Tag Archives: Phase space

A Helium Atom of Space: Dynamical Instability of the Isochoric Pentahedron by Coleman-Smith and Mullery

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

helequ1

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:

helequ2

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:

helequ3

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

helequpq

This may be visualized by representing the polyhedron as a polygon with edges given by the vectors helequvk  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:

helfig1

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

helequ14a

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

helequ15

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

helequ18

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.

helfig5

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.

helfig10

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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Classical and Quantum Polyhedra by John Schliemann

This week I have been  looking again at the Quantum Tetrahedron and quantum polyhedra in general. I’ll be doing further  numerical studies to add to the numerical work done in earlier posts:

Quantum polyhedra constructed from angular momentum operators are the building blocks of space in its quantum description by Loop Quantum Gravity. The author extends results on the semiclassical properties of quantum polyhedra. They compare results from a canonical quantization of the classical system with a recent wave function based approach to the large-volume sector of the quantum system. Both methods agree in the leading order of the resulting effective operator given by an harmonic oscillator, while minor differences occur in higher corrections. Perturbative inclusion of such corrections improves the approximation to the eigenstates. Moreover, the comparison of both methods leads also to a full wave function description of the eigenstates of the (square of the) volume operator at negative eigenvalues of large modulus.

For the case of general quantum polyhedra described by discrete angular momentum quantum numbers the authors formulate a set of quantum operators fullling in the semiclassical regime the standard commutation relations between momentum and position. The position variable here is chosen to have dimension of Planck length squared which facilitates the identication of quantum corrections.

Introduction
The quantum volume operator is pivotal for the construction of space-time dynamics within this Loop Quantum Gravity. Traditionally two versions of such an operator are discussed, due to Rovelli and Smolin, and to Ashtekar and Lewandowski, and more recently, Bianchi, Dona, and Speziale offered a third proposal for a volume operator which is closer to the concept of spin foams. It relies on an older geometric theorem due to Minkowski stating that N face areas Ai with normal vectors ni such that

CCqTequ1

uniquely define a convex polyhedron of N faces with areas Ai.

The approach amounts to expressing the volume of a classical polyhedron in terms of its face areas, which are in turn promoted to be operators. Minkowski’s proof, however, is not constructive,
and a remaining obstacle of this approach  to a volume operator is to actually find the shape of a general polyhedron given its face areas and face normals. Such difficulties do not occur in the simplest case
of a polyhedron, i.e. a tetrahedron consisting of four faces represented by angular momentum operators coupling to a total spin singlet. Indeed, for such a quantum tetrahedron all three definitions of the volume operator coincide. On the other hand, for a classical tetrahedron the general phase space parametrization
devised by Kapovich and Millson results in just one pair of canonical variables, and the square of the volume operator can explicitly formulated in terms of these . Bianchi and Haggard have performed a Bohr-Sommerfeld quantization of the classical tetrahedron where the role of an Hamiltonian generating classical orbits is played by the volume operator squared. The resulting semiclassical eigenvalues agree extremely well with exact numerical data, see the post

 

The above observations make clear that classical tetrahedra, the simplest structures a volume can be ascribed to, should be considered as perfectly integrable systems. In turn, a quantum tetrahedron can be viewed as the hydrogen atom of quantum spacetime, whereas the next complicated case of a pentahedron might be referred to as the helium atom.

Recently, Schliemann put forward another approach to the semiclassical regime of quantum tetrahedra, see the post

Here, by combining observations on the volume operator squared and its eigenfunctions as opposed to the eigenvalues, an effective operator in terms of a quantum harmonic oscillator was derived providing an accurate as well as transparent description of the the large-volume sector.

One of the purposes of this paper is to demonstrate the relation between the different treatments of quantum tetrahedra sketched above.

The outline of this paper is as follows.

  • Summarize the Kapovich-Millson phase space parametrization of general classical polyhedra.
  • Reviewing  the classical tetrahedron and expand of the volume squared around its maximum and minimum in up to quadrilinear order.
  • The quantum tetrahedron.

 Classical Polyhedra

 Kapovich-Millson Phase Space Variables

Viewing the vectors Ai as angular momenta, the Poisson
bracket of arbitrary functions of these variables read

CCqTequ2v1

To implement the closure relation   define

CCqTequ3

resulting in N -3 momenta pi =|pi|. The canonical conjugate variables qi are then given by the angle between the vector

CCqTequ4

These quantities fulfill the canonical Poisson relations

CCqTequ5

The Tetrahedron

The classical volume of a tetrahedron can be expressed
as

CCqTequ6

Look at the quantity,

CCqTequ7

This can be  expressed in terms of the phase space variables p1, q1 using;

CCqTequ8and with

CCqTequ9where Δ(a, b, c) is the area of a triangle with edges a,b, c expressed via Heron’s formula,

CCqTequ10

and

CCqTequ11such that

CCqTequ12

In order to make closer contact to the quantum tetrahedron introduce the notation

CCqTequ13

fullling {p,A} = 1 and

CCqTequ14

with

CCqTequ15

where A varies according to Amin ≤A ≤Amax with

CCqTequ16

β(A) is a nonnegative function with β(Amin) = β (Amax) = 0, and it has a unique maximum at  A between Amin and Amax. Thus, Q has a maximum at A = k and p = 0 while the unique minimum lies at p = . Expanding around the maximum gives

CCqTequ18

with

CCqTequ19

and

CCqTequ20

 

The analogous expansion around the minimum reads

CCqTequ21

 

Concentrating in both cases on the quadratic contributions,
one obtains two harmonic oscillators,

CCqTequ22

 

The Quantum Tetrahedron

General Properties

A quantum tetrahedron is defined by four angular momentum operators ji representing its faces and coupling to a total singlet the Hilbert space consists of all states |k〉 fulling

CCqTequ24

 

 A usual way to construct this space is to couple first the pairs j1,j2 and j3, j4 to two irreducible SU(2) representations of dimension 2k+1 each. For j1, j2 this standard construction reads explicitly

CCqTequ25

CCqTequ27such that

CCqTequ27

where 〈j1m1j2m2|km〉 are Clebsch-Gordan coefficients

Defining analogous states |km〉34 for j3, j4, the quantum number k  becomes restricted by kmin ≤ k ≤kmax with

CCqTequ29

The two multiplets |km〉12, |km〉34 are then coupled to a
total singlet,

CCqTequ31The states jki span a Hilbert space of dimension d = kmax – kmin + 1.

The volume operator of a quantum tetrahedron can be
formulated as

CCqTequ32

where the operators

CCqTequ33

represent the faces of the tetrahedron with CCqTequ33abeing the Planck length squared. Consider the operator

CCqTequ34

which reads in the basis of the states |k〉 as

CCqTequ35

For even d, the eigenvalues of Q come in pairs (q, -q), and since

CCqTequ37

the corresponding eigenstates fulfill

CCqTequ38

For odd d an additional zero eigenvalue occurs.

To make further contact between the classical and the
quantum tetrahedron define

CCqTequ39

fulfilling

CCqTequ41

also

CCqTequ42

and

CCqTequ43

is the projector onto the singlet space.

So far have followed the formalism common to
the literature and parametrized the Hilbert space of the
quantum tetrahedron by a dimensionless quantum number
k, whereas the phase space variable A of the classical
tetrahedron has dimension of area. In order to establish
closer contact between both descriptions, rescale the
involved quantum numbers by the Planck length squared
according to

CCqTequ44

to quantities having also dimension of area.

This gives,

CCqTequ46

with

CCqTequ48

β(a) has a unique maximum at some a.

The Quantum Tetrahedron at Large Volumes

In the post Large-Volume Limit of a Quantum Tetrahedron is a Quantum Harmonic Oscillator

It was shown how to accurately describe the large-volume semiclassical regime of Q or R by a quantum harmonic oscillator in real-space representation with respect to a or k, respectively.

Here the  analysis is extended by taking into account higher order corrections.

label the eigenstates of Q by |n〉, n ∈ {0,1, 2….}, in descending order of eigenvalues with |0〉 being the state of largest eigenvalue. With respect to the basis states |k〉 they can be expressed as

CCqTequ50

Taking the view of the standard Schrodinger formalism of elementary quantum mechanics, the coefficients 〈a|n〉 are the wave function of the state |n〉 with respect to the coordinate a.

Evaluating the matrix elements

CCqTequ51

one obtains up to fourth order in the expansions

CCqTequ53

CCqTequ54

 

Introducing the operators

CCqTequ56

The effective operator expression is

CCqTequ58

Concentrating on the quadratic contributions in gives the harmonic-oscillator expression

CCqTequ61

with eigenvalues

CCqTequ62

and corresponding eigenfunctions

CCqTequ63

where Hn(x) are the usual Hermite polynomials.

fig1

fig2

CONCLUSIONS

The investigation of the semiclassical limit of Loop
Quantum Gravity is one of the key issues in that approach to  quantum gravity. This paper has focussed  on the semiclassical properties of quantum polyhedra. Regarding tetrahedra as their simplest examples, it has been established that there is a connection
between a canonical quantization of the classical system  and the  wave function based approach  to the large-volume sector of the quantum system. In the leading order both routes concur yielding a quantum harmonic oscillator as an effective
description for the square of the volume operator.

A further interesting point is the zero eigenvalue occurring for tetrahedra with odd Hilbert space dimension d. The Bohr-Sommerfeld quantization carried out by Bianchi and Haggard gives accurate results for eigenvalues.

 Related articles

Semiclassical analysis of Loop Quantum Gravity by Claudio Perini

This week I have been reading the PhD thesis Semiclassical analysis of Loop Quantum Gravity by Claudio Perini .

Semiclassical states for quantum gravity

The concept of semiclassical state of geometry is a key ingredient in the semiclassical analysis of LQG. Semiclassical states are kinematical states peaked on a prescribed intrinsic and extrinsic geometry of space. The simplest semiclassical geometry one can consider is the one associated to a single node of a spin-network with given spin labels. The node is labeled by an intertwiner, i.e. an invariant tensor in the tensor product of the representations meeting at the node. However a generic intertwiner does not admit a semiclassical interpretation because expectation values of non-commuting geometric operators acting on the node do not give the correct classical result in the large spin limit. For example, the 4-valent intertwiners defined with the virtual spin do not have the right semiclassical behavior; one has to take a superposition of them with a specific weight in order to construct semiclassical intertwiners.

The Rovelli-Speziale quantum tetrahedron  is an example of semiclassical geometry; there the weight in the linear superposition of virtual links is taken as a Gaussian with phase. The Rovelli-Speziale quantum tetrahedron is actually equivalent to the Livine-Speziale coherent intertwiner with valence 4 more precisely, the former constitutes the asymptotic expansion of the latter for large spins. Coherent intertwiners are as the geometric quantization of the classical phase space associated to the degrees of freedom of a tetrahedron.

In  recent graviton propagator calculations of semiclassical states
associated to a spin-network graph Γ have been  considered. The states used in the definition of semiclassical n-point functions ) are labeled by a spin jo and an angle ξe per link e of the
graph, and for each node a set of unit vectors n, one for each link surrounding that node. Such variables are suggested by the simplicial interpretation of these states: the graph Γ is in fact assumed to be dual to a simplicial decomposition of the spatial manifold, the vectors n are associated to unit-normals to faces of tetrahedra, and the spin jo is the average of the area of a face. Moreover, the simplicial extrinsic curvature is an angle associated to faces shared by tetrahedra and is identified with the label ξe. Therefore, these
states are labeled by an intrinsic and extrinsic simplicial 3-geometry. They are obtained via a superposition over spins of spin-networks having nodes labeled by Livine-Speziale coherent intertwiners.
The coefficients cj of the superposition over spins are given by a Gaussian times a phase as originally proposed by Rovelli.

semi269

Such proposal is motivated by the need of having a state peaked both on the area and on the extrinsic angle. The dispersion is chosen to be given by

semi269a

so that, in the large jo limit, both variables have vanishing relative dispersions. Moreover, a recent result of Freidel and Speziale strengthens the status of these classical labels: they show that the
phase space associated to a graph in LQG can actually be described in terms of the labels (jo , ξe, ne, n′) associated to links of the graph. The states have good semiclassical properties and a clear geometrical
interpretation.

Within the canonical framework, Thiemann and collaborators have strongly advocated the use of complexifier coherent states . Such states are labeled by a graph Γ and by an assignment of a SL(2,C) group element to each of its links. The state is obtained from the gauge-invariant projection of a product over links of modified heat-kernels for the complexification of SU(2). Their peakedness properties have been studied in detail.

Perini’s thesis presents the  proposal of coherent spin-network states: the proposal is to consider the gauge invariant projection of
a product over links of Hall’s heat-kernels for the cotangent bundle of SU(2). The labels of the state are the ones used in Spin Foams: two normals, a spin and an angle for each link of the graph. This set of labels can be written as an element of SL(2,C) per link of the graph. Therefore, these states coincide with Thiemann’s coherent states with the area operator chosen as complexifier, the SL(2,C) labels written in terms of the phase space variables (jo , ξe, ne, n′e) and the heat-kernel time given as a function of jo.

The author shows that, for large jo , coherent spin-networks reduce to the semiclassical states used in the spinfoam framework. In particular that they reproduce a superposition over spins of spin-networks with nodes labeled by Livine-Speziale coherent intertwiners and coefficients cj given by a Gaussian times a phase as originally proposed by Rovelli. This provides a clear interpretation of the geometry these states are peaked on.

Livine-Speziale coherent intertwiners
In ordinary Quantum Mechanics, SU(2) coherent states are defined as the states that minimize the dispersion

semi270

of the angular momentum operator J, acting as a generator of rotations on the representation space Hj ≃C2j+1 of the spin j representation of SU(2). On the usual basis |j,m> formed by simultaneous eigenstates of J2 and J3 we have

semi271

so the maximal and minimal weight vectors |j,±j> are coherent states. Starting from |j, j〉, the whole set of coherent states is constructed through the group action

semi272

One can take a subset of them labelled by unit vectors on the sphere S²:

semi273

where n is a unit vector defining a direction on the sphere S² and g(n) a SU(2) group element rotating the direction z ≡ (0, 0, 1) into the direction n. In other words, a coherent states is a state satisfying

semi274

For each n there is a U(1) family of coherent states and they are related one another by a phase factor. The choice of this arbitrary phase is equivalent to a section of the Hopf fiber bundles :

S2 ≃ SU(2)/U(1) → SU(2).

Explicitly, denoting n = (sin θ cos φ, sin θ sin φ, cos θ), a possible section is

semi275

where m ≡ (sin φ,−cos φ, 0) is a unit vector orthogonal both to z andn. A coherent state can be expanded in the usual basis as

semi276

semi277

Coherent states are normalized but not orthogonal, and their scalar product is

semi278

where A is the area of the geodesic triangle on the sphere S² with vertices z, n and n. Furthermore they provide an overcomplete basis for the Hilbert space Hj of the spin j irreducible representation of SU(2), and the resolution of the identity can be written as

semi278a

with d²n the normalized Lebesgue measure on the sphere S².

The Livine-Speziale coherent intertwiners are naturally defined taking the tensor product of V coherent states (V stands for valence of the node) and projecting onto the gauge-invariant subspace:

semi279

Here the projection is implemented by group averaging. They are labeled by V spins j and V unit vectors n. The states |j, n〉o carry enough information to describe a classical geometry associated to the node. Interpret the vectors j,n as normal vectors to triangles, normalized to the areas j of the triangles.

 Coherent tetrahedron

The case of 4-valent coherent intertwiners is of particular importance for LQG and especially for Spin Foam Models. In fact it is the lowest valence carrying a non-zero volume and most SFM’s are build over a simplicial triangulation, so that the boundary state space has only 4-valent nodes, dual to tetrahedra. A 4-valent coherent intertwiner with normals satisfying the closure condition can be interpreted as a semiclassical tetrahedron. In fact expectation values of geometric operators associated to a node give the correct classical quantities in the semiclassical regime. This regime is identified with the large spin (large areas) asymptotics. In the following we give some details.

Define n = jn as the normals normalized to the area. In terms of them, the volume (squared) of the tetrahedron is given by the simple relation:

semi280

The geometric quantization of these degrees of freedom is based on the identification of generators Ji of SU(2) as quantum operators corresponding to the  ni . This construction gives directly
the same quantum geometry that one finds via a much longer path by quantising the phase space of General Relativity, that is via Loop Quantum Gravity. The squared lengths |ni|² are the SU(2) Casimir
operators C²(j), as in LQG. A quantum tetrahedron with fixed areas lives in the tensor product ⊗Hj .The closure constraint reads:

semi281

and imposes that the state of the quantum tetrahedron is invariant under global rotations. The state space of the quantum tetrahedron with given areas is thus the Hilbert space of intertwiners

semi282

The operators Ji · Jj are well defined on this space, and so is the operator

semi283

Its absolute value |U| can immediately be identified with the quantization of the classical squared volume 36V²,  in agreement with standard LQG results.

To find the angle operators,  introduce the quantities  Jij := Ji+  Jj. Given these quantities, the angle operators θij can be recovered from

semi284

The quantum geometry of a tetrahedron is encoded in the operators Ji ², Jij² and U, acting on Ij₁…j. It is a fact that out of the six independent classical variables parametrizing a tetrahedron, only
five commute in the quantum theory. Indeed while we have

[Jk², Ji · Jj] = 0,

it is easy to see that:

semi285

A complete set of commuting operators, in the sense of Dirac, is given by the operators Ji² , J². In other words, a basis for Ij…j4 is provided by the eigenvectors of any one of the operators Jij² . We write the corresponding eigenbasis as |j〉ij . These are virtual links here we are introducing them via a geometric quantization of the classical tetrahedron. For instance, the basis |j〉 diagonalises the four triangle areas and the dihedral angle θ or, equivalently, the area A of one internal parallelogram. The relation between different basis is  obtained from SU(2) recoupling theory: the matrix describing the change of basis in the space of intertwiners is given by the usual Wigner 6j-symbol,

semi286

so that

semi287

Notice that from the orthogonality relation of the 6j-symbol,

semi288

we have

semi289

The states |j〉 are eigenvectors of the five commuting geometrical operators Ji² , J²    so the average value of the operator corresponding to the sixth classical observable, say J² is on these states maximally spread. This means that a basis state has undetermined classical geometry or, in other words, is not an eigenstate of the geometry.  Then led to consider superpositions of states to be able to study the semiclassical limit of the geometry. Suitable superpositions could be constructed for instance requiring that they minimise the uncertainty relations between non–commuting observables, such as

semi290

States minimising the uncertainty above are usually called coherent states. Coherent intertwiners seem not to verify exactly semi290, but they are such that all relative uncertainties h〈Δ²〉Jij/〈J²ij〉, or equivalently 〈Δθij〉/〈θij〉, vanish in the large scale limit. The limit is defined by taking the limit when all spins involved go uniformly to infinity, namely ji = λki with λ → ∞. Because of these good semiclassical properties we can associate to a coherent intertwiner the geometrical interpretation of a semiclassical tetrahedron; an analogous interpretation should be also valid nodes of higher valence.

 Related articles

Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity by Hal Haggard

This week I have been reading a couple of PhD thesis. The first is ‘The Chiral Structure of Loop Quantum Gravity‘ by Wolfgang  Wieland, the other is Hal Haggard’s PhD thesis, ‘Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity‘ .

I’m going to focus in this post on a small section of  Haggard’s work about  the quantization of space.

In loop gravity quantum states are built upon a spin coloured graph. The nodes of this graph represent three dimensional grains
of space and the links of the graph encode the adjacency of these regions. In the semiclassical description of the grains of space – see the post:

these grains are  interpreted as giving rise to convex polyhedra in this limit. This geometrical picture suggests a natural proposal for the quantum volume operator of a grain of space: it should be an operator that corresponds to the volume of the associated classical polyhedron. This section provides a detailed analysis of such an operator, for a single 4-valent node of the graph. The valency of the node determines the number of faces of the polyhedron and so we will be focusing on classical and quantum tetrahedra.

The space of convex polyhedra with fixed face areas has a natural phase space and symplectic structure.  Using this kinematics we can study the classical volume operator and perform a Bohr-Sommerfeld quantization of its spectrum. This quantization
is in good agreement with previous studies of the volume operator spectrum and provides a simplified derivation.

This section of the thesis looks at:

  • The node Hilbert Spaces Kn for general valency.
  • The classical phase space and  its Poisson structure.
  • The volume operator  in loop gravity.
  •  The quantum tetrahedron
  • Phase space for the classical tetrahedron and its Bohr-Sommerfeld quantization.
  • Wavefunctions for the volume operator

 Setup

Focus on a single node n and its Hilbert space
Kn. The space Kn is defined as the subspace of the tensor
product Hj1⊗ · · ·  ⊗HjF that is invariant under global SU(2) transformations

halequ1

We call Kn the space of intertwiners. The diagonal action is generated by the operatorJ,

halequ2

States of Kn are called intertwiners and can be expanded as:

halequ3

The components i transform as a tensor under SU(2) transformations in such a way that the condition

halequ4

is satisfied. These are precisely the invariant tensors captured graphically by spin networks.The finite dimensional intertwiner space Kn can be understood as the quantization of a classical phase space.

Begin with Minkowski’s theorem, which states the following: given F vectors Ar ∈ R³ (r = 1, . . . , F) whose sum is zero

halequ5

 

then up to rotations and translations there exists a unique convex polyhedron with F faces associated to these vectors. These convex polyhedra are our semiclassical interpretation of the loop gravity grains of space.

The second step  is to associate a classical phase space to these
polyhedra

We call this the space of shapes and denote it P(A1, . . . ,AF ) or more briefly PF.  Interpret the partial sums halequ6

as generators of rotations about the μk = A1 + · · · + Ak+1 axis.This interpretation follows naturally from considering each of the Ar vectors to be a classical angular momentum, so that really Ar ∈ Λr ≅R³.

There is is a natural Poisson structure on each r and this extends to the bracket

halequ7

With this Poisson bracket the μk do in fact generate rotations about the axis A1 + · · ·Ak+1. This geometrical interpretation suggests a natural conjugate coordinate, namely the angle φk of the rotation. Let k be the angle between the vectors

halequ8

then

halequ9

The pairs (μk,φk) are canonical coordinates for a classical phase space of dimension 2(F −3). Thisis called  the space of shapes and denote it P(A1, . . . ,AF ) or more briefly PF.

In the final step we  show that that quantization of PF is the Hilbert space Kn of an F-valent node n. The constraint

halequ11

beomes

halequ12

which is precisely the gauge invariance condition. This is where the classical vector model of angular momentum originates.

 Volume operators in loop gravity

Most of the loop gravity research on the volume operator has been done in the context of the original canonical quantization of general relativity.  In Ashtekar’s formulation of classical general relativity the gravitational field is described in terms of the triad variables E. This triad corresponds to a 3-metric h and is called the electric field. The elementary quantum operator that measures the geometry of space
corresponds to the flux of the electric field through a surface S. When such a surface is punctured by a link of the spin network graph Γ the flux can be parallel transported, back along the link, to the node using the second of Ashtekar’s variables, the
Ashtekar-Barbero connection. This results in an SU(2) operator that acts on the intertwiner space Kn at the node n.The parallel transported flux operator at the node is proportional to the generator of SU(2) transformations

halequ13

where γ is a free parameter of the theory called the Barbero-Immirzi parameter and Pl is the Planck length.

The volume of a region of space R is obtained by regularizing and quantizing the classical expression

halequ14

using the operators Er. The total volume is obtained by summing the contributions from each node of the spin network graph Γcontained in the region R.

There are different proposals for the volume operator at a node. The operator originally proposed by Rovelli and Smolin is

halequ15

A second operator introduced by Ashtekar and Lewandowski is

halequ16

Both the Rovelli-Smolin and the Ashtekar-Lewandowski proposals have classical versions. This results in two distinct functions on phase space:

halequ17

A third proposal for the volume operator at a node has emerged,
Dona, Bianchi and Speziale suggest the promotion of the classical volume of the polyhedron associated to {Ar} to an operator

halequ18

In the case of a 4-valent node all three of these proposals agree.

The volume of a quantum tetrahedron

In the case of a node with four links, F = 4, all the proposals for the volume operator discussed above coincide and match the operator introduced by Barbieri for the volume of a quantum tetrahedron.

– see post Quantum tetrahedra and simplicial spin networks  by A.Barbieri

The Hilbert space K4 of a quantum tetrahedron is the intertwiner space of four representations of SU(2),

halequ19Introduce  basis into this Hilbert space using the recoupling channel Hj1 ⊗Hj2 and call these basis states |k>. The basis vectors are defined as

halequ20

where the tensor ik is defined in terms of the Wigner 3j-symbols as

halequ21The index k ranges from kmin to kmax in integer steps with

halequ22

The dimension d of the Hilbert space K4 is finite and given by

halequ23
The states |k> form an orthonormal basis of eigenstates of the operator Er · Es. This operator measures the dihedral angle between the faces r and s of the quantum tetrahedron.
The operator The operator √Er · Er measures the area of the rth face of the quantum tetrahedron and states in K4 are area eigenstates with eigenvalues halequ24a,

halequ24

The volume operator introduced by Barbieri is

halequ25

and because of the closure relation

halequ26

this operator coincides with the Rovelli-Smolin operator for                  α = 2√2/3. The volume operator introduce by Barbieri can be understood as a special case of the volume of a quantum polyhedron.

In order to compute the spectrum of the volume operator, it is useful to introduce the operator Q defined as

halequ27

It represents the square of the oriented volume. The matrix elements of this operator are  computed in the post:

The eigenstates |q> of the operator Q,

halequ28

are also eigenstates of the volume. The eigenvalues of the volume are simply given by the square-root of the modulus of q,

halequ30

The matrix elements of the operator Q in the basis |k> are given by

halequ31

The function Δ(a, b, c) returns the area of a triangle with sides of length (a, b, c) and is conveniently expressed in terms of Heron’s formula

halequ32

This can be done numerically and we can  compare the eigenevalues calculated in this manner to the results of the Bohr-Sommerfeld quantization.

There are a number of properties of the spectrum of Q and therefore of V that can be determined analytically.

  • The spectrum of Q is non-degenerate: it contains d distinct real eigenvalues. This is a consequence of the fact that the matrix elements of Q on the basis |k> determine a d × d Hermitian matrix of the formhalequ33

with real coefficients ai.

  • The non-vanishing eigenvalues of Q come in pairs ±q. A vanishing eigenvalue is present only when the dimension d of the intertwiner space is odd.
  • For given spins j1, . . . , j4, the maximum volume eigenvalue can be estimated  using Gershgorin’s circle theorem and wefind that it scales as halequ35a   where jmax is the largest of the four spins jr.
  • The minimum non-vanishing eigenvalue (volume gap)
    scales as halequ35b

Tetrahedral volume on shape space

The starting point for the  Bohr-Sommerfeld analysis is the volume of a tetrahedron as a function on the shape phase space P(A1, . . . ,A4)≡  P4.

The Minkowski theorem guarantees the existence and uniqueness
of a tetrahedron associated to any four vectors Ar, (r = 1, . . . , 4) that satisfy A1 + · · · + A4 = 0. The magnitudes Ar ≡ |Ar|, (i = 1, . . . , 4) are interpreted as the face areas. A condition for the existence of a tetrahedron is that A1 +A2 +A3  ≥ A4, equality giving a flat -zero volume tetrahedron. The space of tetrahedra with four fixed face areas P(A1,A2,A3,A4) ≡ P4 is a sphere.

halfig1

Consider the classical volume,

halequ37it will be more straightforward to work with the squared classical volume,

halequ38

Writing the triple product of Q as the determinant of a matrix M = (A1,A2,A3) whose columns are the vectors A1,A2 and A3 and squaring yields,

halequ42

Bohr-Sommerfeld quantization of tetrahedra

The Bohr-Sommerfeld quantization condition is expressed in
terms of the action I associated to the each of these orbits:

halequ47

halfig2

Wavefunctions

Just as the Bohr-Sommerfeld approximation can be used for finding the eigenvalues of the volume operator. Semiclassical techniques can
also be used to find the volume wavefunctions.

halequ97

where,

halequ93

and

halequ94

Conclusions
At the Planck scale, a quantum behaviour of the geometry of space is expected. Loop gravity provides a specific realization of this expectation: it predicts a granularity of space with each grain having a quantum behaviour.

Based on semiclassical arguments applied to the simplest model for a grain of space, a Euclidean tetrahedron, and is closely related to Regge’s discretization of gravity and to more recent ideas about
general relativity and quantum geometry. The spectrum has been computed by applying Bohr-Sommerfeld quantization to the volume of a tetrahedron seen as an observable on the phase space of shapes.
There is quantitative agreement of the spectrum calculated here and
the spectrum of the volume in loop gravity. This result lends credibility to the intricate derivation of the volume spectrum in loop gravity, showing that it matches an elementary semiclassical approach.

Related articles

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Pentahedral volume, chaos, and quantum gravity by Haggard

This week I have been reviewing a couple of papers by Hal Haggard:

  • Pentahedral volume, chaos, and quantum gravity
  • Dynamical Chaos and the Volume Gap

These are related to the posts:

In these papers the author shows that chaotic classical dynamics associated to the volume of discrete grains of space leads to quantal spectra that are gapped between zero and nonzero volume. This strengthens the connection between spectral discreteness in the quantum geometry of gravity and tame ultraviolet behavior.  A detailed analysis of the geometry of a pentahedron, provides new insights into the volume operator and evidence of classical chaos in the dynamics it generates.

An outgrowth of quantum gravity has been the discovery that convex polyhedra can be endowed with a dynamical phase space structure. This structure was utilized to perform a Bohr-Sommerfeld quantization of the volume of a tetrahedron, yielding a novel
route to spatial discreteness and new insights into the spectral properties of discrete grains of space. Many approaches to quantum gravity rely on discretization of space or spacetime. This allows one to control, and limit, the number of degrees of freedom of the gravitational field being studied. Attention is often restricted to simplices. These papers study of grains of space more complex than simplices.

The Bohr-Sommerfeld quantization relied on the
integrability of the underlying classical volume dynamics, that is, the dynamics generated by taking as Hamiltonian the volume, H = Vtet. In general, integrability is a special property of a dynamical system exhibiting a high degree of symmetry. Instead, Hamiltonians with
two or more degrees of freedom are generically chaotic. Polyhedra with more than four faces are associated to systems with two or more degrees of freedom and so ‘Are their volume dynamics chaotic?’

The answer to this question has important physical consequences for quantum gravity. Prominent among these is that chaotic volume dynamics implies that there is generically a gap in the volume spectrum separating the zero volume eigenvalue from its
nearest neighbors. In loop gravity, it is convenient to
work with a polyhedral discretization of space because it
allows concrete study of a few degrees of freedom of the
gravitational fi eld;

polyfig1

however, what is key is the spectral discreteness of the geometrical operators of the theory. This is because the partition functions and transition amplitudes that defi ne the theory are expressed as sums over these area and volume eigenvalues. The generic presence
of gaps in the spectra of these operators ensures that these sums will not diverge as smaller and smaller quanta are considered; such a theory should be well behaved in the ultraviolet regime.

The author looks at the classical volume associated to a single pentahedral grain of space. He provides  evidences that the pentahedral volume dynamics is chaotic.  A new formula for the volume of a pentahedron in terms of its face areas and normals is found and it is shown that the volume dynamics is adjacency changing, General results from random matrix theory are used to argue that a chaotic volume dynamics implies the generic presence of a volume gap.

Consider a single pentahedral grain of space. An examination
of the classical volume dynamics of pentahedra relies on turning the space of convex polyhedra living in Euclidean three-space into a phase space. This is accomplished with the aid of two results:

  • Minkowski’s theorem states that the shape of a polyhedron is completely characterized by the face areas A and face normals n. More precisely, a convex polyhedron is uniquely determined, up to rotations, by its area vectors. The space of shapes of polyhedra with N faces of given areas A is:

pentahedral_equ1b

 

The space PN naturally carries the structure of a phase space, with Poisson brackets,

pentahedral_equ1a

This is the usual Lie-Poisson bracket if the A are interpreted physically as angular momenta, i.e. as generators of rotations.

To study the pentahedral volume dynamics on P5, with
H = Vpent, it is  necessary to find this volume as a function of the area vectors. This can be done as shown in the diagram below:

polyfig2

The main findings of the paper are summarized in a pentahedral phase diagram:

pentahedral_fig3

 

The presence of a volume gap in the integrable case of a tetrahedron has already been established. Furthermore, a chaotic pentahedral volume dynamics strongly suggest that there will be chaos for polyhedra with more faces, which have an even richer structure in their phase space. Consequently, the argument presented in this paper provides a very general mechanism that would ensure a volume gap for all discrete grains of space.

 

In this paper the authors have completely solved the geometry of
a pentahedron speci ed by its area vectors and defined its
volume as a function of these variables. By performing a numerical integration of the corresponding volume dynamics they have given early indicators that it generates a chaotic flow in phase space. These results uncover a new mechanism for the presence of a volume gap in the spectrum of quantum gravity: the level repulsion of quantum
systems corresponding to classically chaotic dynamics. The generic presence of a volume gap further strengthens the expected ultraviolet finiteness of quantum gravity theories built on spectral discreteness.

 

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Numerical work with sagemath 15: Holomorphic factorization

This week I have been  reviewing the new spinfoam vertex in 4d models of quantum gravity. This was discussed in the recent posts:

In this post I explore the large spins asymptotic properties of the overlap coefficients:

holoeeq
characterizing the holomorphic intertwiners in the usual real basis. This consists of the normalization coefficient times the shifted Jacobi polynomial.

In the case  of n = 4. I can study the asymptotics of the shifted Jacobi polynomials in the limit ji → λji, λ → ∞.  A  convenient integral representation for the shifted Jacobi polynomials is given by a contour integral:

holoequ137

This leads to the result that:

This formula relates the two very different descriptions of the phase space of shapes of a classical tetrahedron – the real one in terms of the k, φ parameters and the complex one in terms of the cross-ratio
coordinate Z. As is clear from this formula, the relation between the two descriptions is non-trivial.

In this post I have only worked with the simplest case of this relation when all areas are equal. In this ‘equi-area‘ case where all four representations are equal ji = j, ∀i = 1, 2, 3, 4, as described in the post: Holomorphic Factorization for a Quantum Tetrahedron the overlap function is;

holoequ154

Using sagemath I am able to evaluate the overlap coefficients for various values of j and the cross-ratios z.

holomorphic1

holo4

Here I plot the modulus of the equi-area case state Ck, for j = 20, as a function of the spin label k, for the value of the cross-ratio Z = exp(iπ/3) that corresponds to the equilateral tetrahedron. It is obvious that the distribution looks Gaussian. We also see that the maximum is reached for kc = 2j/√3 ∼ 23, which agrees with an asymptotic analysis.

Here I plot the modulus of the equi-area case state Ck for various j values as a function of the spin label k, for the value of the cross-ratio Z = exp(iπ/3) that corresponds to the equilateral tetrahedron.

cross

Here I have  have plotted the modulus of the j = 20 equi-area state Ck for increasing cross-ratios Z = 0.1i, 0.8i, 1.8i. The Gaussian distribution progressively moving its peak from 0 to 2j. This illustrates how changing the value of Z affects the semi-classical geometry of the tetrahedron.

Conclusions

In this post I we have studied a holomorphic basis for the Hilbert space Hj1,…,jn of SU(2) intertwiners. In particular I have looked at the case of 4-valent intertwiners that can be interpreted as quantum states of a quantum tetrahedron. The formula

holoequ53
gives the inner product in Hj1,…,jn in terms of a holomorphic integral over the space of ‘shapes’ parametrized by the cross-ratio coordinates Zi. In the tetrahedral n = 4 case there is a single cross-ratio Z. The n=4 holomorphic intertwiners parametrized by a single cross-ratio variable Z are coherent states in that they form an over-complete basis of the Hilbert space of intertwiners and are semi-classical states peaked on the geometry of a classical tetrahedron as shown by my numerical studies. The new holomorphic intertwiners are related to the standard spin basis of intertwiners that are usually used in loop quantum gravity and spin foam models, and the change of basis coefficients are given by Jacobi polynomials.

In the canonical framework of loop quantum gravity, spin network states of quantum geometry are labeled by a graph as well as by SU(2) representations on the graph’s edges e and intertwiners on its vertices v. It is now possible to put holomorphic intertwiners at the vertices of the graph, which introduces the new spin networks labeled by representations je and cross-ratios Zv. Since each holomorphic intertwiner can be associated to a classical tetrahedron, we can interpret these new spin network states as discrete geometries. In particular, geometrical observables such as the volume can be expected to be peaked on their classical values as shown in my numerical studies for j=20. This should be of great help when looking at the dynamics of the spin network states and when studying how they are coarse-grained and refined.

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Bohr-Sommerfeld Quantization of Space by E. Bianchi and Hal M. Haggard

This week I have been reading quite a number of papers of about the volume spectrum in loop quantum gravity. One of the most useful is this one – because it gives a  very clear outline of how to actually calculate the volume eigenvalues (see Numerical work with sage 7 Eigenvalues of the volume operator in Loop quantum Gravity)

In this paper the authors introduce semiclassical methods into the study of the volume spectrum in loop gravity. They state that the  classical system behind a 4-valent spinnetwork node is a Euclidean tetrahedron. They  investigate the tetrahedral volume dynamics on phase space and apply Bohr-Sommerfeld quantization to find the volume spectrum. Their analysis shows a remarkable quantitative agreement with the volume spectrum computed in loop gravity. It also provides new geometrical insights into the degeneracy of this spectrum and the maximum and minimum eigenvalues of the volume.

space of shapes for a tetrahedron