Holomorphic Factorization for a Quantum Tetrahedron by Freidel, Krasnov and Livine

I’ve  been reviewing this paper during the last week or so, it’s quite an advanced treatment of the quantum tetrahedron from the point of view of symplectic manifolds and Kahler manifolds. I’ve felt it was well worth doing this because it adds an deeper level to my understanding of the structure and properties of the quantum Tetrahedron. I’ll be updating this post as I work through the paper!

In this paper the authors provide a holomorphic description of the Hilbert space H j1…jn  of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for  the decomposition of identity in Hj1…jn.

The integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. The results provide a new interpretation for this quantity as being, in the limit of large  conformal  dimensions, the exponential of the Kahler potential of the symplectic manifold whose quantization gives Hj1 ,…,jn .

For the case n=4, the symplectic manifold in question has the interpretation of the space of shapes of a geometric tetrahedron with fixed face areas, and the results provide a description for the quantum tetrahedron in terms of holomorphic coherent states. The authors describe how the holomorphic intertwiners are related to the usual real ones. The semi-classical analysis of  overlap coefficients in the case of large spins allows  an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron to be obtained. The results of this are of direct relevance for the subjects of loop quantum gravity and spin foams and also for the  bulk/boundary correspondence.


The main object of interest in the paper is the space

holomorphic equ 1

of SU(2)-invariant tensors (intertwiners) in the tensor product of n irreducible SU(2) representations, dim(Vj) =dj = 2j + 1. This space is naturally a Hilbert space. It is finite-dimensional, with the dimension given by the classical formula:

holomorphic equ 2

The intertwiners play the central role in the quantum geometry or spin foam approach to quantum gravity. In quantum geometry (loop quantum gravity) approach the space of states of  geometry is spanned by the so-called spin network states based on a graph. This space is obtained by tensoring together the Hilbert spaces L2(G)of square integrable functions on the group G = SU(2) – one for every edge e of the underlying graph – while contracting them at vertices v with invariant tensors to form a gauge-invariant state.

Using the Plancherel decomposition the spin network Hilbert space can therefore be written as:

holomorphic equ 2a

which is the product of intertwiner spaces one for each vertex v of the graph.

The first non-trivial case that gives a non trivial dimension of the Hilbert space of intertwiners is n= 4. A beautiful geometric interpretation of states from Hj1,…,j4 has been proposed as seen in the post: Quantum Tetrahedra and Simplicial Spin Networks, where it was seen that the Hilbert space in this case can be obtained via the process of quantisation of the space of shapes of a geometric tetrahedron in R3 whose face areas are fixed to be equal to j1…j4.This space of shapes is a phase space of real dimension two of finite symplectic volume, and its geometric quantization gives rise to a finite-dimensional Hilbert space Hj1,…,j4.

It is known that the space of shapes of a tetrahedron is a Kahler manifold of complex dimension one that is parametrized by Z. This complex parameter is the cross ratio of the four stereographic coordinates zi labelling the direction of the normals to faces of the tetrahedron.The two possible viewpoints on Hj1,…,j4 i.e. that of SU(2) invariant tensors and that of quantization of the space of shapes of a geometric tetrahedron – are equivalent, in line with the general principle of Guillemin and Sternberg saying that geometric quantization commutes with symplectic reduction. An explicit formula for the decomposition of the identity in Hj1,…,j4 in terms of coherent states also exists.

The main aim of this paper is to further develop the holomorphic viewpoint on Hj, both for n = 4 and more generally. It is shown that the Hilbert space Hj of intertwiners can be obtained by quantization of a finite volume symplectic manifold Sj. Where the phase space Sj.
turns out to be a Kahler manifold, with holomorphic coordinates given by a string of n=3 cross-ratios {Z1,..Zn-3}. The paper uses the methods of geometric quantization to get to Hj up to the metaplectic correction occurring in geometric quantization of Kahler manifolds.

In the context of Kahler geometric quantization, one can introduce the coherent states |zi> such that

holomorphic equ 2b

The inner product formula can then be rewritten as a formula for the decomposition of the identity operator in terms of the coherent

holomorphic equ 3

The first main result in this paper is a version the above formula for the identity operator in the Hilbert space Hj, this is:

holomorphic equ 4

The integration kernel Kj here turns out to be just the n-point function of the AdS/CFT duality.

A comparison between the last two equations  shows that, in the semi-classical limit of all spins becoming large, the n-point function K
can be interpreted as an exponential of the Kahler potential on Sj .

holomorphic equ 4

Takes a particularly simple form of an integral over a single cross ratio coordinate in the case n = 4 of relevance for the quantum tetrahedron. The coherent states  are holomorphic functions of Z,  referred  to as holomorphic intertwiners. The resulting
holomorphic description of the Hilbert space of the quantum tetrahedron justifies the title of this paper.

The second part of the paper  characterizes the n = 4 holomorphic intertwiners |j ,Z> by projecting them onto a more familiar real basis in Hj1,…,j4. The real basis |j,k>can be obtained by considering the eigenstates of the operators J(i) · J(j) representing the scalar product of the area vectors between the faces i and j. The overlap between the usual normalised intertwiners  : |j,k> and the holomorphic intertwiner |k,Z> is denoted by:

holomorphic equ 4a

It is given by the “shifted” Jacobi polynomials P(α,β)n :

holomorphic equ 5

where jij = ji − jj , and Nk  is a normalisation constant.

This result can be used to express the Hj1,…,j4 norm of the holomorphic intertwiner |j,Z> in a holomorphically
factorised form

holomorphic equ 6

The last step of the paper’s analysis is to discuss the asymptotic properties of the (normalized) overlap coefficients

holomorphic equ 4a for large spins and the related geometrical interpretation. This asymptotic analysis gives a relation between the real and holomorphic description of the phase space of shapes of a geometric tetrahedron. The authors report that the normalized overlap coefficient is sharply picked both in k and in Z around a value k(Z) determined by the classical geometry of a tetrahedron.

The results of this paper are important for the field of quantum gravity the n = 4 intertwiner characterized in this paper plays a very important role in both the loop quantum gravity and the spin foam approaches. These intertwiners have so far been characterized using the real basis |j,k> In particular, the main building blocks of the spin foam models – the (15j)symbols and their analogs – arise as simple pairings of 5 of such intertwiners

The main result of this paper is a holomorphic description of the space of intertwiners, and, in particular, an explicit basis in Hj1,…,j4 given by the holomorphic intertwiners |j.Z>. The basis |j,k> being discrete, is convenient for some purposes, but the underlying geometric interpretation is quite hidden. Recalling the interpretation of the intertwiners from Hj1,…,j4
as giving the states of a quantum tetrahedron, the states |j,k>describe a tetrahedron whose shape is maximally uncertain. In contrast, the intertwiners |j.Z> being holomorphic, are coherent states in that they manage to contain the complete information about the shape of the tetrahedron coded into the real and imaginary parts of the cross-ratio coordinate Z. With the holomorphic intertwiners |j,Z> the quantum geometry can be characterized much more completely than it was possible before. It is possible to build the spin networks – states of quantum geometry – using the holomorphic intertwiners, and then the nodes of these spin networks receive a well-defined geometric interpretation of corresponding to tetrahedra of particular shapes. Similarly, the spin foam model simplex amplitudes can now be built using the coherent intertwiners, and then the basic object becomes not the (15j)-symbol of previous studies, but the (10j)-(5Z)-symbol with a well-defined geometrical interpretation.




Numerical work with sagemath 13: 1+1 causal dynamical triangulations

This week I have been studying and working on several things. Firstly I have been reading a long paper on ‘Holomorphic Factorization for a Quantum Tetrahedron’ which I’ll review in my next post. This is quite a sophisticated paper with some quite high level mathematics so I’ve taken my time with it.

Then I been upgrading my work on the 3d toy model including the presentation of the equilateral tetrahedron and hyperplanes as seen in the posts:



The other thing I have been working on is 1 +1 dimensional causal dynamical triangulations, starting with two main references quantum gravity ( in addition to Loll, Amborn papers) on a laptop: 1+1 cdt by Normal S Israel and John F Linder and  Two Dimensional Causal Dynamical Triangulation by Normal S Israel. So far I have set up a rough framework including initial conditions, detailed balance for monte carlo analysis and initial graphical work.

cdt1+1 fig

cdt1+1 figb


What I want to be able to do is independently construct a working CDT code and build it up from 2d to 3d and finally 4d. For the 1+1 CDT code I would be aiming to be able to independently construct simulated universes which exhibit quantum fluctuations as shown below:

cdt1+1 fig1

Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model by Bonzom, Livine,Smerlak and Speziale

This week I have been doing further work on the Quantum Tetrahedron as an Quantum Harmonic Oscillator – which I’ll review in a later post and also looking at the 3d toy model in more detail. In particular I have been studying ‘Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model.’ In this paper the authors consider an exact expression for the 6j-symbol for the isosceles tetrahedron, involving SU(2) group integrals, and use it to write the two-point function of 3d gravity on a single tetrahedron as a group integral. The perturbative expansion of this expression can then be performed with respect to the geometry of the boundary using a simple saddle-point analysis. They can then derive the complete expansion in inverse powers of the length scale and evaluate explicitly the quantum corrections up to second order. They also  use the same method to provide the complete expansion of the isosceles 6j-symbol with the next-to-leading correction to the Ponzano-Regge asymptotics.

Considering for simplicity the Riemannian case, the spinfoam amplitude for a single tetrahedron is the 6j-symbol of the Ponzano-Regge model. Its large spin asymptotics is dominated by exponentials of the Regge action for 3d general relativity. This is a key result, since the quantization of the Regge action is known to reproduce the correct free graviton propagator around flat spacetime.

This paper considers the simplest possible setting given by the 3d toy model introduced in the post Towards the graviton from spinfoams: the 3d toy model and studies analytically the full perturbative expansion of the 3d graviton. The results are based on a reformulation of the Wigner 6j symbol and the graviton propagator as group integrals. The authors compute explicitly
the leading order then both next-to-leading and next-to-next analytically. They also calculate  a formula for the next-to-leading order of the  Ponzano-Regge asymptotics of the 6j-symbol in the case of an isosceles tetrahedron.

Applying the same methods and tools to 4d spinfoam models would  allow a more thorough study of the full non-perturbative spinfoam graviton propagator and its correlations in 4d quantum gravity.

The boundary states and the kernel

Consider a triangulation consisting of a single tetrahedron. To define transition amplitudes in a background independent context for a certain region of spacetime, the main idea is to perform a perturbative expansion with respect to the geometry of the boundary. This classical geometry acts as a background for the perturbative expansion. To do so  – have to specify the values of the intrinsic and extrinsic curvatures of such a boundary, that is the edge lengths and the dihedral angles for a single tetrahedron in spinfoam variables. As in the post Towards the graviton from spinfoams: the 3d toy model, attention is restricted  to a situation in which the lengths of four edges have been measured, so that their values are fixed, say to a unique value jt + 1/2 . These constitute the time-like boundary and we are then interested in the correlations of length fluctuations between the two remaining and opposite edges which are the initial and final spatial slices. This setting is referred to as the time-gauge setting. The two opposite edges e1 and e2 have respectively lengths, j1 + 1/2 and j2 + 1/2 .

towards graviton fig1

Physical setting to compute the 2-point function. The two edges whose correlations of length fluctuations will be
computed are in fat lines, and have length j1 + 1/2 and j2 + 1/2 . These data are encoded in the boundary state of the tetrahedron. In the time-gauge setting, the four bulk edges have imposed lengths jt + 1/2 interpreted as the proper time of a particle propagating along one of these edges. Equivalently, the time between two planes containing e1 and e2 has been measured to be T = (jt + 1/2 )/sqrt(2).

In the spinfoam formalism, and in agreement with 3d LQG, lengths are quantized so that jt, j1 and j2 are half-integers.

The lengths and the dihedral angles are conjugated variables with regards to the boundary geometry, and have to satisfy the classical equations of motion. Here, it simply means that they must have admissible values to form a genuine flat tetrahedron. The dimension of the SU(2)-representation of spin j, dj ≡ 2j +1 is twice the edge length.

towards graviton equ 1

To assign a quantum state to the boundary, peaked on the classical geometry of the tetrahedron. Since jt is fixed, we only need such a state for e1, peaked on the length j1 + 1/2 , and for e2, peaked on        j2 + 1/2 .

Previous work have used a Gaussian ansatz for such states. However, it is more convenient to choose states which admit a well-defined Fourier transform on SU(2). In this perspective, the dihedral angles of the tetrahedron are interpreted as the class angles of SU(2) elements. So the Gaussian ansatz can be replaced for the edges e1 and e2 by the following Bessel state:

towards graviton equ 3a

The role of the cosine  is to peak the variable dual to j, i.e. the dihedral angle, on the value αe. Then the boundary state admits a well-defined Fourier transform, which is a Gaussian on the group SU(2). The SU(2) group elements are parameterized as:

towards graviton equ 4

These states carry the information about the boundary geometry necessary to induce a perturbative expansion around it. Interested in the following correlator,

towards graviton equ 6

W1122 measures the correlations between length fluctuations for the edges e1 and e2 of the tetrahedron, and it can be interpreted as
the 2-point function for gravity , contracted along the directions of e1 and e2. The 6j-symbol emerges from the usual spinfoam models for 3d gravity as the amplitude for a single tetrahedron.

towards graviton equ4043

Perturbative expansion of the isosceles 6j-symbol

The procedure described above can be applied directly to the isosceles 6j-symbol, obtaining the known Ponzano-Regge formula and its corrections.

This is interesting for a number of reasons. The corrections to the Ponzano-Regge formula are a key difference between the spinfoam perturbative expansion  and the one from quantum Regge calculus. The 6j-symbol is also the physical boundary state of 3d gravity
for a trivial topology and a one-tetrahedron triangulation. In 4d, it appears as a building block for the spin-foams amplitudes, such as the 15j-symbol. So for many aspects of spin-foams in 3d and 4d, in particular for the quantum corrections to the semiclassical limits, it good to have a better understanding of this object.

.The expansion of this isosceles 6j-symbol is;

towards graviton equ80

The leading order asymptotics, given by the original Ponzano-Regge formula is:

towards graviton equ80a

It is possible to compute analytically the two-point function – the graviton propagator – at all orders in the Planck length for the 3d toy model -the Ponzano-Regge model for a single isoceles tetrahedron as in Towards the graviton from spinfoams: the 3d toy model.

Related articles

The Quantum Tetrahedron and the 6j symbol in quantum gravity

This week as well as working on calculations and modelling the quantum tetrahedron in Lorentzian 3d quantum gravity I have been reading more about the Wigner{6j} symbol, in particular a great paper: Quantum Tetrahedra by Mauro Carfora .

Tetrahedra and 6j symbols in quantum gravity

The Ponzano–Regge asymptotic formula for the 6j symbol and  Regge Calculus are the basis of all discretized approaches to General Relativity, both at the classical and at the quantum level.

sagemath 13 equ0

In Regge’s approach the edge lengths of a triangulated spacetime are taken as discrete counterparts of the metric, a tensor  which encodes the dynamical degrees of freedom of the gravitational field and appears in the classical Einstein–Hilbert action for General Relativity through its second derivatives combined in the  Riemann scalar curvature.

A Regge spacetime is a piecewise linear  manifold of dimension D dissected into simplices; triangles in D = 2, tetrahedra in D = 3, 4 simplices in D = 4 and so on. Inside each simplex either an Euclidean or a Minkowskian metric can be assigned: manifolds obtained by gluing together D–dimensional simplices acquire a metric of Riemannian or Lorentzian signature 2.The Regge action is given explicitly by ( G = 1)

sagemath 13 equ1

where the sum is over (D − 2)–dimensional simplices (hinges), the Vol(D−2)are their (D − 2)–dimensional volumes expressed in
terms of the edge lengths and the deficit angles. A discretized
spacetime is flat inside each D–simplex, while curvature is concentrated at the hinges.  The limit of the Regge action  when the edge lengths become smaller and smaller gives the usual Einstein–Hilbert action for a spacetime which is smooth everywhere, the curvature being distributed continuously.

Regge equations –the discretized analog of Einstein field equations– can be derived from the classical action by varying it with respect to the dynamical variables, i.e. the set  of edge lengths , according to Hamilton principle of classical field theory – see the post: Background independence in a nutshell: the dynamics of a tetrahedron by Rovelli
Regge Calculus gave rise to an approach to quantization of General Relativity known as Simplicial Quantum Gravity. The quantization procedure most commonly adopted is the Euclidean path–sum approach, which is a discretized version of Feynman’s path integral describing D–dimensional Regge geometries undergoing quantum fluctuations.

The Ponzano–Regge asymptotic formula for the 6j symbol;

sagemath 13 equ0

represents the semiclassical limit of a path–sum over all quantum fluctuations, to be associated with the simplest 3–dimensional ‘spacetime’, an Euclidean tetrahedron T. In fact the argument in the exponential reproducesthe Regge action S3 for T.

In general, we denote by T3(j) (a particular triangulation of a closed 3–dimensional Regge manifoldM3 (of fixed topology) obtained by assigning SU(2) spin variables {j} to the edges of T 3. The assignment must satisfy a number of conditions,  illustrated if we introduce the state functional associated with T3(j), namely

sagemath 13 equ2

where No, N1, N3 are the number of vertices, edges and tetrahedra in T3(j).The Ponzano–Regge state sum is obtained by summing over triangulations corresponding to all assignments of spin variables {j} bounded by the cut–off L.

sagemath 13 equ3

where the cut–off is formally removed by taking the limit in front of the sum.

The state sum ZPR [M3] is a topological invariant of the manifold M3, owing to the fact that its value is actually independent of the particular triangulation, namely does not change under suitable combinatorial transformations. These moves are expressed algebraically in terms of  the
Biedenharn-Elliott identity  –representing the moves (2 tetrahedra) <-> (3 tetrahedra)– and of both the Biedenharn–Elliott identity and the orthogonality conditions  for 6j symbols, which represent the barycentric move together its inverse, namely (1 tetrahedra) <-> (4 tetrahedra).

A well–defined quantum invariant for closed 3–manifolds3 based on representation theory of a quantum deformation of the group SU(2)is given by

sagemath 13 equ4
where the summation is over all {j} labeling highest weight irreducible representations of SU(2)q (q = exp{2i/r}, with {j = 0, 1/2, 1 . . . , r − 1}).

The Wigner 6j symbol and its symmetries – The features of the ‘quantum tetrahedron’

Given three angular momentum operators J1, J2, J3 –associated with three kinematically independent quantum systems– the Wigner–coupled Hilbert space of the composite system is an eigenstate of the total angular momentum

J1 + J2 + J3.= J

and of its projection Jz along the quantization axis. The degeneracy can be completely removed by considering binary coupling schemes such as;

(J1 + J2) + J3 and J1 + (J2 + J3),

and by introducing intermediate angular momentum operators defined by;

(J1 + J2) = J12; J12 + J3 = J
(J2 + J3) = J23; J1 + J23 = J

respectively. In Dirac notation the simultaneous eigenspaces of the two complete sets of commuting operators are spanned by basis vectors

|j1j2j12j3> and |j1j2j3j23>

where j1, j2, j3 denote eigenvalues of the corresponding operators, and j is the eigenvalue of J and m is the total magnetic quantum number with range −j < m < j in integer steps.

The j1, j2, j3 run over {0, 1/2 , 1, 3/ 2 , 2, . . . } (labels of SU(2) irreducible representations), while;

|j1 −j2| < j12 < j1 +j2


|j2 −j3| < j23 <j2 + j3.

The Wigner 6j symbol expresses the transformation between the two schemes;

sagemath 13 equ11
Apart from a phase factor, it follows that the quantum mechanical probability;

sagemath 13 equ12

represents the probability that a system prepared in a state of the coupling scheme;

(J1 + J2) = J12; J12 + J3 = J

will be measured to be in a state of the coupling scheme;

(J2 + J3) = J23; J1 + J23 = J

The 6j symbol may be written as sums of products of four Clebsch–Gordan coefficients or their symmetric counterparts, the Wigner 3j symbols. The relations between 6j and 3j symbols are given by;

sagemath 13 equ13
The 6j symbol is invariant under any permutation of its columns or under interchange the upper and lower arguments in each of any two columns. These algebraic relations involve
3! × 4 = 24 different 6j with the same value. The 6j symbol is naturally endowed with a geometric symmetry, the tetrahedral symmetry. In the three–dimensional picture introduced by Ponzano and Regge the 6j is thought of as a real solid tetrahedron T with edge lengths;

L1 = a+ 1/2, L2 ,2 = b+ 1/2 …L6 = f + 1/2

This implies that the quantities;

q1 = a+b+c, q2 = a+e+f, q3 = b+d+f, q4 = c + d + e

(sums of the edge lengths of each face) are all integer. The conditions addressed are sufficient to guarantee the existence of a non–vanishing 6j symbol, but they are not enough to ensure the existence of a geometric tetrahedron T living in Euclidean 3–space with the given edges. More precisely, T exists only if its square volume V evaluated by means
of the Cayley–Menger determinant, is positive.

Ponzano–Regge asymptotic formula
The Ponzano–Regge asymptotic formula for the 6j symbol reads;

sagemath 13 equ0

where the limit is taken for all entries >> 1 and Lp=  j + 1/2,with {jr} = {a, b, c, d, e, f}. V is the Euclidean volume of the tetrahedron T and theta is the angle between the outer normals to the faces which share the edge r.

From a quantum mechanical viewpoint, the above probability amplitude has the form of a semiclassical (wave) function since the factor sagemath 13 equ14ais slowly varying with respect to the spin variables while the exponential is a rapidly oscillating dynamical phase. This  asymptotic behaviour complies with Wigner’s semiclassical estimate for the probability;

sagemath 13 equ14b
compared with the quantum probability:

sagemath 13 equ12

According to Feynman path sum interpretation of quantum mechanics , the argument of the exponential  must represent a classical action, and  it can be read as

sagemath 13 equ14cfor pairs (p, q) of canonical variables, angular momenta and conjugate angle.

Numerical work with Vpython and sagemath12: The Quantum Tetrahedron

In a number of recent posts including:

I have been studying the ‘time-fixed’ quantum tetrahedron in which a quantum tetrahedron is used to model the evolution of a state a into state b as shown below:

physical boundary fig 1

In this toy model the quantum tetrahedron evolves from a flat shape:

Background independence fig3

via an equilateral quantum tetrahedron

Background independence fig1

towards a long stretched out quantum tetrahedron in the limit of large T.

Background independence fig2

This can be displayed on a phase space diagram,


Where “Minkowski vacuum states” are the states which minimize the energy.

Using sagemath I am able to model the quantum tetrahedron during this evolution by varying the lenght of c – which is used to measure the time T variable;


By using Vpython I am able to model this same system as shown in these animated gifs, the label in the diagram indicated the value of T.

An animated gif showing the edge a




An animated gif showing a different view of the quantum tetrahedron:


Analyzing the {6j} kernel

An important object in the modelling of the quantum tetrahedron is the Wigner {6j} symbol. As we saw in the post: Physical boundary state for the quantum tetrahedron by Livine and Speziale The quantum dynamics can be studied as by Ponzano-Regge, by  associating with the tetrahedron the amplitude

physical boundary equ 4

Using sagemath I was able to evaluate the value of this in some extremal cases: when b=0 and b=2j.

Case b=0



Case b=2j



In the next post I will be looking at the mathematics behind the Wigner{6j} symbol in more detail.

Physical boundary state for the quantum tetrahedron by Livine and Speziale

In this paper Levine and Specziale consider the stability under evolution as a criterion to select a  physical boundary state for the spinfoam formalism. They apply this to the simplest spinfoam defined by a single quantum tetrahedron and solve the associated eigenvalue problem at leading  order in the large spin limit.

They show that this fixes uniquely the free parameters entering the boundary state. The state obtained this way gives a correlation between edges which varies inversely with the distance  between the edges, in agreement with the linearized continuum theory. They argue that this  correlator represents the propagation of a pure gauge which is consistent with the absence of  physical degrees of freedom in 3d general relativity.

The covariant spinfoam formulation of Loop Quantum Gravity gives a regularized expression for the path integral of the full theory. The state of the art in spinfoams is the proposal for computing the graviton propagator in non-perturbative quantum gravity by working with a bounded region of space-time and introducing a semiclassical state peaked on a given boundary geometry. This boundary state gauge-fixes the spinfoam amplitude and allows to compute the
(two-point) correlations for the gravitational field ,so inducing a non-trivial bulk geometry. The boundary state is a taken as a Gaussian state with a phase factor in the Hilbert space of boundary spin networks.

This paper addresses issues in 3d Riemannian quantum gravity, because  the theory is much simpler in three dimensions. Gravity is topological and we know how to spinfoam quantize it exactly as the Ponzano-Regge model. We can solve this toy model explicitly and we know the physical states.

In this paper the authors consider the smallest 3d triangulation, that is a single tetrahedron. The corresponding Ponzano-Regge spinfoam amplitude is simply Wigner’s {6j} symbol for the unitary group SU(2).

Following the setting introduced in the post Towards the graviton from spinfoams: The 3d toy model, the authors consider the
‘time-fixed’ tetrahedron: out of its six edges, study the correlations between the fluctuations of two opposite edges while freezing the lengths of the remaining four edges. The fixed-length edges define the time interval associated to that piece of 3d spacetime, while the two fluctuating edges are distinguished as the initial and final edges.

By defining a physical state as a wave function unaffected by time evolution,  the phased Gaussian ansatz is shown to be a physical state at first order in the asymptotical regime. The physical state criterion uniquely fixes the width of the Gaussian. From a covariant point of view looking at the tetrahedron, there is a unique physical state, namely the flat connection boundary state.



From the propagation kernel to the boundary state

Consider a scalar φ in the Schrodinger picture , and introduce two spacelike hyperplanes in Minkowski spacetime, separed by a time T. Denote by  φ1 and φ2 two classical field configurations associated with these planes, and ψn[φ] a complete basis of energy eigenstates. Given a state:

physical boundary equ 1

on the initial plane, the evolution to the final state can be written in terms of the propagation kernel K[φ1, φ2, T] through:physical boundary equ 2

The Quantum Tetrahedron

Consider the quantum tetrahedron introduced in the post Towards the graviton from spinfoams: The 3d toy model

physical boundary fig 1
Given a tetrahedron of edge lengths ℓe, fix four opposite ones to j, and the remaining two to a and b., orient it in such a way that we can think of it as representing the evolution of the edge a into the edge b, in a time j.

The dynamics of this model was studied at both the classical and quantum level in the post: Background independence in a nutshell: The dynamics of a tetrahedron.

The classical dynamics is encoded in the Regge action:

physical boundary equ 2a

where θe are the dihedral angles of the tetrahedron. The quantum dynamics can be studied as by Ponzano-Regge, by making the assumption that the lengths in the quantum theory can only have half-integer values ℓe = je+ 1/2 , and associating with the tetrahedron the amplitude

physical boundary equ 4

In this expression da ≡ 2a + 1 is the dimension of the spin-a representation of SU(2) and the {6j} is Wigner’s 6j-symbol for the recoupling theory of SU(2).

With this simple form of amplitude K[φ1, φ2, T] the stability condition for physical states  simply is:

physical boundary equ 4a

If we view this tetrahedron as part of a triangulation of flat 3d space between the two planes, we can also introduce the “asymptotic time”

physical boundary equ 5

In the isosceles p case a = b = jo, we have

physical boundary equ 6

Diagonalizing the kernel
Consider the kernel as a dj -by-dj matrix Kab[j] = K[a, b, j]. This matrix
satisfies K^2 ≡ 1 for all j, thus the evolution generated by it is unitary, K is diagonalizable and all its eigenvalues are ±1. Using the notation ψn(a) = <a|n> to indicate the n-th eigenvector in the basis a, we have

physical boundary equ 7
The presence here of two possible eigenvalues ±1, is because the  Ponzano-Regge model sums over both orientation of the tetrahedron.

The next step is to work out the eigenvectors solving the associate eigenvalue problem,

physical boundary equ 8

The general explicit solutions to this equation are not known at the moment. So at this stage we can not study exact physical semiclassical states as linear combinations of eigenstates. The equation can however  be solved approximately.

Semiclassical states
The perturbative expansion considered is the large spin limit , where the {6j} symbol is dominated by exponentials of the Regge action

physical boundary equ 9
This is the property that makes it possible to show that the quantum theory based upon the {6j} has a  semiclassical limit.

To the lowest order of the wavefunction should satisfy

physical boundary equ 8

with the linearized kernel and be peaked around q. In the simple setting considered here, the intrinsic and extrinsic geometry of q are specified giving a value jo for edge length and a value θo for its dihedral angle. The latter is chosen  in such a way that the complete background tetrahedron is given by the isosceles configuration a = b = jo. For the exterior dihedral angles associated to jo and j, elementary geometry gives respectively:

physical boundary equ 10

By analogy with the continuum, we expect a physical semiclassical state to be implemented in this approximation by a Gaussian state around q = (jo, θo), for which we make the following assumption:

physical boundary equ 12

Here N is the normalization, σ the width and φ is a phase that is undetermined for the moment. For σ scaling linearly with the spins, this Gaussian is peaked on q in the large spin limit as seen in the post: A semiclassical tetrahedron

These kind of states have been extensively used as ansatz for leading order semiclassical states in the recent spinfoam literature. A semiclassical state typically has a precise width uniquely fixed by the dynamics (e.g. the factor ω in the exponent of the vacuum Gaussian of the harmonic oscillator).

Thus a crucial question is whether also in spinfoams the width can be fixed requiring the state to solve the dynamics.

physical boundary equ 12

Is an eigenstate of the kernel for a unique choice of the width, given by:

physical boundary equ 12a

and for two choices of φ, corresponding to eigenvalues ±1:
physical boundary equ 12b

Here ~ means at first order in the large spin limit. There is a clear restriction that should be kept in mind:

physical boundary equ 9

holds only if the volume V appearing in it is real. For a generic configuration (a, b, j) this is not always the case. The quantum range of a and b is [0, 2j], and the condition V real is violated when the endpoints are approached by one of the variables. Specifically,  have V ~0 for a and/or b close to zero, and V unreal for a and/or b close to 2j. The calculations here only apply in the regime a ~b ~ j.

The end result of these calculations being that the physical boundary state is given by:

physical boundary equ 13

Physical state in the general boundary formalism

In the so-called general boundary formalism, the four bulk edges are varied freely.  The boundary state then has to carry information on the background value of the (intrinsic and  extrinsic) geometry of all six edges, and correlations between all six of them can be computed.
Such a general boundary formalism  is used for the 4d spinfoam graviton calculations.

In the particular 3d case the fact that the theory is topological strongly simplifies this analysis,  because once the topology and the triangulation are fixed, there is a single physical state.  Assuming trivial topology, this is given in the group representation by:

physical boundary equ 14

where gf, represents the gravitational holonomy on a closed path ∂f. The product is over the independent faces, and the condition F = 0 is ensured everywhere.

In the case considered here, the triangulation is given by a single tetrahedron. Then gf is a product of four deltas ensuring the flatness of each face. Only three faces are independent, so can get rid of one delta. Fixing the orientation of the edges as in the diagram below.

physical boundary fig 2

The state can be written as:

physical boundary equ 15

The physical boundary state coincides with the kernel, so that correlations now read:

physical boundary equ 16

This result  can be understood as follows. The boundary state ψo(j1…j6) is the state induced by the (exterior) bulk geometry onto the tetrahedron. Having assumed a trivial boundary topology (homomorphic to S2) and a trivial bulk topology – obtain a spinfoam amplitude in {6j} which is naturally associated to the triangulation of the closed S3 manifold with two tetrahedra.

The quantum tetrahedron as a quantum harmonic oscillator

It is known that the large-volume limit of a quantum tetrahedron is a quantum  harmonic oscillator. In particular the volume operator of a quantum tetrahedron is, in the sector of large eigenvalues, accurately described by a quantum harmonic oscillator.

Using  Vpython, it is possible to visualise the quantum tetrahedron as a  semi classical quantum simple harmonic oscillator.  The implementation of this is shown below;

Quantum simple harmonic oscillator

The quantum tetrahedron as a  Quantum  harmonic oscillator .

quantum tetrahedron as quantum SHM

Click to view animated gif

Review of the basic mathematical structure of a quantum tetrahedron

A quantum tetrahedron consists of four angular momenta Ji, where i = {1,2,3, 4} representing its faces and coupling to a vanishing total angular momentum – this means that the Hilbert space consists of all states ful filing:

j1 +j2 +j3 +j4 = 0

In the coupling scheme both pairs j1;j2 and j3;j4 couple frst to two irreducible SU(2)representations of dimension 2k+1 each, which are then added to give a singlet state.

The quantum number k ranges from kmin to kmax in integer steps with:
kmin = max(|j1-j2|,|j3 -j4|)
kmax = min(j1+j2,j3 +j4)

The total dimension of the Hilbert space is given by:
d = kmax -kmin + 1.

The volume operator can be formulated as:

large vol equ 1

where the operators

large vol equ 2

represent the faces of the tetrahedron with

large vol equ 3and being the Immirzi parameter.

It is useful to use the operator

large vol equ 4

which, in the basis of the states can be represented as

large vol equ 5


large vol equ 6

and where

large vol equ 7

is the area of a triangle with edges a; b; c expressed via Herons formula,

These choices mean that the matrix representation of Q is real and antisymmetric and  the spectrum of Q consists for even d of pairs of eigenvalues q and -q differing in sign. This makes it much easier to find the eigenvalues as I found in  Numerical work with sage 7.

The large volume limt of Q is found to be given by:

large vol equ 8

The eigenstates of Q are just the well-known wave functions diagonalizing the harmonic oscillator in real-space representation:

large vol equ 9

For a monochromatic quantum tetrahedron we can obtain closed analytical expressions:


Below is shown a sagemath program which displays the wavefunction  of a Quantum simple harmonic oscillator using Hermite polynomials Hn:


This an really important step in the development of Loop Quantum Gravity, because now we have  a  quantum field theory of geometry. This leads from combinatorial or simplicial description of geometry, through spin foams and the  quantum tetrahedron, to many particle states of quantum tetrahedra and the emergence of spacetime as a Bose-Einstein condensate.

quantum tetrahedron concepts