# Ising Spin Network States for Loop Quantum Gravity by Feller and Livine

This week I have been studying a great paper by Feller and Livine on Ising Spin Network States for Loop Quantum Gravity. In the context of loop quantum gravity, quantum states of geometry are defined as
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:

1.  Use a fixed graph, discarding graph superposition and graph changing dynamics and  work with a fixed regular
lattice.
2. 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.
3. 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 ZIsing:

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.
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# Calculations on Quantum Cuboids and the EPRL-FK path integral for quantum gravity

This week I have been studying a really great paper looking at Quantum Cuboids and path-integral calculations for the EPRL vertex in LQG and also beginning to write some calculational software tools for performing these calculations using Sagemath.

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
jf ∈ 1/2 N of SU(2) to the faces, as well as a collection of intertwiners ιe on edges.

The actual sum is given by

where Af , Ae and Av 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 jf in the EPRL-FK model are such
that  are both also half-integer spins.

The face amplitudes are either

The edge amplitudes Ae 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 jf
and intertwiners ιe.

Intertwiners ιe can be given a geometric interpretation in terms of polyhedra in R³. Given a collection of spins j1, . . . jn and vectors n1, . . . nn which close . Can define the coherent polyhedron

The geometric interpretation is that of a polyhedron, with face areas jf and face normals ni. 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 = lp² 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, V4 = dxdydzdt

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 ιj1,j2,j3 is given by:

The vertex amplitude for a Barbero-Immirzi parameter γ < 1 factorizes as Av = A+vAv 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 jl. There are two distinct stationary and critical points, satisfying the equations.

for all links ab . Using the convention shown below

having fixed g0 = 1, the two solutions Σ1 and Σ2 are

The amplitudes A±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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# Platonic polyhedra tune the 3-sphere: Harmonic analysis on simplices by Kramer

This week I have been reviewing the paper: Platonic polyhedra tune the 3-sphere: Harmonic analysis on simplices by Kramer. This Harmonic analysis can be applied to the cosmic microwave background observed in astrophysics. The Selection rules found in this analysis can detect the multiple connectivity of spherical
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 π1(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) = |π1(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 S0(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 S0(3). Its internal point symmetry group is S(4). The homotopy group π1( S0(3)) of the Platonic tetrahedron is described by a graph algorithm. Its prime dimension 5 identifies it and the group of deck transformations as the cyclic group C5.  The group/subgroup analysis can be used to characterize the harmonic analysis on S0(3).

The reduction S(5) > C5

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 D0(C5) is straightforward and we include it
in the last column of the table below. The representation D0 is contained once in the representations f = [5] , [11111] , [32] , [221], twice in the representation f = [311], but not in the representations f = [41] , f = [2111].

Harmonic analysis on S0(3)
Summary the basis construction for the harmonic analysis on S0(3) in terms of C5-periodic states on the sphere S³.

The spherical harmonics for fixed degree 2j = 0, 1, 2, . . . are the Wigner Dj(u) functions. The Wigner Dj 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 S0(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) > Cn+1 eliminate complete representations Df  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.

# U(N) Coherent States for Loop Quantum Gravity by Freidel and Livine

This week I have been reading a paper by Freidel and Livine which investigates the geometry of the space of N-valent SU(2)intertwiners. In this paper, the authors propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian GrN,2, and have  a geometrical interpretation in terms of framed polyhedra and are related to  coherent intertwiners – see the post

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 je on its edges and intertwiners iv on its vertices. The spins define quanta of area while the intertwiners describe chunks of space volume. The dynamics then acts on the spins je and intertwiners iv, 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 iv 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 Eij 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 HN is established as a Fock space by introducing annihilation and creation operators  Fij , Fij , which allow transitions between intertwiners with different total areas.  These creation operators can be used to define U(N) coherent states |J,  zi 〉 ∝ Fz| 0〉labeled by the total area J and a set of N spinors zi . These states turn out to have very interesting properties.

• They transform simply under U(N)-transformations                             u |J,  zi 〉 = |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,zi〉 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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# A Helium Atom of Space: Dynamical Instability of the Isochoric Pentahedron by Coleman-Smith and Mullery

This week I have been reviewing a paper on the  Isochoric  Pentahedron. In this paper, the authors present an analysis of the dynamics of the equifacial pentahedron on the Kapovich-Millson phase space under a volume preserving Hamiltonian. The classical dynamics of polyhedra under this Hamiltonian may arise from the classical limit of the node volume operators in loop quantum gravity. The pentahedron is the simplest nontrivial polyhedron for which the dynamics may be chaotic.  Canonical and microcanonical estimates of the Kolmogorov-Sinai entropy suggest that the pentahedron is a strongly chaotic system. The presence of chaos is further suggested by calculations of intermediate time Lyapunov exponents which saturate to non-zero values.

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 Al and normals nl of each face are sufficient to uniquely characterize a polyhedron. The polyhedral closure relationship

is a sufficient condition on Al 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.

# Causal cells: spacetime polytopes with null hyperfaces by Neiman

This week I have been reading a paper about polyhedra and 4-polytopes in Minkowski spacetime – in particular, null polyhedra
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 Hyperplanes

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 ℓμ to the hyperplane, ℓμ (1, 0, 0, 1) is a null vector, i.e. ℓμμ = 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 {ni}
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  ημν(i)μ(j)νof 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  Lij =(3!)²ημν(i)μ(j)ν of their scalar products. The diagonal elements of Lij 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.

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# Deformations of Polyhedra and Polygons by the Unitary Group by Livine

This week whilst preparing some calculations I have been looking at the paper ‘Deformations of Polyhedra and Polygons by the Unitary Group’. In the paper the author inspired by loop quantum gravity, the spinorial formalism and the structures of twisted geometry discusses the phase space of polyhedra in three dimensions and its quantization, which serve as basic building of the kinematical states of discrete geometry.

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 C2N//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 C2N//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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