4-valent intertwiners are interpreted as representing quantum tetrahedron, which becomes the building block of the quantum geometry in loop quantum gravity and spinfoam models. U takes the following form in the s channel basis:

The smallest possible value of a chunk of space is the square volume ±√3/4 in Plank units.

The two oriented volume states of û, | u _{↑,↓}〉 , can be considered as

the two levels of an effective qubit. Let’s now define a pure spin network state which maps its quantum fluctuations on the thermal fluctuations of a given classical statistical model such as the Ising model by

This state represents a particular configuration of the spin network and the full state is a quantum superposition of them all. Defined as such, the state is unnormalized but its norm is easily computed using the Ising partition function Z_{Ising}:

The intertwiner states living at each vertex are now entangled and carry non-trivial correlations. More precisely this state exhibits Ising correlations between two vertex i, j:

Those correlations are between two volume operators at different vertex which are in fact components of the 2- point function of the gravitational field. So understanding how those correlations can behave in a non-trivial way is a first step toward understanding the behavior of the full 2-point gravity correlations and for instance recover the inverse square law of the propagator.

The generalization to 3d is straightforward. Keeping the requirement that the lattice be 4-valent the natural

regular lattice is the diamond lattice

Using the usual geometrical interpretation of loop quantum gravity, this lattice can be seen as dual to a triangulation of the 3d space in terms of tetrahedra dual to each vertex. This can be seen as an extension of the more used cubic lattice better suited to loop quantum gravity. The Ising spin network state and the whole

set of results which followed are then identical :

- The wave function

- The Hamiltonian constraints and their algebra are the same.

In 3d, the Ising model also exhibits a phase transition

Information about the 2-points correlation functionssuch as long distance behavior at the phase transition or near it can be obtained using methods of quantum field theory. In d dimensions we have

where K(r) are modified Bessel functions and ξ is the

correlation length. For the three-dimensional case, we

have the simple and exact expression

**Conclusions**

In this paper, the authors have introduced a class of spin network

states for loop quantum gravity on 4-valent graph. Such 4-valent graph allows for a natural geometrical interpretation in terms of quantum tetrahedra glued together into a 3d triangulation of space, but it also allows them to be map the degrees of freedom of those states to effective qubits. Then we can define spin network states corresponding to known statistical spin models, such as the Ising model, so that the correlations living on the spin network are exactly the same as those models.

**Related articles**

**The Universe, where space-time becomes discrete (spacedaily.com)****Bounds On Discreteness In Quantum Gravity From Lorentz Invariance (dispatchesfromturtleisland.blogspot.com)**

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In this work the authors investigate the 4d path integral for Euclidean quantum gravity on a hypercubic lattice, as given by the EPRL-FK model. To tackle the problem, they restrict the path to a set of quantum geometries that reflects the lattice symmetries. In particular, the sum over intertwiners is restricted to quantum cuboids, that is, coherent intertwiners which describe a cuboidal

geometry in the large-j limit.

Using asymptotic expressions for the vertex amplitude, several interesting properties of the state sum are found.

- The value of coupling constants in the amplitude functions determines whether geometric or non-geometric configurations dominate the path integral.
- There is a critical value of the coupling constant α, which separates two phases. In one phase the main contribution

comes from very irregular and crumpled states. In the other phase, the dominant contribution comes from a highly regular configuration, which can be interpreted as flat Euclidean space, with small non-geometric perturbations around it. - States which describe boundary geometry with high

torsion have exponentially suppressed physical norm.

**The symmetry-restricted state sum**

Will work on a regular hypercubic lattice in 4d. On this lattice consider only states which conform to the lattice symmetry. This is a condition on the intertwiners, which corresponds to cuboids.

A cuboid is completely determined by its three edge lengths, or equivalently by its three areas.

All internal angles are π/2 , and the condition of regular cuboids on all dual edges of the lattice result in a high degree of symmetries on the labels: The area and hence the spin on each two parallel squares of the lattice which are translations perpendicular to the squares, have to be equal.

The high degree of symmetry will make all quantum geometries flat. The analysis carried out here is therefore not suited for describing local curvature.

**Introduction**

The plan of the paper is as follows:

- Review of the EPRL-FK spin foam model
- Semiclassical regime of the path integral
- Construction of the quantum cuboid intertwiner
- Full vertex amplitude, in particular describe its asymptotic expression for large spins
- Numerical investigation of the quantum path integral

The spin foam state sum employed is the Euclidean EPRL-FK model with Barbero-Immirzi parameter γ < 1. The EPRL-FK model is defined on an arbitrary 2-complexes. A 2-complex is determined by its vertices v, its edges e connecting two vertices, and faces f which are bounded by the edges.

The path integral is formulated as a sum over states. A state in this context is given by a collection of spins – irreducible representations

j_{f} ∈ 1/2 N of SU(2) to the faces, as well as a collection of intertwiners ι_{e} on edges.

The actual sum is given by

where A_{f} , A_{e} and A_{v} are the face-, edge- and vertex- amplitude functions, depending on the state. The sum has to be carried out over all spins, and over an orthonormal orthonormal basis in the intertwiner space at each edge.

The allowed spins j_{f} in the EPRL-FK model are such

that are both also half-integer spins.

The face amplitudes are either

The edge amplitudes A_{e} are usually taken to be equal to 1.

In Sagemath code this looks like:

**Coherent intertwiners**

In this paper, the space-time manifold used is M∼ T³×[0, 1] is the product of the 3-torus T3 and a closed interval. The space is compactified toroidally. M is covered by 4d hypercubes, which

form a regular hypercubic lattice H.There is a vertex for each hypercube, and two vertices are connected by an edge whenever two hypercubes intersect ina 3d cube. The faces of are dual to squares in H, on which four hypercubes meet.The geometry will be encoded in the state, by specification of spins j_{f}

and intertwiners ι_{e}.

Intertwiners ι_{e} can be given a geometric interpretation in terms of polyhedra in R³. Given a collection of spins j_{1}, . . . j_{n} and vectors n_{1}, . . . n_{n} which close . Can define the coherent polyhedron

The geometric interpretation is that of a polyhedron, with face areas j_{f} and face normals n_{i}. The closure condition ensures that such a polyhedron exists.

We are interested in the large j-regime of the quantum cuboids. In this limit, these become classical cuboids which are completely specified by their three areas. Therefore, a

semiclassical configuration is given by an assignment of

areas a = l_{p}² to the squares of the hypercubic lattice.

Denote the four directions in the lattice by x, y, z, t. The areas satisfy

The two constraints which reduce the twisted geometric

configurations to geometric configurations are given by:

For a non-geometric configuration, define the 4-volume of a hypercube as:

Define the four diameters to be:

then we have, V_{4} = d_{x}d_{y}d_{z}d_{t}

We also define the non- geometricity as:

as a measure of the deviation from the constraints.

In sagemath code this looks like:

**Quantum Cuboids**

We let’s look at the quantum theory. In the 2-complex, every edge has six faces attached to it, corresponding to the six faces of the cubes. So any intertwiner in the state-sum will be six-valent, and therefore can be described by a coherent polyhedron with six faces. In our setup, we restrict the state-sum to coherent cuboids, or quantum cuboids. A cuboid is characterized by areas on opposite sides of the cuboid being equal, and the respective normals being negatives of one another

The state ιj_{1},j_{2},j_{3} is given by:

The vertex amplitude for a Barbero-Immirzi parameter γ < 1 factorizes as A_{v} = A^{+}_{v}A^{–}_{v} with

with the complex action

where, a is the source node of the link l, while b is its target node.

**Large j asymptotics**

The amplitudes A^{±}_{v} possess an asymptotic expression for large j_{l}. There are two distinct stationary and critical points, satisfying the equations.

for all links ab . Using the convention shown below

having fixed g_{0} = 1, the two solutions Σ_{1} and Σ_{2} are

The amplitudes A^{±}_{v }satisfy, in the large j limit,

In the large j-limit, the norm squared of the quantum cuboid states is given by:

For the state sum, in the large-j limit on a regular hypercubic lattice:

In sagemath code this looks like:

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**harmonic functions (johndcook.com)**

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3-manifolds on the space part of cosmic space-time.

Harmonic analysis on topological 3-manifolds has been invoked

in cosmological models of the space part of space-time. A direct experimental access to the topology from the autocorrelation of the cosmic matter distribution is difficult. As an alternative, the data from fluctuations of the Cosmic Microwave Background radiation can be examined by harmonic analysis. It is hoped to find in this way the characteristic selection rules and tuning for a specific nontrivial

topology, distinct from the standard simply-connected one.

**Introduction**

Viewed on its universal cover S^{(n−1)}, a spherical topological manifold of dimension n − 1 forms a prototile on its cover the (n-1)-sphere. The tiling is generated by the fixpoint free action of the group of deck transformations. This group is isomorphic to the first homotopy group π_{1}(*M*) and hence is a topological invariant.

A basis for the harmonic analysis on the (n-1)-sphere is given by the spherical harmonics which transform according to irreducible representations of the orthogonal group. Multiplicity and selection rules appear in the form of reduction of group representations.

The deck transformations form a subgroup, and so the representations of the orthogonal group can be reduced to those of this subgroup. Upon reducing to the identity representation of the subgroup, the reduced subset of spherical harmonics becomes periodic on the tiling and tunes the harmonic analysis on the (n-1)-sphere to the manifold.

A particular class of spherical 3-manifolds arises from the five Platonic polyhedra. The harmonic analysis on the Poincare dodecahedral 3-manifold was analyzed along these lines. The authors construct the harmonic analysis on simplicial spherical manifolds of dimension n = 1, 2, 3.

Below is listed the five polyhedra and the known order of their homotopy group and the volume fraction frac(*M*) = |π_{1}(*M*))|^{−1} of the prototile with respect to the volume of the 3-sphere. .

The tetrahedron and the dodecahedron display extremal values of the frac(*M*).

The harmonic analysis on these manifolds can be started from S^{(n−1)}. There its basis is the complete, orthonormal set 〈Y ^{λ }〉 of spherical harmonics, the square integrable eigenmodes of S^{(n−1)}. To pass

to a 3-manifold *M* universally covered by S^{(n−1)}, the author considers the maximal subset 〈Y ^{λ0 }〉 of this basis periodic with respect to deck transformations. Due to the periodicity, it can be restricted to the prototile M and forms its eigenmodes. These periodic eigenmodes tune the sphere S^{(n−1) }to the topology of M.

Of the Platonic 3-manifolds, the Poincare dodecahedral manifold of minimal volume fraction and its eigenmodes have found particular attention. Representation theory was applied to the harmonic analysis on Poincare’s dodecahedral 3-manifold. A comparative study of the harmonic analysis, tuned to different topological 3-manifolds, can provide clues for future applications.

**The Platonic tetrahedral 3-manifold**

With regard to simplicial manifolds on S^{(n−1)}, where n− 1 = 1, 2, 3, the diagram below illustrates symbolically the tilings and simplicial manifolds for n − 1 = 1, 2, 3.

**The tetrahedral 3-simplex S _{0}(3) on the sphere S³**

Consider the 3-sphere S³ < E4 and an inscribed regular 4-simplex with its vertices enumerated as 1, 2, 3, 4, 5. The full point symmetry of the 4-simplex is S(5). Central projection of the 3-faces of this simplex to S³ yields a tiling with 5 tetrahedral tiles. Choose the tetrahedron obtained by dropping the vertex 5 as the simplicial manifold **S _{0}(3)**. Its internal point symmetry group is S(4). The homotopy group π

**The reduction S(5) > C _{5}**

The cyclic group C5 has the elements

They belong to the classes (4)(1) or (5) of S(5).

The computation of the multiplicity m(f, 0) of the identity representation D^{0}(**C _{5}**) is straightforward and we include it

in the last column of the table below. The representation D

**Harmonic analysis on S0(3)**

Summary the basis construction for the harmonic analysis on S0(3) in terms of **C _{5}**-periodic states on the sphere S³.

The spherical harmonics for fixed degree 2j = 0, 1, 2, . . . are the Wigner D^{j}(u) functions. The Wigner D^{j} functions are the irreducible representations of SU(2,C).

which are homogeneous polynomials with real coefficients of degree 2j in the complex matrix elements they are explicitly given by

**Conclusion**

The methods of group theory allow the construction and analysis of the harmonic analysis on topological manifolds. This is demonstrated for the simplicial manifold S_{0}(3). The multiplicities provide the specific selection rules for the chosen simplex

topology. The symmetric group S(5) plays a key role. Its representations f = [41] , [2111] are eliminated from the harmonic analysis.

In general, the harmonic analysis on two different manifolds *M*, *M*′ covered by the sphere S^{(n−1)} is unified by the spherical harmonics and corresponding representations. The differences between topologies appear in the form of different subgroups of deck

transformations. In the harmonic analysis these involve different group/subgroup representations and reductions in O(n,R) > *deck*(M), O(n,R) > *deck*(M′).

Intermediate subgroups as S(n + 1) in can dominate the harmonic analysis on spherical manifolds. The reduction O(n,R) > S(n + 1), n > 2 for simplicial manifolds may require generalized Casimir operators.

Selection rules for S(n + 1) > C_{n+1} eliminate complete representations D^{f} of the group S(n + 1) from the harmonic analysis on the sphere S^{(n−1)} when restricted to the simplicial manifold.

To see the topological variety of the harmonic analysis, compare the tetrahedral Platonic 3-manifold M analyzed here with the dodecahedral Platonic 3-manifold M′. The homotopy group of Poincare’s dodecahedral 3-manifold M′ is, compare the binary icosahedral group. It ha been found that the isomorphic group deck(M′) acts exclusively as a subgroup of SU(2,C)^{r} from the right on the sphere S³ in the coordinates,

with the consequence of a degeneracy of the dodecahedral eigenmodes. The multiplicity in the reduction from O(4,R) to the

subset of eigenmodes for the dodecahedral 3-manifold is completely resolved by a generalized Casimir operator. Multiplicity analysis in shows that the lowest dodecahedral eigenmodes are of degree (2j) = 12.

Comparison with the harmonic analysis for the simplicial 3-manifold demonstrates a dependence of the selection rules and the spectrum of eigenmodes on the topology and on the topologically invariant subgroups involved. Corresponding implications can be drawn for the use of harmonic analysis in the cosmic topology of 3-space.

- Learning Motion Manifolds with Convolutional Autoencoders (theorangeduck.com)

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Loop quantum gravity is a canonical quantization of general relativity where the quantum states of geometry are the so-called spin network states. A spin network is based on a graph dressed up with half-integer spins j_{e} on its edges and intertwiners i_{v} on its vertices. The spins define quanta of area while the intertwiners describe chunks of space volume. The dynamics then acts on the spins j_{e} and intertwiners i_{v}, and can also deform the underlying graph .

In this paper, the authors focus on the structure of the space of intertwiners describing the chunk of space. They focus their study on a region associated with a single vertex of a graph and arbitrary high valency. Associated with this setting there is a classical geometrical description. To each edge going out of this vertex there is associated a dual surface element or face whose area is given by the spin label. The collection of these faces encloses a 3-dimensional volume whose boundary forms a 2-dimensional polygon with the topology of a sphere. This 2d-polygon is such that each vertex is trivalent. At the quantum level the choice of intertwiner i_{v} attached to the vertex describes the shape of the full dual surface and gives the volume contained in that surface.

The space of N-valent intertwiners carries an irreducible representation of the unitary group U(N). These irreducible representations of U(N) are labeled by one integer: the total area of the dual surface – defined as the sum of the spins coming

through this surface. The U(N) transformations deform of the shape of the intertwiner at fixed area. This provides a clean geometric interpretation to the space of intertwiners as wavefunctions over the space of classical N-faced polyhedron. It also leads to a clearer picture of what the discrete surface dual to the intertwiner should look like in the semi-classical regime.

In this work the authors present the explicit construction for new coherent states which are covariant under U(N), then compute their norm, scalar product and show that they provide an overcomplete

basis. They also compute their semi-classical expectation values and uncertainties and show that they are simply related to the Livine-Speziale coherent intertwiners used in the construction of the Engle-Pereira-Rovelli-Livine (EPRL) and Freidel-Krasnov (FK) spinfoam models and their corresponding semi-classical boundary states. These new coherent states confirm the polyhedron interpretation of the intertwiner space and show the relevance of the U(1) phase/frame attached to each face, which appears very similar to the extra phase entering the definition of the discrete twisted geometries for loop gravity.

Considering the space of all N-valent intertwiners, they decompose it separating the intertwiners with different total area :

Each space H^{(J)}_{N} at fixed total area carries an irreducible representation of U(N). The u(N) generators E_{ij} are quadratic operators in the harmonic oscillators of the Schwinger representation of the su(2)-algebra. the full space HN as a Fock space by introducing annihilation and creation operators

The full space H_{N} is established as a Fock space by introducing annihilation and creation operators F_{ij} , F^{†}_{ij} , which allow transitions between intertwiners with different total areas. These creation operators can be used to define U(N) coherent states |J, z_{i} 〉 ∝ F^{†}_{z}| 0〉labeled by the total area J and a set of N spinors z_{i} . These states turn out to have very interesting properties.

- They transform simply under U(N)-transformations u |J, z
_{i}〉 = |J(u z)_{i}〉

- They get simply rescaled under global GL(2,C) transformation acting on all the spinors: In particular, they are invariant under global SL(2,C) transformations.

- They are coherent states and are obtained by the action of U(N) on highest weight states. These highest weight vectors correspond to bivalent intertwiners such as the state defined by
- For large areas J, they are semi-classical states peaked around the expectation values for the
**u**(N) generators: - The scalar product between two coherent states is easily computed:
- They are related to the coherent and holomorphic intertwiners, writing |j,z
_{i〉 }for the usual group-averaged tensor product of SU(2) coherent states defining the coherent intertwiners, we have:

The authors believe that this U(N) framework opens the door to many applications in loop quantum gravity and spinfoam models.

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**Introduction**

Black holes act as thermodynamic systems whose entropy is proportional to the area of their horizon and a temperature that is inversely proportional to their mass. They may be fast scramblers and show deterministic chaos. Einstein’s field equations suggest that dynamical chaos, and the tendency to lose information is a generic property of classical gravitation. For microscopic black holes with masses near the Planck mass, which possess only a small number of degrees of freedom – we need to consider if there a smallest black hole that can act as a thermal system and what mechanism drives the thermal equilibration of black holes at the microscopic level. The pursuit of these questions requires a quantum theory of gravity.

The authors consider the problem of the microscopic origin of the thermal properties of space-time in the framework of Loop Quantum Gravity . In LQG the structure of space-time emerges naturally from the dynamics of a graph of SU(2) spins. The nodes of this graph can be thought of as representing granules of space-time, the spins connecting these nodes can be thought of as the faces of these granules. The volume of these granules, along with the areas of the connected faces are quantized. A recent focus has been on finding a semi-classical description of the spectrum of the volume operator at one of these nodes. There have been several reasonable candidates for the quantum volume operator and a semi-classical limit may pick out a particular one of these forms. The volume preserving deformation of polyhedra has recently emerged as a candidate for this semi-classical limit. In this scheme the black hole thermodynamics can be derived in the limit of a large number N of polyhedral faces. Here the deformation dynamics of the polyhedron is a secondary contribution after the configuration entropy of the polyhedron, which can be readily developed from the statistical mechanics of polymers.

The dynamics of the elementary polyhedron, the tetrahedron, can be exactly solved and semi-classically quantized through the Bohr-Sommerfeld procedure. The volume spectrum arising from quantizing this classical system has shown agreement with full quantum calculations. If the tetrahedron is the hydrogen atom of space, the next complex polyhedron, the pentahedron (N = 5), can be considered as the analogue of the helium atom. The dynamical system corresponding to the isochoric pentahedron with fixed face areas has a four-dimensional phase space compared with two dimensional phase space of the tetrahedron. Non-integrable Hamiltonian systems exhibit behaviors including Hamiltonian chaos.

There are two distinct classes of polyhedra with five faces, the triangular prism and a pyramid with a quadrilateral base. The latter forms a measure zero subset of allowed configurations as its construction requires reducing one of the edges of the triangular prism to zero length.

This article reviews the symplectic Kapovich-Millson phase space of polyhedral configurations and a method by which it is possible to uniquely construct a triangular prism or quadrilateral pyramid for each point in the four-dimensional phase space. It also reviews a method for computing the volume of any polyhedron from its face areas and their normals.

**Polyhedra and Phase Space**

A convex polyhedron is a collection of faces bounded with any number of vertices. The areas A_{l} and normals n_{l} of each face are sufficient to uniquely characterize a polyhedron. The polyhedral closure relationship

is a sufficient condition on A_{l} to uniquely define a polyhedron with N faces. The space of shapes of polyhedra is defined as the space of all

polyhedra modulo to their orientation in three-dimensional space:

The shape space of convex polyhedra with N faces is 2(N – 3) dimensional; in particular, the shape space of the tetrahedron (N = 4) is two-dimensional and that of the pentahedron (N = 5) is four dimensional. This space admits a symplectic structure, which can be defined by introducing a Poisson bracket:

Canonical variables with respect to this Poisson bracket are defined by setting firstly . Then the canonical momenta in the Kapovich-Millson space are defined as and the conjugate positions are given by the angle q given by and we have:

This may be visualized by representing the polyhedron as a polygon with edges given by the vectors this generally gives a non-planar polygon. Now systematically triangulate this polygon, the inserted edges are the conjugate momenta p and q the angles between each of these edges are the conjugate positions. An illustration of the pentagon associated with a pentahedron in shown below:

*An example configuration of the system in the polygon representation, the phase space coordinates plotted here **are z = {0.3, 0.4, 0.9, 0.91}. The normal vectors are plotted as the red solid arrows and the momentum vectors are plotted as **the dashed blue arrows. The associated polyhedron is also shown. All polyhedral faces have area fixed to **one, so all polygonal edges have unit length .*

**The shape of the phase space**

The geometric structure of the polyhedron itself, particularly the fixed face areas, induces certain restrictions upon the phase space. The position space is 2π periodic by construction. The momentum space is restricted by the areas of the faces, from the triangle inequality

Heron’s formula for the area of a triangle can be used to simplify the above inequalities:

where a, b,c are the edges of the triangle.

Considering the triangles Δ1 and Δ2 then inorder for the system to be in a reasonable configuration we require that the area of each of these triangles be non zero.

**Hamiltonian and Polyhedral Reconstruction**

We can use the volume of the pentahedron at a given point in the phase space as the Hamiltonian. This ensures that trajectories generated by Hamilton’s equations will deform the pentahedron while maintaining a constant volume. Consider a vector field F(x) = ⅓x, using the divergence theorem we can find the volume of a polyhedron

We can compute the volume of a polyhedron specied as a set of normals and areas once we know the location of a point upon each face.

*A section in the q1, q2 plane through the Hamiltonian evaluated at p1 = p2 = 0:94, all face areas are fixed to 1. The contours are *isochors*, the color scheme is brighter at larger volumes.*

In the investigation of the phase space of the unit area triangular prism the authors found a great deal of structure in the Hamiltonian and in the distribution of configurations. The phase space contains moderate regions of local stability and large regions of local dynamical instability.

The distribution of local Lyapunov exponents appears to be correlated with the boundaries in the configuration space. They calculated the average dynamical instability measures in the canonical and microcanonical ensembles and obtained values that are comparable to those found in well-known chaotic systems.

*The density of the positive real components of LLE’s plotted against the volume of the system*

**Conclusions**

The large degree of dynamical instability found in the isochoric pentahedron with unit area faces provides a starting point for a bottom-up investigation of the origin of thermal behavior of gravitational field configurations in loop quantum gravity. That the dynamical instability occurs in the simplest polyhedron where

it can suggests that it will be a generic property of more complex polyhedra. Any coupling to other polyhedral configurations can be expected to enhance the degree of instability. At low energies, the pentahedron appears to be a fast scrambler of information.

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with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces.

The paper presents the basic properties of several classes of null faced 4-polytopes: 4-simplices, tetrahedral diamonds and 4-parallelotopes. A most regular representative of each class is proposed.

The most-regular parallelotope is of particular interest: its edges, faces and hyperfaces are all congruent, and it features both null hyperplanes and null segments. A tiling of spacetime with copies of this polytope can be viewed alternatively as a lattice with null edges, such that each point is at the intersection of four light rays in a tetrahedral pattern. This construct may have relevance for discretizations of curved spacetime and for quantum gravity.

In this paper, the author studies the properties of some special 4-polytopes in spacetime. The main qualitative difference between spacetime and Euclidean space is the existence of null i.e. lightlike directions. So, there exist line segments with vanishing length, plane elements with vanishing area, and hyperplane elements with vanishing volume. 3d null hyperplane elements are especially interesting. In relativistic physics, null hypersurfaces play the role of causal boundaries between spacetime regions. They also function as characteristic surfaces for the differential equations of relativistic field theory.

Important examples of null hypersurfaces include the lightcone of an event and the event horizon of a black hole.

The prime example of a closed null hypersurface is a causal diamond – the intersection of two light cones originating from two timelike-separated points.

Null 3d polyhedra or polyhedra with vanishing volume reside in null hyperplanes, such as the hyperplane t = z. let’s look at the geometry of these hyperplanes. The normal ℓ^{μ} to the hyperplane, ℓ^{μ} (1, 0, 0, 1) is a null vector, i.e. ℓ_{μ}ℓ^{μ} = 0. As a result, it is also tangent to the hyperplane. It’s integral lines form null geodesics. The hyperplane

is foliated into light rays. All intervals within the hyperplane are spacelike, except the null intervals along the rays.

**Null Polyhedra**

In 3d Euclidean space, each area element has a normal vector n. When discussing polyhedra, it is convenient to define the norm of n to equal the area of the corresponding face. The orientation of the normals is chosen to be outgoing. Not every set of area normals {n_{i}}

describes the faces of some polyhedron. For this to be true, the normals must sum up to zero:

This can be understood as the requirement that the flux of any constant vector field through the polyhedron vanishes. In loop quantum gravity, this condition encodes the local SO(3) rotation symmetry.

**Null tetrahedra**

The simplest null polyhedron is a tetrahedron. Up to reflections along the null axis, null tetrahedra come in two distinct types: (1,3) and (2,2). The pairs of numbers denote how many of the tetrahedron’s four faces are past-pointing and future- pointing, respectively.

**Null-faced 4-simplices**

Null-faced 4-simplices have hyperfaces which have zero volume. A 4-simplex has five tetrahedral hyperfaces, which in this case will be null tetrahedra,

The scalar products η^{μν}ℓ^{(i)}_{μ}ℓ^{(j)}_{ν}of the null volume normals are directly related to the spacetime volume of the 4-simplex and to the areas of the 2d faces. To express the spacetime volume, we must choose a set of four volume normals ℓ^{(i)}_{μ}. The time-orientation of the

normals should be correlated with the past/future status of their hyperfaces. Next, we construct a symmetric 4 × 4 matrix L^{ij }=(3!)²η^{μν}ℓ^{(i)}_{μ}ℓ^{(j)}_{ν }of their scalar products. The diagonal elements of L^{ij } are zero. Elements corresponding to past- future pairs ij are positive, while those for past-past and future-future pairs are negative.

The spacetime volume can then be found as:

The area of the face at the intersection of the i’th and j’th hyperplanes can be found as:

Can also define the 4-volume directly in terms of triangle areas:

**Null parallelepipeds**

The six faces of a null parallelepiped are spacelike parallelograms. There are three pairs of opposing faces, such that each pair is parallel and congruent. In a given pair of opposing faces, one is past-pointing, and the other future-pointing.

**Tetrahedral diamonds**

Beginning with an arbitrary spacelike tetrahedron, situated at t = 0 hyperplane, this is the base tetrahedron. For each of the base tetrahedron’s four faces, the lightcross of two null hyperplanes orthogonal to it are drawn. The tetrahedral diamond is then defined by the convex hull of the intersections of these null hyperplanes.

The 4-volume of a tetrahedral diamond can be found as twice the volume of a 4-simplex, with the spacelike tetrahedron as its base and the inscribed radius r as its height. The result is:

where V is the base tetrahedron’s volume.

**Related posts**

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They show that the Grassmannian space U(N)/(U(N − 2) × SU(2)) is the space of framed convex polyhedra with N faces up to 3d rotations. The framing consists in the additional information of a U(1) phase per face. This provides an extension of the Kapovich-Milson phase space for polyhedra with fixed number of faces and fixed areas for each face – see the post:

Polyhedra in loop quantum gravity

They describe the Grassmannian as the symplectic quotient C^{2N}//SU(2), which provides canonical complex variables for the Poisson bracket. This construction allows a natural U(N) action on the space of polyhedra, which has two main features. First, U(N) transformations act non-trivially on polyhedra and change the area and shape of each individual face. Second, this action is cyclic: it allows us to go between any two polyhedra with fixed total area – sum of the areas of the faces.

On quantization, the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners, which is interpreted as the space of quantum polyhedra. By performing a canonical quantization from the complex variables of C^{2N}//SU(2) all the classical features are automatically exported to the quantum level. Each face carries now a irreducible representation of SU(2) – a half-integer spin j, which defines the area of the face. Intertwiners are then SU(2)-invariant states in the tensor product of these irreducible representations. These intertwiners are the basic

building block of the spin network states of quantum geometry in loop quantum gravity.

The U(N) action on the space of intertwiners changes the spins of the faces and each Hilbert space for fixed total area defines an irreducible representation of the unitary group U(N). The U(N) action is cyclic and allows us to generate the whole Hilbert space from the action of U(N) transformation on the highest weight

vector. This construction provides coherent intertwiner states peaked on classical polyhedra.

At the classical level, we can use the U(N) structure of the space of polyhedra to compute the averages of polynomial observables over the ensemble of polyhedra and to use the Itzykson-Zuber formula from matrix models as a generating functional for these averages. It computes the integral over U(N) of the exponential of the matrix elements of a unitary matrix tensor its complex conjugate.

At the quantum level the character formula, giving the trace of unitary transformations either over the standard basis or the coherent intertwiner basis, provides an extension of the Itzykson-Zuber formula. It allows us in principle to generate the expectation values of all polynomial observables and so their spectrum.

This paper defines and describe the phase space of framed polyhedra, its parameterization in terms of spinor variables and the action of U(N) transformations. Then it shows how to compute the averages and correlations of polynomial observables using group integrals over U(N) and the Itzykson-Zuber integral as a generating function. It discusses the quantum case, with the Hilbert space

of SU(2) intertwiners, coherent states and the character formula.

The paper also investigates polygons in two dimensions and shows that the unitary group is replaced by the orthogonal group and that

the Grassmannian Ø(N)/(Ø(N −2)×SO(2)) defines the phase space for framed polygons. It then discusses the issue of gluing such polygons together into a consistent 2d cellular decomposition, as a toy model for the gluing of framed polyhedra into 3d discrete manifolds. These constructions are relevant to quantum gravity in 2+1 and 3+1 dimensions, especially to discrete approaches based on a description of the geometry using glued polygons and polyhedra such as loop quantum gravity and dynamical

triangulations.

The paper’s goal is to clarify how to parametrize the set of polygons or polyhedra and their deformations, and to introduce mathematical tools to compute the average and correlations of observables over the ensemble of polygons or polyhedra at the classical level and then the spectrum and expectation values of geometrical operators on

the space of quantum polygons or polyhedra at the quantum level.

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This looks great when animated:

below is the python program for these diagrams.

from scipy.special import jv, legendre, sph_harm, jacobi

from scipy.misc import factorial, comb

from numpy import floor, sqrt, sin, cos, exp, power

from math import pi

def wignerd(j,m,n=0,approx_lim=10):

”’

Wigner “small d” matrix. (Euler z-y-z convention)

example:

j = 2

m = 1

n = 0

beta = linspace(0,pi,100)

wd210 = wignerd(j,m,n)(beta)

some conditions have to be met:

j >= 0

-j <= m <= j

-j <= n <= j

The approx_lim determines at what point

bessel functions are used. Default is when:

j > m+10

and

j > n+10

for integer l and n=0, we can use the spherical harmonics. If in

addition m=0, we can use the ordinary legendre polynomials.

”’

if (j < 0) or (abs(m) > j) or (abs(n) > j):

raise ValueError(“wignerd(j = {0}, m = {1}, n = {2}) value error.”.format(j,m,n) \

+ ” Valid range for parameters: j>=0, -j<=m,n<=j.”)

if (j > (m + approx_lim)) and (j > (n + approx_lim)):

#print ‘bessel (approximation)’

return lambda beta: jv(m-n, j*beta)

if (floor(j) == j) and (n == 0):

if m == 0:

#print ‘legendre (exact)’

return lambda beta: legendre(j)(cos(beta))

elif False:

#print ‘spherical harmonics (exact)’

a = sqrt(4.*pi / (2.*j + 1.))

return lambda beta: a * conjugate(sph_harm(m,j,beta,0.))

jmn_terms = {

j+n : (m-n,m-n),

j-n : (n-m,0.),

j+m : (n-m,0.),

j-m : (m-n,m-n),

}

k = min(jmn_terms)

a, lmb = jmn_terms[k]

b = 2.*j – 2.*k – a

if (a < 0) or (b < 0):

raise ValueError(“wignerd(j = {0}, m = {1}, n = {2}) value error.”.format(j,m,n) \

+ ” Encountered negative values in (a,b) = ({0},{1})”.format(a,b))

coeff = power(-1.,lmb) * sqrt(comb(2.*j-k,k+a)) * (1./sqrt(comb(k+b,b)))

#print ‘jacobi (exact)’

return lambda beta: coeff \

* power(sin(0.5*beta),a) \

* power(cos(0.5*beta),b) \

* jacobi(k,a,b)(cos(beta))

def wignerD(j,m,n=0,approx_lim=10):

”’

Wigner D-function. (Euler z-y-z convention)

This returns a function of 2 to 3 Euler angles:

(alpha, beta, gamma)

gamma defaults to zero and does not need to be

specified.

The approx_lim determines at what point

bessel functions are used. Default is when:

j > m+10

and

j > n+10

usage:

from numpy import linspace, meshgrid

a = linspace(0, 2*pi, 100)

b = linspace(0, pi, 100)

aa,bb = meshgrid(a,b)

j,m,n = 1,1,1

zz = wignerD(j,m,n)(aa,bb)

”’

return lambda alpha,beta,gamma=0: \

exp(-1j*m*alpha) \

* wignerd(j,m,n,approx_lim)(beta) \

* exp(-1j*n*gamma)

if __name__ == ‘__main__’:

”’

just a bunch of plots in (phi,theta) for

integer and half-integer j and where m and

n take values of [-j, -j+1, …, j-1, j]

Note that all indexes can be any real number

with the conditions:

j >= 0

-j <= m <= j

-j <= n <= j

”’

from matplotlib import pyplot, cm, rc

from numpy import linspace, arange, meshgrid, real, imag, arccos

rc(‘text’, usetex=False)

ext = [0.,2.*pi,0.,pi]

phi = linspace(ext[0],ext[1],200)

theta = linspace(ext[2],ext[3],200)

pphi,ttheta = meshgrid(phi,theta)

# The maximum value of j to plot. Will plot real and imaginary

# distributions for j = 0, 0.5, … maxj

maxj = 2.0

for j in arange(0,maxj+.1,step=0.5):

fsize = (j*2+3,j*2+3)

title = ‘WignerD(j,m,n)(phi,theta)’

if j == 0:

fsize = (4,4)

else:

title += ‘, j = ‘+str(j)

figr = pyplot.figure(figsize=fsize)

figr.suptitle(r’Real Part of ‘+title)

figi = pyplot.figure(figsize=fsize)

figi.suptitle(r’Imaginary Part of ‘+title)

for fig in [figr,figi]:

fig.subplots_adjust(left=.1,bottom=.02,right=.98,top=.9,wspace=.02,hspace=.1)

if j == 0:

fig.subplots_adjust(left=.1,bottom=.1,right=.9,top=.9)

if j == 0.5:

fig.subplots_adjust(left=.2,top=.8)

if j == 1:

fig.subplots_adjust(left=.15,top=.85)

if j == 1.5:

fig.subplots_adjust(left=.15,top=.85)

if j == 2:

fig.subplots_adjust(top=.87)

if j != 0:

axtot = fig.add_subplot(1,1,1)

axtot.axesPatch.set_alpha(0.)

axtot.xaxis.set_ticks_position(‘top’)

axtot.xaxis.set_label_position(‘top’)

axtot.yaxis.set_ticks_position(‘left’)

axtot.spines[‘left’].set_position((‘outward’,10))

axtot.spines[‘top’].set_position((‘outward’,10))

axtot.spines[‘right’].set_visible(False)

axtot.spines[‘bottom’].set_visible(False)

axtot.set_xlim(-j-.5,j+.5)

axtot.set_ylim(-j-.5,j+.5)

axtot.xaxis.set_ticks(arange(-j,j+0.1,1))

axtot.yaxis.set_ticks(arange(-j,j+0.1,1))

axtot.set_xlabel(‘n’)

axtot.set_ylabel(‘m’)

nplts = 2*j+1

data_j=[]

data_zz=[]

for m in arange(-j,j+0.1,step=1):

for n in arange(-j,j+0.1,step=1):

print j,m,n

zz = wignerD(j,m,n)(pphi,ttheta)

data_j.append(j)

data_zz.append(zz)

i = n+j + nplts*(j-m)

for fig,data in zip((figr,figi), (real(zz),imag(zz))):

ax = fig.add_subplot(nplts, nplts, i+1, projection=’polar’)

plt = ax.pcolormesh(pphi,ttheta,data.copy(),

cmap=cm.jet,

#cmap=cm.RdYlBu_r,

vmin=-1., vmax=1.)

if j == 0:

ax.grid(True, alpha=0.5)

ax.set_title(r’j,m,n = (0,0,0)’, position=(0.5,1.1), size=12)

ax.set_xlabel(r’$\phi$’)

ax.set_ylabel(r’$\theta$’, rotation=’horizontal’, va=’bottom’)

ax.xaxis.set_ticks([0,.25*pi,.5*pi,.75*pi,pi,1.25*pi,1.5*pi,1.75*pi])

ax.xaxis.set_ticklabels([‘0′,r’$\frac{\pi}{4}$’,r’$\frac{\pi}{2}$’,r’$\frac{3 \pi}{4}$’,r’$\pi$’,r’$\frac{5 \pi}{4}$’,r’$\frac{3 \pi}{2}$’,r’$\frac{7 \pi}{4}$’], size=14)

ax.yaxis.set_ticks([0,.25*pi,.5*pi,.75*pi,pi])

ax.yaxis.set_ticklabels([‘0′,r’$\frac{\pi}{4}$’,r’$\frac{\pi}{2}$’,r’$\frac{3 \pi}{4}$’,r’$\pi$’], size=14)

else:

ax.xaxis.set_ticks([])

ax.yaxis.set_ticks([])

ax.set_xlim(ext[0],ext[1])

ax.set_ylim(ext[2],ext[3])

if j == 0:

fig.colorbar(plt, pad=0.07)

# uncomment the following if you want to save these to image files

#figr.savefig(‘wignerD_j’+str(j)+’_real.png’, dpi=150)

#figi.savefig(‘wignerD_j’+str(j)+’_imag.png’, dpi=150)

#pyplot.show()

#pl.plot(j_data,zz_data)

print ‘j_data=’,data_j

#print ‘zz_data=’, data_zz

**Related posts**

- The wigner d matrix
- Numerical work with sagemath 23: wigner reduced rotation matrix elements as limits of 6j symbols
- Review of wigner nj symbols
- Exact computation and asymptotic approximations of 6j symbols illustration of their semiclassical limits

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The starting point is to take the action of the area operator in the spin representation, and to consider the following normalized trace of the area operator:

If this give a well-defined operator on either the kinematical Hilbert space or on the Hilbert space of fully gauge-invariant wave functions, and also if we took the limit Λ_{fix} →∞ it is possible to read off the spectrum from the representation. There, the oscillatory behaviour of the sine function is suppressed by a factor of 1/d_{j }which leads to a discrete spectrum for sufficiently small spins j.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.1 and j=1…100.

And below the figure from the original paper, A new realization of quantum geometry.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.3 and j=1…100.

And below the figure from the original paper, A new realization of quantum geometry.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.05, 0.1 and 0.3 and j=1…100

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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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