Learning about Quantum Gravity with a Couple of Nodes by Boria, Garay and Vidotto

This week I have looking at  paper about Quantum gravity on a graph with just two nodes. This has great technical information on performing quantum cosmological calculations which I’ll review in another post. In this post I just want to give a brief qualitative outline of the paper.

Loop Quantum Gravity provides a natural truncation of the infinite degrees of freedom of gravity by setting the theory on a finite graph. In this paper the authors review this procedure  and  present the construction of the canonical theory on a simple graph, formed by only two nodes.

They  review the U(N) framework, which provides a powerful tool for the canonical study of this model, and a formulation of the system based on spinors. They also consider  the covariant theory, which allows the  derivation of the  the model from a more complex formulation.

Why graphs?

Discrete gravity

The essential idea behind the graph truncation can be traced to
Regge calculus, which is based on the idea of approximating spacetime with a triangulation, where the metric is everywhere flat except on the triangles. On a fixed spacelike surface, Regge
calculus induces a discrete 3-geometry defined on a 3d triangulation, where the metric is everywhere flat except on the bones. The two-skeleton of the dual of this 3d cellular decomposition is a graph Γ, obtained by taking a point -a node- inside each cell, and connecting it to the node in an adjacent cell by a link, puncturing the triangle shared by the two cells.

Spin networks

In Loop Quantum Gravity, the spinnetwork basis |Γ, j, ni 〉 is an orthonormal basis that diagonalizes the area and volume operators. The states in this basis are labelled by a graph Γand two quantum numbers coloring it: a spin j at each link  and a volume eigenvalue  at each node n. The  Hilbert space HΓ obtained by considering only the states on the  graph Γis precisely the Hilbert space of an SU(2) Yang-Mills theory on this lattice. Penrose’s spin-geometry theorem connects this Hilbert space with the description of the geometry of the cellular decomposition: states in this Hilbert space admit a geometrical interpretation as a quantum version of the 3-geometry. That is, a Regge 3-geometry defined on a triangulation with dual graph Γ can be approximated by semiclassical state in HΓ.


In the canonical quantization of General Relativity, in order to implement Dirac quantization, it’s convenient to choose the densitized inverse triad Ea –Ashtekar‘s electric field and the Ashtekar-Barbero connection Ai as conjugate variables, and then use the flux of Ei and the the holonomy h as fundamental variables for the quantization. The holonomies can be taken along the links of the graph, and the densitized inverse triad can be smeared over the faces of the triangulation. This connects the holonomy triad variables to the discrete geometry picture.

The common point of these different derivations is 3d coordinate gauge invariance. This invariance is the reason for the use of abstract graphs: it removes the physical meaning of the location of the graph on the manifold. Therefore the graph is just a combinatorial object, that codes the adjacency of the nodes. Each node describe a quantum of space, and the graph describes the relations between different pieces of space. The Hilbert subspaces associated to distinct but topologically equivalent embedded graphs are identified, and each graph space contains the Hilbert spaces of all the subgraphs.


Doing physics with few nodes

The restriction to a fixed triangulation or a fixed graph amounts only to a truncation of the theory, by cutting down the theory to an approximate theory with a finite number of degrees of freedom. Truncations are always needed in quantum field theory, in order to extract numbers from the theory -in appropriate physical
regimes even a low-order approximation can be effective.

Discretizing a continuous geometry by a given graph is nothing but coarse graning the theory. The discreteness introduced by this process is different from the fundamental quantum discreteness
of the theory. The first is the discreteness of the abstract graphs; the later is the discreteness of the spectra of the area and volume operator on each given HΓ.

There are also  two different expansions:

  • The graph expansion obtained by a refinement of the graph, valid at scales smaller that the curvature scale R.
  • The semiclassical expansion,  a large-distance limit on each graph, valid at scales larger that the Planck scale Lp


Interesting physics can arise even by considering a simple graph, with few nodes, and comparing our results with classical discrete gravity. This is true for FLRW cosmologies. It has in fact been proven numerically  that the dynamics of a closed universe, with homogeneous and isotropic geometry, can be captured by 5, 16 and 600 nodes –  the regular triangulations of a 3-sphere.


 On the left, a 4d building block of spacetime and, on the right, the evolution of 5, 16 and 500 of these building block (dashed lines), modelling a closed universe, compared whit the continuous analytic
solution (solid line).

The cosmological interpretation

The most striking example, where this kind of approximation applies, is given by cosmology itself. Modern cosmology is based on the cosmological principle, that says that the dynamics of a homogeneous and isotropic space approximates well our universe. The presence of inhomogeneities can be disregarded at a first order approximation, where we consider the dynamics as described at the scale of the scale factor, namely the size of the universe.

Working with a graph corresponds to choosing how many degrees of freedom to describe. A graph with a single degree of freedom is just one node: in a certain sense, this is the case of usual Loop Quantum Cosmology . To add degrees of freedom, we add nodes and links with a colouring. These further degrees of freedom are a natural way to
describe inhomogeneities and anisotropies , present in our universe.

The easiest thing that can be done is to pass from n = 1 to n = 2 nodes. We choose to connect them by L = 4 links, because in this way the dual graph will be two tetrahedra glued together, and this can be viewed as the triangulation of a 3-sphere .


Choosing a graph corresponding to a triangulation is useful when we want to associate an intuitive interpretation to our model. In order to understand how this can be concretely use to do quantum gravity and quantum cosmology, we need to place on the graph the SU(2) variables.

LQG can describe large semiclassical geometries also over a small number of nodes and links. This paper  reviewed a number of constructions in Loop Quantum Gravity, based on the idea of truncating the Hilbert space of the theory down to the states supported on a simple graph with two nodes.


Below there is a diagram of the solutions found using a 2-node graph. The quantum Hamiltonian of the 2-node model is mathematically analogous to the gravitational part of the Hamiltonian in LQC. Following this analogy, we can interpret the results obtained as the classical analogous of the quantum big bounce found in LQC.


The restriction of the full LQG Hilbert space to a simple graph is a truncation of the degrees of freedom of the full theory. It defines an approximation where concrete calculations can be performed. The approximation is viable in physical situations where only a small number of the degrees of freedom of General Relativity are relevant.

A characteristic example is cosmology. The 2-node graphs with 4 links defines the simplest triangulation of a 3-sphere and can accommodate the anisotropic degrees of freedom of a Bianchi IX model, plus some inhomogeneous degrees of freedom.  In this context, a Bohr-Oppenheimer approximation provides a tool to separate heavy and light degrees of freedom, and extract the
FLRW dynamics. This way of deriving quantum cosmology from LQG is different from the usual one: in standard loop quantum cosmology, the strategy is to start from a symmetry-reduced system, and quantize the single or the few degrees of freedom that survive in the symmetry reduction. In this paper  a truncated version of the full quantum theory of gravity is consided.


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