The Spinfoam Framework for Quantum Gravity by Livine

This post looks at  the part of Livine’s  great PhD thesis about the quantum tetrahedron.

Below is a table from this thesis looking at the types of structures found in quantum gravity:


I’ll be looking at the quantum tetrahedron, intertwiner and group field structures.

Loop quantum gravity provides a mathematically  rigorous quantisation of  operators for geometric observables such as area and volume and also shows that the spectra of these operators are discrete in Planck units. This leads to a clear picture of discrete quantum geometry.


There is  a straightforward geometric interpretation of spin network states: the edges e of the graph are dual to elementary surfaces whose area is given by the spin j carried to the edge, and the vertices v are dual to elementary chunks of 3d space bounded by those elementary surfaces and whose volume is determined by the intertwiner  living at the vertex.



Spin network states


This interpretation points towards the reconstruction of a discrete geometry dual to the spin network state, with classical polyhedra reconstructed around each vertex whose faces are dual to the edges attached to the vertex and whose exact shape would depend on the explicit intertwiner living at the vertex. This point of view has been particularly developed from the perspective of geometric quantization. It is possible to see intertwiners as quantum polyhedra.

See the post: Polyhedra in loop quantum gravity

In particular, a lot of  work has focused on the interpretation of 4-valent intertwiners as quantum tetrahedron. This point of view has been particularly useful to build spinfoam models as quantized 4-dimensional triangulations



The quantum tetrahedron

In order to  identify intertwiners as quantum polyhedra and spin network states as discrete geometries, we need to be able to build semi-classical intertwiner states whose shape would be peaked on classical polyhedra and then to glue them together in order to build semi-classical spin network states peaked on classical discrete geometries.

There has  been a lot of research work done on developing concepts such as complexifier coherent states introduced by Thiemann, the related holomorphic spin network states  and coherent intertwiner states introduced by Livine Speziale.

Particular recent lines of research which seems to re-unify these works and viewpoints are the twisted geometry framework  and the U(N) framework for intertwiners which actually converge themselves to a unified picture of coherent spin network states as semiclassical discrete geometries . These frameworks are partly inspired from the picture of coherent intertwiners and allow to define explicit variables which control the shape of intertwiners and also parameterize classical polyhedra, thus creating an explicit bridge between the two.

 Related articles


Quantum cosmology of loop quantum gravity condensates: An example by Gielen

This week I have mainly been studying the work done during the Google Summer of Code workshops, in particular that on sagemath knot theory at:

This work looks great and I’ll be using the results in some of my calculations later in the summer.

Another topic I’ve been reviewing is the idea of spacetime as a Bose -Einstein condensate. This together with emergent, entropic and  thermodynamic gravitation seem to be an area into which the quantum tetrahedron approach could naturally fit via statistical mechanics.

In the paper, Quantum cosmology of loop quantum gravity condensates, the author reviews the idea that spatially homogeneous universes can be described in loop quantum gravity as condensates of elementary excitations of space. Their treatment by second-quantised group field theory formalism allows the adaptation of techniques from the description of Bose–Einstein condensates in condensed matter physics. Dynamical equations for the states can be derived directly from the underlying quantum gravity dynamics. The analogue of the Gross–Pitaevskii equation defines an anisotropic  quantum cosmology model, in which the condensate wavefunction becomes a quantum cosmology wavefunction on minisuperspace.

The spacetimes relevant for cosmology are to a very good approximation spatially homogeneous. One can use this fact and perform a symmetry reduction of the classical theory – general relativity coupled to a scalar field or other matter – assuming spatial
homogeneity, followed by a quantisation of the reduced system. Inhomogeneities are usually added perturbatively. This leads to models of quantum cosmology which can be studied  without the need for a full theory of quantum gravity.

Loop quantum gravity (LQG) has some of the structures one would expect in a full theory of quantum gravity: kinematical states corresponding to functionals of the Ashtekar–Barbero connection can be rigorously defined, and geometric observables such
as areas and volumes exist as well-defined operators, typically with discrete spectrum. The use of the LQG formalism in quantising symmetry-reduced gravity leads to loop quantum cosmology (LQC).

Because of the well-defined structures of LQG, LQC allows a rigorous analysis of issues that could not be addressed within the Wheeler– DeWitt quantisation used in conventional quantum cosmology, such as a definition of the physical inner product. More recently, LQC has made closer contact with CMB observations, and the usual inflationary scenario is now discussed within LQC.

A new approach towards addressing the issue of how to describe cosmologically relevant universes in loop quantum gravity uses the group field theory (GFT) formalism, itself a second quantisation formulation of the kinematics and dynamics of LQG: one has a Fock space of LQG spin network vertices or tetrahedra, as building blocks of a simplicial complex, annihilated and created by the field operator ϕ and its Hermitian conjugate ϕ†, respectively. The advantage of using this reformulation is that field-theoretic techniques are available, as a GFT is a standard quantum field theory on a curved group manifold. In particular, one can define coherent or squeezed states for the GFT field, analogous to states used in the physics of Bose– Einstein condensates or in quantum optics; these represent quantum gravity condensates. They describe a large number of degrees of freedom of quantum geometry in the same microscopic quantum state, which is the analogue of homogeneity for a differentiable metric geometry. After embedding a condensate of tetrahedra into a smooth manifold representing a spatial hypersurface, one shows that the spatial metric in a fixed frame reconstructed from the quantum state is compatible with spatial homogeneity. As the number of tetrahedra is taken to infinity, a continuum homogeneous metric can be approximated to a better and better degree.

At this stage, the condensate states defined in this way are kinematical. They are gauge-invariant by construction, and represent geometric data invariant under spatial diffeomorphisms. The strategy followed for extracting information about the dynamics of these states is the use of Schwinger–Dyson equations of a given GFT model. These give constraints on the n point functions of the theory evaluated in a given condensate state – approximating a non-perturbative vacuum, which can be translated into differential equations for the condensate wavefunction used in the definition of the state. This is analogous to condensate states in many-body quantum physics, where such an expectation value gives, in the simplest case, the Gross–Pitaevskii equation for the condensate
wavefunction. The truncation of the infinite tower of such equations to the simplest ones is part of the approximations made. The effective dynamical equations obtained can be viewed as defining a quantum cosmology model, with the condensate wavefunction interpreted as a quantum cosmology wavefunction. This provides a general procedure for deriving an effective cosmological dynamics directly from the underlying theory of quantum gravity. It canbe shown that  a particular quantum cosmology equation of this type, in a semiclassical WKB limit and for isotropic universes, reduces to the classical Friedmann equation of homogeneous,
isotropic universes in general relativity.

See posts:

Let’s  analyse more carefully the quantum cosmological models derived from quantum gravity condensate states in GFT. In particular, the formalism identifies the gauge-invariant configuration space of a tetrahedron with the minisuperspace of homogeneous generally anisotropic geometries.

Using a convenient set of variables the gauge-invariant geometric data, can be mapped to the variables of a general anisotropic Bianchi model it is possible to  find simple solutions to the full quantum equation, corresponding to isotropic universes.

They can only satisfy the condition of rapid oscillation of the WKB approximation for large positive values of the coupling μ in the GFT model. For μ < 0, states are sharply peaked on small values for the curvature, describing a condensate of near-flat building blocks, but these do not oscillate. This supports the view that rather than requiring semiclassical behaviour at the Planck scale, semiclassicality should be imposed only on large-scale observables.

 From quantum gravity condensates to quantum cosmology

Review the relevant steps in the construction of effective quantum cosmology equations for quantum gravity condensates. Use group field theory (GFT) formalism, which is a second quantisation formulation of loop quantum gravity spin networks of fixed valency, or their dual interpretation as simplicial geometries.

The basic structures of the GFT formalism in four dimensions are a complex-valued field ϕ : G⁴ → C, satisfying a gauge invariance property


and the basic non-relativistic commutation relations imposed in the quantum theory


These relations  are analogous to those of non-relativistic scalar field theory, where the mode expansion of the field operator defines annihilation operators.

In GFT, the domain of the field is four copies of a Lie group G, interpreted as the local gauge group of gravity, which can be taken to be G = Spin(4) for Riemannian and G = SL(2,C) for Lorentzian models. In loop quantum gravity, the gauge group is the one given by the classical Ashtekar–Barbero formulation, G = SU(2). This property encodes invariance under gauge transformations acting on spin network vertices.

The Fock vacuum |Ø〉 is analogous to the diffeomorphism-invariant Ashtekar–Lewandowski vacuum of LQG, with zero expectation value for all area or volume operators. The conjugate  ϕ acting on the Fock vacuum |Ø〉  creates a GFT particle, interpreted as a 4-valent spin network vertex or a dual tetrahedron:


The geometric data attached to this tetrahedron, four group elements gI ∈ G, is interpreted as parallel transports of a gravitational connection along links dual to the four faces. The LQG interpretation of this is that of a state that fixes the parallel transports of the Ashtekar–Barbero connection to be gI along the four links given by the spin network, while they are undetermined everywhere else.

In the canonical formalism of Ashtekar and Barbero, the canonically conjugate variable to the connection is a densitised inverse triad, with dimensions of area, that encodes the spatial metric. The GFT formalism can be translated into this momentum space formulation by use of a non-commutative Fourier transform


The geometric interpretation of the variables B ∈ g is as geometric bivectors associated to a spatial triad e, defined by the integral triadover a face △ of the tetrahedron. Hence, the one-particle state


Defines a tetrahedron with minimal uncertainty in the fluxes, that is the oriented area elementstriad given by B . In the LQG interpretation this state completely determines the metric variables for one tetrahedron, while being independent of all other degrees of freedom of geometry in a spatial hypersurface.

The idea of quantum gravity condensates is to use many excitations over the Fock space vacuum all in the same microscopic configuration, to better and better approximate a smooth homogeneous metric or connection, as a many-particle state can contain information about the connection and the metric at many different points in space. Choosing this information such that it is compatible with a spatially homogeneous metric while leaving the particle number N free, the limit N → ∞ corresponds to a continuum limit in which a homogeneous metric geometry is recovered.

In the simplest case, the definition for GFT condensate states is


where N(σ) is a normalisation factor. The exponential creates a coherent configuration of many building blocks of geometry. At fixed particle number N, a state of the form σⁿ|Ø〉 would be interpreted as defining a metric (or connection) that looks spatially homogeneous when measured at the N positions of the tetrahedra, given an embedding into space usually there is a sum over all possible particle numbers. The condensate picture does not use a fixed graph or discretisation of space.

The GFT condensate is defined in terms of a wavefunction on G⁴
invariant under separate left and right actions of G on G⁴ . The strategy is then to demand that the condensate solves the GFT quantum dynamics, expressed in terms of the Schwinger–Dyson equations which relate different n-point functions for the condensate. An important approximation is to only consider the simplest Schwinger– Dyson equations, which will give equations of the form


This is analogous to the case of the Bose–Einstein condensate where the simplest equation of this typegives the Gross–Pitaevskii equation.

In the case of a real condensate, the condensate wavefunction Ψ (x), corresponding to a nonzero expectation value of the field operator, has a direct physical interpretation: expressing it in terms of amplitude and phase, psi one can rewrite the
Gross–Pitaevskii equation to discover that ρ(x) and v(x) = ∇θ(x) satisfy hydrodynamic equations in which they correspond to the density and the velocity of the quantum fluid defined by the condensate. Microscopic quantum variables and macroscopic classical variables are directly related.

The wavefunction σ or ξ of the GFT condensate should play a similar role. It is not just a function of the geometric data for a single tetrahedron, but equivalently a function on a minisuperspace of spatially homogeneous universes. The effective dynamics for it, extracted from the fundamental quantum gravity dynamics given by a GFT model, can then be interpreted as a quantum cosmology model.

Minisuperspace – gauge-invariant configuration space of a tetrahedron

Condensate states are determined by a wavefunction σ, which is
a complex-valued function on the space of four group elements for given gauge group G which is invariant under


is a function on G\G⁴/G. This quotient space is a smooth manifold
with boundary, without a group structure. It is the gauge-invariant configuration space of the geometric data associated to a tetrahedron. When the effective quantum dynamics of GFT condensate states is reinterpreted as quantum cosmology equations, G\G⁴/G becomes a minisuperspace of spatially homogeneous geometries.

Condensate states in group field theory can be used to derive effective quantum cosmology models directly from the dynamics of a quantum theory of discrete geometries. This can be illustrated by the interpretation of the configuration space of gauge-invariant geometric data of a tetrahedron, the domain of the condensate
wavefunction, as a minisuperspace of spatially homogeneous 3-metrics.

I’ll also looking at the calculations behind this paper in more detail in a later post.











Quantum states of elementary three–geometry by Carbone, Carfora , and Marzuoli

This is a relatively old paper but its so clearly written that it is well worth reviewing and understanding the material.

In this paper the authors introduce a quantum volume operator K in three–dimensional Quantum Gravity by taking into account a symmetrical coupling scheme of three SU(2) angular momenta. The spectrum of K is discrete and defines a complete set of eigenvectors
which is an alternative with respect to the complete sets employed when the usual binary coupling schemes of angular momenta are considered. Each of these states,  called quantum bubbles, represents an interference of extended configurations which provides a rigorous meaning to the heuristic notion of quantum tetrahedron. They study the generalized recoupling coefficients connecting the symmetrical and the binary basis vectors, and provide an explicit recursive solution for such coefficients by analyzing also its asymptotic limit.


Ponzano–Regge gravity is based on an asymptotic formula for theSU(2) 6j symbols,


The physical interpretation follows if we recognize that the exponential includes the classical Regge action – the discretized version of the Hilbert– Einstein action for the tetrahedron T. The
presence of the slowly varying volume term in front of the phase factor means that {6j}as can be interpreted as a probability amplitude in the approach to the classical limit. The probability amplitude for an elementary block of Euclidean three–geometry to emerge from the recoupling of quantum angular momenta or from a spin network.

Physical reality can only be ascribed to the tetrahedron T if and only if its volume –written in terms of the squares of the edges through the Cayley determinant– satisfies the condition (V (T))² > 0. The triangle inequalities on the four triples of spin variables  – associated with the four faces of the tetrahedron – which ensure the existence of the 6j symbol are weaker than the condition that T exists as a realizable solid.

The authors try to answer the following questions:

  • Why does a classical three–geometry emerge from a recoupling of angular momenta?
  • What does it mean to give the quantum state of an elementary block of geometry?
  • What are the degrees of freedom of a quantum tetrahedron?

Quantum bubbles

The theory of the coupling of states of three SU(2) angular momenta operators J₁, J₂,  J₃ to states of sharp total angular momentum J is usually developed in the framework of binary couplings. Starting from the ordered triple,  denote symbolically the admissible schemes according to



The corresponding state vectors may be written as


In this framework the Wigner 6j symbol is just the recoupling coefficient relating the two sets  and , namely


where Φ ≡ J₁+J₂+J₃

In spite of the tetrahedral symmetry of the 6j symbol, at the quantum level we cannot recognize something like a quantum tetrahedron: it is only in the approach to the classical limit  that both the coupling schemes coexist and the tetrahedron may take shape.

This situation changes  if we bring into play the symmetrical coupling
scheme for the addition of three SU(2) angular momenta to give a fourth definite angular momentum J with projection J₀ symbolically

3gequ7 such a coupling is characterized by the simultaneous diagonalization of the six Hermitian operators

J²₁,J²₂,J²₃,  K, J² and J₀, where K is a scalar operator built from the irreducible tensor operators J₁, J₂,  J₃ and defined according to


Since K is the mixed product of three angular momentum vectors we can call it a volume.

Each of the eigenvectors of this new set of operators is denoted by 3gequ8

the set of eigenvalues in each subspace with (jm) fixed is discrete and
consists of pairs (k,−k), with at most one zero eigenvalue.

From a geometrical point of view a state |kjm >, for k ≠ 0, represents a quantum volume of size kħ³ and k2 > 0 is the quantum counterpart of the condition (V (T))² > 0.

According to the Correspondence Principle in the region of large quantum numbers the states state |kjm >characterize angular momentum vectors confined to narrow ranges
around specific values: the narrower are the ranges, the closer we approach the classical regime. This implies that the classical limit of the operator coincides with the expression of the volume of the classical tetrahedron T spanned by the counterparts of the operators  J₁, J₂,  J₃.

The state |kjm > for fixed (jm) provides a proper description of the quantum state of an elementary block of Euclidean three–geometry. However |kjm > cannot be directly interpreted as a quantum tetrahedron, but rather as a quantum interference of extendend configurations from which information about all significant geometrical quantities can be extracted.

 Generalized recoupling theory

The 6j symbol can be interpreted as a propagator between two states belonging to alternative binary bases. The asymptotic expression:  3gequ1is actually the generating functional of a path–sum evaluated at the semiclassical level and the physical information about the underlying classical theory is encoded in the form of the action.

For the quantitative analysis of the relations between one of the sets of eigenvectors arising from a binary coupling parametrized by the eigenvalue of the intermediate momentum J₁₂ and the symmetrical basis  we introduce a unitary transformation defined as


The generalized recoupling coefficient


and its inverse defined as


A three–term recursion relation for the generalized recoupling coefficient is


The functions α (l =  J₁₂, J₁₂ + 1) can be cast in the form


Since 3gequ21the conditions at the extrema fixed by  are


The formal solution of 3gequ19



Asymptotic limit

To explore the generalized recoupling coefficient we use a semiclassical (WKB) approximation to the recursion relation


Which  gives an ordinary second order difference equation
of the type


we search for a solution of this for each k, in the form


with ρ and A to be determined. By substitution we get a pair of differential equations corresponding respectively to the imaginary and real parts;


Finally, since the classical counterpart of the operator K is the volume V ≡ V (T) of the Euclidean tetrahedron T we see that the generalized recoupling coefficient in the asymptotic limit behaves as



where θ₁₂ is the dihedral angle between the faces of the tetrahedron T which share J₁₂.

 Related articles

Sagemath21: Knot theory calculations

This week I have been working on a variety of topics, but I’d like to post about some sagemath knot theory investigations Ive been doing.  This is connected to one of my longer term aims, which  is to  perform SU(2) recoupling calculations based on  Temperley-Lieb algebras using Sage Mathematics Software. Essentially I would like to to write a set of tools for doing SU(2) recoupling theory, Penrose binor calculus and  Temperley—Lieb algebra calculations using sagemath. 

Lets see what sagemath can do:

knots code

knot output1




animate knot

animated knot