Tag Archives: Hamiltonian

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

Curvature operator and gravity coupled to a scalar field: the physical Hamiltonian operator by Alesci, Assanioussi and Lewandowski

This week I been reviewing some conference proceedings the FFP14 Conference, Marseilles 2014. One paper I particularly like is this one by Alesci, Assanioussi and Lewandowski. I have been doing a some collaboration with Alesci and Assanoussi on the Hamiltonian Operator.

Gravity (minimally) coupled to a massless scalar field

The theory of 3+1 gravity (Lorentzian) minimally coupled to a free massless scalar field Φ(x) is described by the action

Picture1

where,

Picture2

Assuming that

Picture3

The Hamiltonian constraint is solved for π using the diff. constraint

Picture4

Φ becomes the emergent time.

In the region (+,+), an equivalent model could be obtained by keeping the Gauss and Diff. constraints and reformulating the scalar constraints

Picture5

Where

Picture6

Quantization of the model:Hilbert space and Gauss constraint

The kinematical Hilbert space is defined as

Picture11

 

Where its elements are

Picture12

The gauge invariant subspace

Picture13

 

Quantization of the model: Hilbert space of gauge & diff. invariant states

The space of the Gauss & vector constraints is defined as

Picture14

Quantization of the model: Physical Hamiltonian

Solving the scalar constraint C‘ in the quantum theory is equivalent to finding solutions to

Picture15

given a quantum observable  ,  the dynamics in this quantum theory is generated by

Picture16

where

Picture17

Picture18

Quantization of the model: construction of the Hamiltonian operator

Interested in constructing the quantum operator corresponding to the classical quantity

Picture19

Consider

Picture20

Lorentzian part:

Picture22

The final quantum operator corresponding to the Lorentzian part:

Picture23

where,

Picture24

Properties of this operator:

  • Gauge & Diffeomorphism invariant
  • Cylindrically consistent  if the averaging used in defining the curvature operator is restricted to non zero contributions;
  • Self-adjoint;
  • Discrete spectrum & compact expression for the matrix elements expressed explicitly in terms of the colouring on the spin network basis

Euclidean part:

Picture25

The resulting operator:

Picture27

The action of the full H on a spin network state can be expressed as

Picture28

Introduce the adjoint operator of  H

Picture29

 

For two spin network states have

Picture30 Picture31

Define a symmetric operator

Picture32

Which acts on s-n states as

Picture33

Define the physical Hamiltonian for this deparametrized model

Picture34

Properties of the final operator   hphys[N]   :

  • Gauge and Diffeomorphism invariant
  • Cylindrically consistent  if the averaging used in defining the operator is restricted to only non zero contributions
  • Symmetric /Self-adjoint
  • Discrete spectrum  andcompact expression for the matrix elements expressed explicitly in terms of the colouring on the spin network basis – volume operator not involved

 

The paper presented a way of implementing the Hamiltonian operator in the case of the deparametrized model of gravity with a scalar field using:

  • The simple and well defined curvature operator;
  • A new regularization scheme that allows to define an adjoint operator for the regularized expression and hence construct a symmetric Hamiltonian operator;

This operator verifies the properties of gauge symmetries and cylindrical consistency could be imposed.

Related articles

Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint by Alesci, Thiemann, and Zipfel

This week I have been continuing my work on the Hamiltonian constraint in Loop Quantum Gravity,  The main paper I’ve been studying this week is ‘Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint’. Fortunately enough linking   covariant and canonical LQG was also the topic of a recent seminar by Zipfel in the ilqgs spring program.

The authors of this paper emphasize that spin-foam models could realize a projection on the physical Hilbert space of canonical Loop Quantum Gravity (LQG). As a test  the authors analyze the one-vertex expansion of a simple Euclidean spin-foam. They find that for fixed Barbero-Immirzi parameter γ= 1 the one vertex-amplitude in the KKL prescription annihilates the Euclidean Hamiltonian constraint of LQG. Since for γ = 1 the Lorentzian part of the Hamiltonian constraint does not contribute this gives rise to new solutions of the Euclidean theory. Furthermore, they fi nd that the new states only depend on the diagonal matrix elements of the volume. This seems to be a generic property when applying the spin-foam projector.

To circumvent the problems of the canonical theory, a
covariant formulation of Quantum Gravity, the so-called spin-foam model was introduced. This model is mainly based on the observation that the Holst action for GR  de fines  a constrained BF-theory. The strategy is first to quantize discrete BF-theory and then to implement the so called simplicity constraints. The main building block of the model is a linear two-complex  κ embedded into 4-dimensional space-time M whose boundary is given by an initial and final gauge invariant spin-network, Ψi respectively Ψf , living on the initial respectively final spatial hyper surface of a
foliation of M. The physical information is encoded in the spin-foam amplitude.

linkingequ1.1

where Af , Ae and Av are the amplitudes associated to the internal faces, edges and vertices of  κ and B contains the boundary amplitudes.

Each spin-foam can be thought of as generalized
Feynman diagram contributing to the transition amplitude from an ingoing spin-network to an outgoing spin-network. By summing over all possible two-complexes one obtains the complete transition amplitude between ψi and ψf .

The main idea in this paper is that if f spin-foams provide a rigging map  the physical inner product would be given by

linkingequ1.6

and the rigging map would correspond  to

linkingequ1.7

Since all constraints are satis ed in Hphys the physical scalar product must obey

linkingequ1.8

for all ψout, ψn ∈ Hkin.

As a test  the authors consider an easy spin-foam amplitude and show that

linkingequ1.9

where  κ is a two-complex with only one internal vertex such that  Φ is a spin-network induced on the boundary of  κ and Hn is the Hamiltonian constraint acting on the node n.

 Hamiltonian constraint

The classical Hamiltonian constraint is

linkingequ2.1

 

where,

linkingequ2.1a

The constraint can be split into its Euclidean part H = Tr[F∧e] and Lorentzian part HL = C- H.

Using,

linkingequ2.2

where V is the volume of an arbitrary region  ∑ containing the point x. Smearing the constraints with lapse function N(x) gives

linkingequ2.3

This expression requires a regularization in order to obtain a well-de fined operator on Hkin. Using a triangulation T of the manifold  into elementary tetrahedra with analytic links adapted to the graph Γ of an arbitrary spin-network.

linkingfig1

Three non-planar links de fine a tetrahedron . Now  decompose H[N] into a sum of one term per each tetrahedron of the triangulation,

linkingequ2.5

To define  the classical regularized Hamiltonian constraint as,

linkingequ2.6

The connection A  and the curvature are regularized  by the holonomy h in SU (2), where in the fundamental representation m = ½. This gives,

linkingequ2.7

which converges to the Hamiltonian constraint if the triangulation is sufficiently fine.

As seen in the post

This can be generalized with a trace in an arbitrary irreducible representation m leading to

linkingequ2.8

this converges to H[N] as well.

Properties

The important properties of the Euclidean Hamiltonian constraint are;

when acting on a spin-network state, the operator reduces to a sum over terms each acting on individual nodes. Acting on nodes of valence n the operator gives

linkingequ2.9

The Hamiltonian constraint on di ffeomorphism invariant states is independent from the refi nement of the triangulation.

linkingequ2.10

Since the Ashtekar-Lewandowsk volume operator annihilates coplanar nodes and gauge invariant nodes of valence three H does not act on the new nodes – the so called extraordinary nodes.

Action on a trivalent node

To ompute the action of the operator Hmdeltaon a trivalent node where all links are outgoing, denote a trivalent node by |n(ji,jj , jk)> ≡ |n3>, whereas ji, jj ,jk are the spins of the adjacent links ei, ej , ek:

linkingfig2

 

To quantize  Hmdelta[N] the holonomies and the volume are replaced by their corresponding operators and the Poisson bracket is replaced by a commutator. Since the volume operator vanishes on a gauge invariant trivalent node  only need to compute;

linkingequ2.12

so, h(m) creates a free index in the m-representation located at the node , making it non-gauge invariant and a new node on the link ek:

linkingequ2.13

so we get,

linkingequ2.14

where the range of the sums over a, b is determined by the Clebsch-Gordan conditions and

linkingequ2.15

The complete action of the operator on a trivalent state |n(ji, jj , jk)> can be obtained by contracting the trace part with εijk. So, H projects on a linear combination of three spin networks which differ by exactly one new link labelled by m between each couple of the oldlinks at the node.

Action on a 4-valent node

The computation for a 4-valent node |n4> is

linkingequ2.16

where i labels the intertwiner – inner link.

The holonomy h(m) changes the valency of the node and the Volume therefore acts on the 5-valent non-gauge invariant node. Graphically this corresponds to

linkingequ2.18

and finally we get:

linkingequ2.19

This can be simpli ed to;

linkingequ2.20

SPIN-FOAM

Using  the de finition of an Euclidean spin-foam models as suggested by Kaminski, Kisielowski and Lewandowski (KKL) and since we,re   only interested in the evaluation of a spin-foam amplitude we choose a combinatorial de finition of the model:

Consider an oriented two-complex  de fined as the union of the set of faces (2-cells) F, edges (1-cells) E and vertices (0-cells) V such that every edge e is a 1-face of at least one face f (e ∈ ∂f) and every vertex V is a 0-face of at least one edge e (v ∈∂e).

We call edges which are contained in more than one face f internal and denote the set of all internal edges by Eint.  All vertices adjacent to more than one internal edge are also called internal and denote the set of these vertices by Vint. The boundary ∂κ is the union of all external vertices (-nodes) n ∉ Vint and external edges (links) l ∉Eint.

A spin-foam is a triple (κ, ρ; I) consisting of a proper foam whose faces are labelled by irreducible representations of a Lie-group G, in this case here SO(4) and whose internal edges are labeled by intertwiners I. This induces a spin-network structure ∂(κ, ρ; I) on the boundary of κ.

The BF partition function can be rewritten as

linkingequ3.3

Av de fines an SO (4) invariant function on the graph Γ induced on the boundary of the vertex v

linkingequ3.4

In the EPRL model  the simplicity constraint is imposed weakly so,

linkingequ3.5

It follows immediately that

linkingequ3.6

defi nes the EPRL vertex amplitude with

linkingequ3.7

Expanding the delta function in terms of spin-network function and integrating over the group elements gives

linkingequ3.8

In order to evaluate the fusion coefficients by graphical calculus it is convenient to work with 3j-symbols instead of Clebsch-Gordan coefficients. When replacing the Clebsch-Gordan coefficients we have to multiply by an overall factor cc.

Spin-foam projector

Given any couple of ingoing and outgoing kinematical states ψout, ψin, the Physical scalar product can be
formally de fined by

linkingequ3.10

where η is a projector – Rigging map onto the Kernel of the Hamiltonian constraint.

Suppose that the transition amplitude Z

linkingequ3.10

can be expressed in terms of a sum of spin-foams, then

linkingequ3.11

this can be interpreted as a function on the boundary graph ∂κ;

linkingequ3.12

in the EPRL sector.

Restricting the boundary elements h ∈ SU (2)  ⊂ SO (4) then;

linkingequ3.13

where |S>N is a normalized spin-network function on SU(2). This fi nally implies;

 

 

linkingequ3.14

NEW SOLUTIONS TO THE EUCLIDEAN HAMILTONIAN CONSTRAINT

Compute new solutions to the Euclidean Hamiltonian constraint by employing spin-foam methods. Show thatlinkingequ3.15


linkingfig3

in the Euclidean sector with γ= 1 and s = 1, where κ is a 2-complex with only one internal vertex.

Trivalent nodes

Consider the simplest possible case given by an initial and final state  |Θ>, characterized by two trivalent nodes joined by three links:

linkingequ3.16

 

the only states produced by the HamiltonianHm acting on a node, are given by a linear combination of spin-networks that diff er from the original one by the presence of an extraordinary (new) link. In particular the term sHwill be non vanishing only if |s> is of the kind:

linkingequ3.17

The simplest two-complex κ(Θ,s) with only one internal vertex de fining a cobordism between   |Θ> and |s> is a tube  Θ x[0,1] with an additional face between the internal vertex and the new link m,

Since the space of three-valent intertwiners is one-dimensional and all labelings jf are fixed by the states   |Θ> , |s>  the fi rst sum  is trivial.

linkingequ3.18

Γv= s and therefore we have,

linkingequ3.19

where the sign factor is due to the orientation of s, The fusion coefficients contribute four 9j symbols since,

linkingequ3.20

The full amplitude is

linkingequ4.7

and

linkingequ4.8

This yields,

linkingequ4.10

The last two terms are equivalent to the first term when exchanging jk↔ jj. The EPRL spin-foam reduces just to the SU(2) BF amplitude that is just the single 6j  in the first line.

Now using the defi nition of a 9j in terms of three 6j’s,

linkingequ4.11

The 9j’s involved in this expression can be reordered using the permutation symmetries  giving,

linkingequ4.12

the statessphysare solutions of the Euclidean Hamiltonian constraint

The spin-foam amplitude selects only those terms which depend on the diagonal elements on the volume. This  simplifies the calculation since we do not have to evaluate the volume explicitly.

Four valent nodes

The case with ψ in = ψout = |n4>where

linkingequ4.13

The matrix element <s| Hm|n4>  is non-vanishing if |s>is of the form

linkingequ4.14

Choose again a complex  of the form

linkingfig3

with one additional face jl.

The vertex trace  can be evaluated by graphical calculus

linkingequ4.15

The fusion coefficients give two 9j symbols for the two trivalent edges and two 15j- symbols for the two four-valent edges.The fusion coefficients reduce to 1 when γ=1.Taking the scalar product  we obtain,

linkingequ4.16

Taking the scalar product with the Hamiltonian gives,

linkingequ4.17

Summing over a and using the orthogonality relation gives,

linkingequ4.18

The three 6j’s in the two terms defi ne a 9j ,summing over the indexes and b respectively gives:

linkingequ4.19

the final result is,

linkingequ4.20

As for the trivalent vertex the spin-foam amplitude just takes those elements into account which depend on the diagonal Volume elements.

CONCLUSIONS

LQG is grounded on two parallel constructions; the canonical and the covariant ones. In this paper the authors  construct a simple spin-foam amplitude which annihilates the Hamiltonian constraint .They found that in the euclidean sector with signature s = 1 and Barbero-Immirzi parameter γ = 1 the Euclidean Hamiltonian constraint is annihilated by a spin-foam amplitude Z for a simple two-complex with only one internal vertex. The one vertex amplitudes of BF theory are explicit analytic solutions of the Hamiltonian theory.

Also the 6j symbol associated to every face is annihilated by the Euclidean scalar constraint. This is a generalization of the work by Bonzom-Freidel in the context of 3d gravity where they found that the 6jis annihilated by a suitable quantization of the 3d scalar constraint F = 0 . The spin-foam amplitude diagonalizes the Volume.

 

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