Tag Archives: Quantum Mechanics

Numerical indications on the semiclassical limit of the flipped vertex by Magliaro, Perini and Rovelli

This week is I’ve reviewing an old but interesting paper on the flipped vertex. I’m working on replicating and extesnding  the calculations in this paper and will post about them next week.

This is related to the posts

In this paper the authors take  the propagation kernel Wt(x,y) of a one-dimensional nonrelativistic quantum system defined by a hamiltonian operator H:


Then they  consider a semiclassical wave packet centered on its initial values and compute its  evolution under the kernel Wt(x,y):


and see whether or not the initial state  evolves into a semiclassical wave packet centered on the correct final values.

The flipped vertex W(jnm,in) is a function of ten spin variables jnm where n,m = 1,…,5 and five intertwiner variables in:

The process described by one vertex can be seen as the dynamics of a single cell in a Regge triangulation of general relativity. This gives a simple and direct geometrical interpretation to the dynamical variables entering the vertex amplitude and a simple formulation of
the dynamical equations. The boundary of a Regge cell is formed by five tetrahedra joined along all their faces, forming a closed space with the topology of a 3-sphere.


  • Anm be the area of the triangle(nm) that separates the tetrahedra n and m.
  • αn(mp,qr) be the angle between the triangles (mp) and (qr) in the tetrahedron n.
  • Θnm be the angle between the normals to the tetrahedra n and m.

These quantities determine entirely the intrinsic and extrinsic classical geometry of the boundary surface.

The ten spins jnm are the quantum numbers of the areas Anm . The five intertwiners in are the quantum numbers associated to the angles αn(mp,qr) . They are the eigenvalues of the quantity:


In general relativity, the Einstein equations can be seen as constraints on boundary variables Anm ,  αn(mp,qr)   and Θnm. These can be viewed as the ensemble of the initial, boundary and final data for a process happening inside the boundary 3-sphere.

In general finding a solution to these constraints is complicated but one is easy –  that corresponding to a flat space and to the boundary
of a regular 4-simplex. This is given by all equal areas Anm = j0, all equal angles in = i0, and Θnm = Θ, where elementary geometry gives:


A boundary wave packet centered on these values must be correctly propagated by the vertex amplitude, if the vertex amplitude is to give the Einstein equations in the classical limit.

The simplest wave packet is a diagonal gaussian wave packet:


normal is the normalization factor. The constants σ and θ are fixed by the requirement that the state is peaked on the value in = i0  so all angles of the tetrahedron are equally peaked on in = i0 :


The state considered is formed by a gaussian state on the spins, with phases given by the extrinsic curvature and by a coherent tetrahedron state:


for each tetrahedron.

We can also write the wave packet


as an initial state times a final state:


We can then test the classical limit of the vertex amplitude by computing the evolution of the four incoming tetrahedra generated by the vertex amplitude and comparing φ(i) with ψ(i).:


So compare the evolved state flippedequ10with the coherent tetrahedron stateflippedequ7



If the function φ(i) turns out to be  close to the coherent tetrahedron state ψ(i), we can say that the flipped vertex amplitude appears to evolve four coherent tetrahedra into one coherent tetrahedron, consistently with the at solution of the classical Einstein equations.

The flipped vertex  in the present case is:



We compared the two functions ψ(i) -coherent tetrahedron and φ(i)  -evolved state for the cases jn = 2 and jn = 4. The numerical results are shown below:



The agreement between the evolved state and the coherent tetrahedron state is quite good. Besides the overall shape of the state, there is a concordance of the mean values and the widths of the wave packet.

Related articles


Hamiltonian dynamics of a quantum of space

In this post I follow up on some of the work reviewed in the post:

The action of the quantum mechanical volume operator plays a fundamental role in discrete quantum gravity models, can be given as a second order difference equation. By a complex phase change this can be  turned into a discrete Schrödinger-like equation.

The introduction of discrete potential–like functions reveals the role of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols.

I’ll look at the underlying geometric features. When the spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary quantum of space, and an asymptotic picture emerges of the semiclassical and classical regimes.

The definition of coordinates adapted to Regge symmetry is used to construct a set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.


For an elementary spin network as shown below


A quadrilateral and its Regge “conjugate”illustrating the elementary spin network representation of the symmetric coupling scheme: each quadrilateral is dissected into two triangles sharing, as a common side, the diagonal l.


the volume operator K = J1 .J2 × J3,  acts democratically on vectors J1; J2 and J3 plus a fourth one, J4, which closes a not necessarily planar quadrilateral vector diagram J1 + J2 + J3 + J4 = 0.

The matrix elements of K are computed to provide a Hermitian representation, whose features have been studied by many see posts:

By a suitable complex change of phase we can transform the imaginary antisymmetric representation into a real, time-independent Schrödinger equation which governs the Hamiltonian dynamics as a function of a discrete variable denoted l. The Hilbert space spanned by the eigenfunctions of the volume operator can be constructed combinatorially and geometrically, applying polygonal relationships to the two quadrilateral vector diagram which are conjugated by a hidden symmetry discovered by Regge .

Discrete schrodinger equation and Regge symmetry

Eigenvalues k and eigenfunctions Ψl(k) of the volume operator are obtained through the three–terms recursion relationship – see post:

Applying a change of phaseΨl (k)=(-i)lΦl(k) to obtain a real, finite difference Schrodinger–like equation

hamiltonianequ1The Ψl are the eigenfunctions of the volume operator expanded in the J12 = J1 +J2 basis. The matrix elements αl in  are given in terms of geometric quantities, namely

hamiltonianequ2αl is proportional to the product of the areas of the two triangles sharing the side of length l and forming a quadrilateral of sides J1 +½, J2 +½, J3 +½ and J4 +½.

The requirement that the four vectors form a (not necessarily planar) quadrilateral leads to identify the range of l with


which is also the dimension of the Hilbert space where the volume operator acts.

Hamiltonian Dynamics

The Hamiltonian operator for the discrete Schrödinger equation


can be written, in terms of the shift operator


The two-dimensional phase space (l, φ) supports the corresponding classical Hamiltonian function given by

hamiltonianequ7This is  illustrated in below for the two Regge conjugate quadrilaterals of the diagram above.



The quadrilaterals are now allowed to fold along l with φ seen as a torsion angle.

The classical regime occurs when quantum numbers j are large and l can be considered as a continuous variable. This limit for l permits us to draw the closed curves in the (l, k) plane when φ = 0 or φ=π . These curves have the physical meaning of torsional-like potential functions


viewing the quadrilaterals as mechanical systems.

hamiltonianfig4Potential functions U+ and U in  hamiltonianequ8are shown for two cases where the conjugated tetrahedra coincide.

  • left panel:   j1,j2, j3, j4=100,110,130,140 the tangential quadrilateral
  • right panel: j1= j2=j3=j4 =120 the ex-tangential quadrilateral.

During the classical motion, the diagonal l changes its value preserving the energy of the system. The result is a geometric configuration  a tetrahedron changing continuously its shape but preserving its volume as a constant of motion.

Quantum mechanics extends the domain of the canonical variables to regions of phase space classically not allowed. Boundaries of these regions are the so-called potential-energy curves particularly important in applications. They are defined as turning points, namely the points where for each value of energy the classical  changes sign. This happens when the momentum φ is either 0 or π.


The above conditions define closed curved in the l-energy plane. These curves have the physical meaning of torsional like potential functions.

At each value of the possible values E’ of the Hamiltonian are bounded by


and the eigenvalues λk of the quantum system are bounded by



The sagemath code and output for this work is shown below:





Exact Computation and Asymptotic Approximations of 6j Symbols: Illustration of Their Semiclassical Limits by Mirco Ragni et al

In recent posts I have been doing some numerical work based on a series of papers on the ‘Exact Computation and Asymptotic Approximations of 6j Symbols’.

and a review of their use in spin networks

In this posts I’ll be looking at a paper which looks and their semiclassical properties.

In this paper the authors describe a direct method for the exact computation of 3nj symbols from the defining series. The properties and asymptotic formulas of the 6j symbols or Racah coefficients are discussed. Relationships with families of hypergeometric orthogonal polynomials are presented and the asymptotic behaviour is studied both from the viewpoints of the basic geometrical significance and as a source of accurate approximation formulas.

Numerical aspects are specifically investigated in detail, regarding the relationship between functions of discrete and of continuous variables, exhibiting the transition in the limit of large angular momenta toward both Wigner’s reduced rotation matrices or Jacobi
polynomials and harmonic oscillators or Hermite polynomials.

This paper contains a presentation of properties, useful formulas, and  illustrations for a basic mathematical tool, the 6j-symbol, also known in quantum mechanical angular momentum theory as Racah coefficient and the building block of 3nj-symbols and spin networks.

There are basic connections among the 6j symbols of angular momentum theory with both the theory of superposition coefficients of hyperspherical harmonics and the theory of discrete orthogonal polynomials. There is a connection between the Askey
scheme of orthogonal polynomials and the tools of angular momentum theory  such as 6j, 3j, rotation d-matrices. Going down the Askey scheme corresponds in quantum mechanics to the semiclassical limit, while going up provides discretization algorithms for quantum mechanical calculations for example the hyperquantization algorithm.


Explicit expressions for the 6j coefficients can be written according to the series expressions of the Racah type, or as generalized hypergeometric series, or in connection with the so-called Racah polynomials. Orthogonal polynomials of a discrete variable are important tools of numerical analysis for the representation of functions on grids.

Computation of Mathematical Functions and Angular Momentum

We can calculate the 3nj-symbols and Wigner d functions
by directly summing the defining series using multiprecision arithmetic. The multiple precision arithmetic allows convenient calculation of hypergeometric functions, pFq, of small
and large argument by their series definition.



d functions with 2F1:


Clebsch–Gordan coefficients and 3j-symbols with  3F2


6j-symbols with 4F3


 Semiclassical Limits and Schulten–Gordon Approach

In the Askey scheme for orthogonal polynomials
of hyperspherical family and its counterpart for the
tools of angular momentum theory shown arrows pointing out downwards are asymptotic connections.

A basic role is played by the relationship which relates
three 6j symbols with an argument differing by one


In this formula, one can introduce a quantity R such
that either and m1=j23−j, m2=j3−j23, m3 = j −j3.

We have


So that when R goes to infinity, we obtain a three terms
recurrence relationship for the 3j symbols as a function of j1.



when R goes to infinity, we obtain another three term recurrence relationship for the 3j symbols as function of m2.

Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols

The post Numerical work with sagemath 23: Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols deals with this section.

Geometrical interpretation

The equation:


has an interesting geometrical interpretation, based on the vector model visualization of quantum angular momentum coupling by
the triangle of vectors that we would draw in classical mechanics.


In view of this when we consider 6j properties as correlated to those of the tetrahedron,


we use the substitution

Jx =jx

which greatly improves all asymptotic formulas down to surprisingly low values of the entries.

The square of the area of each triangular face is given by the formula:


where a, b, c are the sides of the face. Similarly, the square of the volume of an irregular tetrahedron,can be written as the Cayley-Menger determinant:


When the values of J1, J2, J12, J3, and J are fixed, the maximum value for the volume as a function of J23 is given at:


The corresponding volume is


Therefore, the two values of J23 for which the volume is zero are:

hyperequ18They mark the boundaries between classical and nonclassical regions.

Introduce a parameter λ indicating the growth of the angular momentum. Consider the Schulten-Gordon relationships:

For λ =1




where F(a, b, c) is area of abc triangle from


The coefficients in


are connected to the geometry of the tetrahedron:


In terms of the finite difference operator:

hyperequ22We have


From these formulas, and from that of the volume,
we have that

  • V=0 implies cosθ1 =±1 and establishes the classical domain between J1min and J1max
  • F(J1, J2, J3)=0 or F(J1, L2, L3)=0 establish the definition limits j1min and j1max

For a Schrodinger type equation


its discrete analog in a grid having one as a step


and so comparing this with hyperequ22a

we have


allowing the identification


Plots corresponding to the three cases are given below:


On the closed loop, we can enforce Bohr–Sommerfeld phase space


where the role of q is played by j12. This is an eigenvalue equation for allowed L1. The illustration of these formulas is below:

hyperfig6Illustration of phase space for semiclassical quantization with  j1=92, j2 =47, j3 =80, j =121 for j12 =139 (dots), j12 =134 (triangles) ,j12 =129 (plus signs)


Values for the integral hyperequ29 for different
number of nodes n.

  • j12 = 139 ⇒ n = 0,
  • j12 =134 ⇒n =5,
  • j12 =129 ⇒n =10.

The values of the integrals are connected by the line while the dots are evaluated with p given by hyperequ29 – see table below:


 The 6j symbol and the oscillator wavefuncions
The Askey scheme and its counterpart point out at the connection in the angular momentum case between the top, the 6j symbol, and
the bottom, the harmonic oscillator. The geometrical insight of the Ponzano–Regge theory and its implementation in the Schulten–Gordon asymptotic formulas consistently lead to the expected Airy function behaviour astride of the transitions between classical and nonclassical regions of the ranges of elongations of the oscillator.

There is a  connection between the harmonic oscillator wavefunctions and 6j coefficients for large angular momentum arguments. Ponzano and Regge have approximated 6j-coefficients with sine and cosine functions as well as Airy functions. The formulas and extensions by Schulten and Gordon are excellent for the uniform semi-classical approximation for 6j-coefficients.

These semi-classical approximations  rely on the volume, surface areas, and angles that characterize the tetrahedron that corresponds to each 6j-coefficient that is required. A simple method connects a large set of special 6j-coefficients to harmonic oscillator wavefunctions by using only three parameters that are uniquely given from a simple algebraic analysis of the volumes of some tetrahedra related to the desired set of 6j-coefficients.

Consider the approximation of the 6j-coefficients:


as a function of j12 and j23. These discrete functions are orthonormal with relations:


Compare the 6j-coefficients with one dimensional quantum mechanical harmonic oscillator wavefunctions which belong to
an orthogonal set. Consider weighted 6j-coefficients:

hyperequ32Draw the connection with the harmonic oscillator wave-functions by noting that for given j1, j2, j12, j3, j there will be a value of j23max that will yield the maximum volume for the corresponding tetrahedron. The volume is given by


The maximum V² is obtained by finding the appropriate value of j23
that gives d (V²)/dj23=0. Consider the particular 6j-symbol:


where j2 = j1 and j= j3. This symbol has n nodes as j23 is varied. Set up the approximation using harmonic oscillator wave functions:



Looking at the 6j-symbol:


which gives the values for j23max and α. The figures
show n = 0, n =2, and n =7.


Representation of the 6j symbols by the harmonic oscillator wavefuntion for the case j1=1000, j2=1000, j12 =200, j3 =100,
j =100.


Representation of the 6j symbols by the harmonic oscillator wavefunction for the case j1 = 1000, j2 = 1000, j12 =198, j3 = 100, j =100.


Representation of the 6j symbols by the harmonic oscillator wavefunction for the case j1 = 1000, j2 =1000, j12 =193, j3 = 100, j =100.



Representation of the 6j symbols by the harmonic oscillator wavefunction for the case for j1 = 4000, j2 = 4000, j12 = 200, j3 =100, j =100


Representation of the 6j symbols by the harmonic oscillator wavefunction for the case for j1 = 8000, j2 =8000, j12 = 200, j3 =100, j = 100.

The harmonic oscillator parameters obtained from the two parameters:  j23max  and α provide a  representation of the behaviour of specific 6j-symbols by harmonic oscillator wavefunctions. The present state of the theory shows that the agreement should get better with increasing j.

 Related articles

Numerical work with sagemath 23: Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols

This work is based on the paper “Exact Computation and Asymptotic Approximations of 6j Symbols: Illustration of Their Semiclassical Limits by Mirco Ragni et al which I’ll be reviewing in my next post.

The 6j symbols tend asymptotically to Wigner dlnm functions when some angular momenta are large where θ assumes certain discrete values.







These formulas are illustrated below:




This can be modelled using sagemath.


The routine gives some great results:

For N=320, M=320, n=0, m=0, l=20, L=0,  Lmax=640

Wigner 6j vs cosθL


For N=320, M=320, n=0, m=0, l=10, L=0,  Lmax=640

Wigner 6j vs cosθL


For N=320, M=320, n=0, m=0, l=5, L=0,  Lmax=640

Wigner 6j vs cosθL



Exact and asymptotic computations of elementary spin networks

This week I have been following up some work which I was introduced to in Dimitri Marinelli’s PhD thesis ‘Single and collective dynamics of discretized geometries’. Essentially this involves the analysis of the volume operator.  This is really exciting for me as it is in my specialist research area –  the numerical analysis of Quantum geometric operators and their spectra. I’ll be following up the literature survey with numerical work in sagemath.

The paper I’ll look at this week is ‘Exact and asymptotic computations of elementary spin networks: classification of the quantum–classical boundaries’ by  Bitencourt, Marzuoli,  Ragni, Anderson and and Aquilanti.

There has been increasing interest to the issues of exact computations and asymptotics of spin networks. The large–entries regimes – semiclassical limits, occur in many areas of physics and in particular in discretization algorithms of applied quantum mechanics.

The authors extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient,  by exploiting its self–dual properties and studying it as a function of two discrete variables. This arises from its original definition as an orthogonal angular momentum recoupling matrix. Progress comes
from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational
advances –based on traditional and new recurrence relations– which allow an interpretation of the observed behaviors in terms of an underlying Hamiltonian formulation.

The paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines – caustics in the plane of the two matrix labels. This analysis marks the evolution of the turning points of relevance for the semiclassical regimes and highlights the key role of the Regge symmetries of the 6j.


The diagrammatic tools for spin networks were developed by the Yutsis school and  in connection with applications to discretized models for quantum gravity after Penrose, Ponzano and Regge.

The basic building blocks of all spin networks are the Wigner 6j symbols or Racah coefficients, which are studied here by exploiting their self dual properties and looking at them as functions of two variables. This approach is natural in view of their origin as matrix elements describing recoupling between alternative angular momentum binary coupling schemes, or between alternative hyperspherical harmonics.

Semiclassical and asymptotic views are introduced to describe the dependence on parameters. They originated from the association due to Racah and Wigner to geometrical features, respectively a dihedral angle and the volume of an associated tetrahedron, which is the starting point of the seminal paper by Ponzano and Regge . Their results provided an impressive insight into the functional dependence of angular momentum functions showing a quantum mechanical picture in terms of formulas which describe classical and non–classical discrete wavelike regimes, as well as the transition between them.

The screen: mirror, Piero and Regge symmetries

The 6j symbol becomes the eigenfunction of the Schrodinger–like equation in the variable q, a continuous generalization of j12:


where Ψ(q) is related to


and p² is related with the square of the volume V of the associated tetrahedron.

exactequtetraThe Cayley–Menger determinant permits to calculate the square of the volume of a generic tetrahedron in terms of squares of its edge lengths according to:


The condition for the tetrahedron with fixed edge lengths to exist as a polyhedron in Euclidean 3-space amounts to require V²> 0, while the V²= 0 and V²< 0 cases were associated by Ponzano and Regge to “flat” and nonclassical tetrahedral configurations respectively.

Major insight is provided by plotting both 6js and geometrical functions -volumes, products of face areas – of the associated tetrahedra in a 2-dimensional j12 -􀀀 j23 plane , in whch the square “screen” of allowed ranges of j12 and j23 is used in all the pictures

  •  The mirror symmetry. The appearance of squares of tetrahedron edges entails that the invariance with respect to the exchange J ↔− 􀀀J implies formally j ↔ – 􀀀j 􀀀-1 with respect to the entries of the 6j symbol.
  • Piero line. In general, an exchange of opposite edges of a tetrahedron corresponds to different tetrahedra and different symbols. In Piero formula, there is a term due to this difference that vanishes when any pair of opposite edges are equal.
  • Regge symmetries. The these arises through connection with the projective geometry of the elementary quantum of space, which
    is associated to the polygonal inequalities -triangular and quadrilateral in the 6j case -, which have to be enforced in
    any spin networks.

The basic Regge symmetry can be written in the following form:


The range of both J12 and J23, namely the size of the screen, is given by 2min (J1, J2, J3, J, J1 +ρ , J2 +ρ ,J3 +ρ, J + ρ).

Features of the tetrahedron volume function

Looking at the volume V as a function of  x=J12 and  y=J23 we get the expressions for the xVmaxand yVmax that correspond to the maximum of the volume for a fixed value of x or y:


The plots of these are  called “ridge” curves on the x,y-screen. Each one marks configurations of the associated tetrahedron when two specific pairs of triangular faces are orthogonal. The corresponding values of the volume (xVmax,xand yVmax,y) are

exactequ8F is the area of the triangle with sides a, b and c.Curves corresponding to V = 0, the caustic curves, obey the equations:





Symmetric and limiting cases

When some or all the j’s are equal, interesting features appear in the screen. Similarly when some are larger than others.


Symmetric cases



exactequfig4Limiting cases

We can discuss the caustics of the 3j symbols as the limiting case of the corresponding 6j where three entries are larger than the other ones:






The extensive images of the exactly calculated 6j’s on the square screens illustrate how the caustic curves studied in this paper separate the classical and nonclassical regions, where they show wavelike and evanescent behaviour respectively. Limiting
cases, and in particular those referring to 3j and Wigner’s d matrix elements can be analogously depicted and discussed. Interesting also are the ridge lines, which separate the images in the screen tending to qualitatively different flattening of the quadrilateral,
namely convex in the upper right region, concave in the upper left and lower right ones, and crossed in the lower left region.

Related articles

Quanta in a Quantum Spacetime

The Frontiers of Fundamental Physics 14 conference again captures my interest this week and I’ve been looking at a paper, ‘ How Many Quanta are there in a Quantum Spacetime?

In this paper the authors develop a technique for describing quantum states of the gravitational field in terms of coarse grained spin networks. They show that the number of nodes and links and the values of the spin depend on the observables chosen for the description of the state. So in order to say how many quanta are in a quantum spacetime further information about what has been measured has to be given.


The electromagnetic field can be viewed as formed by individual photons. This is a consequence of quantum theory. Similarly, quantum theory is likely to imply a granularity of the gravitational field, and therefore a granularity of space.

How many quanta form a macroscopic region of space? This question has implications for the quantum physics of black holes, scattering calculations in non perturbative quantum gravity and quantum cosmology. It is related to the question of the number of nodes representing a macroscopic geometry in a spin network state in loop gravity. In this context, it takes the following form: what is the relation between a state with many nodes and small spins, and a state with few nodes but large spins?

Quanta of space

A quanta of space may be  a quanta of energy from the excitation of the gravitational field. In loop quantum gravity, each quanta is a  quantum polyhedron. The geometry of quantum polyhedron defined by graph. We associate a state or element of Hilbert space for each quanta of space. The basis which spanned this Hilbert space is the spin network basis.

A quantum tetrahedron and its dual space geometry: the graph 

quantum tetrahedron

A graph γ is a finite set N of element n called nodes and a set of L of oriented couples called links l = (n, n’). Each node corresponds to one quantum tetrahedron. Four links pointing out from the node correspond to each triangle of the tetrahedron.

How many quanta in a field?

Consider a free scalar field in a finite box, in a classical configuration φ(x,t). The standard quantum-field-theoretical number operator, which sums the number the quanta on each mode, has a well defined classical limit. The number operator is

quantaequ1where an and an are the annihilation and creation operators for the mode n of the field and the sum is over the modes, namely the Fourier components, of the field. Since the energy can be expressed as a sum over modes as


where ωn is the angular frequency and En its energy of the mode n, it follows that the number of particles is given


which is a well defined classical expression that can be directly obtained from φ(x,t) by computing the energy in each mode. Therefore each classical configuration defines a total particle-number N and a distribution of these particles over the modes


Subset graphs

The state space of loop quantum gravity contains subspaces Hγ associated to abstract graphs γ. A graph γ is defined by a finite set N of |N| elements n called nodes and a set L of |L| oriented couples l = (n, n’) called links.

A pure state |ψ〉 determines the density matrix ργ= |ψ〉〈ψ|. A generic state can be written in the form


In the loop gravity the operators defined on Hγ can be interpreted as the description of the geometry of |N| quantum polyhedra connected to one another when there is a link between the corresponding nodes.

Given a graph , define a subset graph Γ which partitions N into subsets N such that each N is a set of nodes connected among themselves by sequences of links entirely formed by nodes in N.

quantafig1 We define the area of the big link by


and the volume of the big node by


where we recall that v is the expression for the classical volume of a polyhedron. The operators AL and VN commute, so they can be diagonalized together. The quantum numbers of the big areas are half integers JL and the quantum numbers of the volume are VN.

Course graining spin networks

Coarse-graining the entire graph into a graph Γ formed by a single node N with legs b

quantafig2A  set of small links l that are contained in a single large link L.

quantafig3Any general coarse-graining is a combination of collecting nodes and summing links


The geometry of the subset graph

The geometrical interpretation of the coarse grained states in HΓ is that these describe the geometry of connected polyhedra. The partition that defines the subset graph Γ is a coarse-graining of the polyhedra into larger chunks of space. The surfaces that separate these larger chunks of space are labelled by the big links L and are formed by joining the individual faces labelled by the links l in L.

In general, it is clearly not the case that the area AL is equal to the sum of the areas Al of all l in L. However, this is the case if all these faces are parallel and have the same orientation. Similarly, in general, it is clearly not the case that the volume VN is equal to the sum of the volumes Vn for the n in N. However, this is true if in gluing n polyhedra one obtains a at polyhedron with flat faces.

The two operators,


provide a good measure of the failure of the geometry that the state associates to Γto be flat.

To have a good visualization of the coarse-grained geometries, it is helpful to consider the classical picture. In the 4-dimensional theory, the graph is defined at the boundary of a 3-dimensional hypersurface, the spin operator on the links is related to the area operator byquantaequ48a

Given a 3-valent graph with spins operators Jla, Jlb , and Jlc on each link, the dihedral angle between Jlb and Jlc can be obtained from the angle operator, defined by


Applying this operator to the spin network state gives the dihedral angle between Jland Jlc on the internal links lb and lc.


The Regge intrinsic curvature of a discretized manifold is given by the deficit angle on the hinges, the (n -2) dimensional simplices of the n-dimensional simplex. Thus, given a loopgraph with n-external links, the deficit angle for a general n-polytope (n-valent loop graph) is:


Coarse grained area

The boundary of spacetime is a 3-dimensional space. Triangulation on the boundary is defined using flat polyhedra. Every closed, flat, n polyhedron satisfies the closure relation on the node given by


Consider the net of a polyhedron:


Since the interior of polyhedron is at, the closure relation can be written as


Then the area operator on the base is


and we can define the coarse-grained area as:


Thus, for a 2-dimensional surface, we can always think the coarse grained area AL as the area of the base of a polyhedron, while the total sum of area Al is the area around the hat – the area of n triangle which form the net of the polyhedron.

The differences between the coarse-grained and the fine-grained area gives a good measurement on how the space deviates from being flat. It is possible to obtain the explicit relation between the Regge curvature with these area differences in some special cases.

The Regge curvature for a 2-dimensional surface is defined as 2 minus the sum of all dihedral angle surrounding a point of the triangulation, which is n. The Regge curvature as a function of the coarse-grained and the fine-grained area is:


In the classical limit, it is clear that there can be states where ε = 0 or AL = 0. These correspond to geometries where the normals to the facets forming the large surface L are parallel. However, this is only true in the classical limit, namely disregarding Planck scale effects. If we take Planck-scale effects into account, we have the  result that


quantaequ63where n + 1 is the number of facets. Therefore the fine grained area is always strictly larger than the coarse grained area. There is a Planck length square contribution for each additional facet. It is as if there was an irreducible Planck-scale fluctuation in the orientation of the facets.

 Coarse-grained volume


In the same manner as the surface’s coarse-graining, we triangulate a 3-dimensional chunk of space using n symmetric tetrahedra. The Regge curvature is defined by the dihedral angle on the bones of the tetrahedra. Using the volume of one tetrahedron,



we obtain the fine-grained volume, which is


The coarse-grained volume is the volume of the 3-dimensional base, which is the volume of the n-diamond:


so the Regge curvature is


Notice that this is just a classical example. In the quantum picture, adding two quantum tetrahedra does not gives only a triangular bipyramid, it could give other possible geometries which have 6 facets, i.e., a parallelepiped, or a pentagonal-pyramid.


The number of quanta is not an absolute property of a quantum state: it depends on the basis on which the state is expanded. In turn, this depends on the way we are interacting with the system. The quanta of the gravitational field we interact with, are those described by the quantum numbers of coarse-grained operators like AL and V, not the maximally fine-grained ones.

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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.

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


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


To implement the closure relation   define


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


These quantities fulfill the canonical Poisson relations


The Tetrahedron

The classical volume of a tetrahedron can be expressed


Look at the quantity,


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,



CCqTequ11such that


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


fullling {p,A} = 1 and




where A varies according to Amin ≤A ≤Amax with


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







The analogous expansion around the minimum reads



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



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



 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


CCqTequ27such that


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


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


where the operators


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


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


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


the corresponding eigenstates fulfill


For odd d an additional zero eigenvalue occurs.

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








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


to quantities having also dimension of area.

This gives,




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


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


one obtains up to fourth order in the expansions




Introducing the operators


The effective operator expression is


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


with eigenvalues


and corresponding eigenfunctions


where Hn(x) are the usual Hermite polynomials.




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.

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