Constructing spacetime from the quatum tetrahedron: Spacetime as a Bose-Einstein Condensate

This post is based on the paper Cosmology from Group Field Theory Formalism for Quantum Gravity by Gielen, Oriti and Sindoni. In this paper the authors identify a class of condensate states in the group field theory formulation of quantum gravity that can be interpreted as macroscopic homogeneous spatial geometries. They then extract the dynamics of the condensate states directly from the quantum GFT dynamics, following the procedure used in the study of quantum fluids. They find the effective dynamics to be a non-linear and non-local extension of quantum cosmology. They also show that any GFT model with a kinetic term of Laplacian type gives rise, in a semi-classical (WKB) approximation and in the isotropic case, to a modified Friedmann equation. Their procedure is the first concrete, general procedure for extracting an effective cosmological dynamics directly from a theory of quantum geometry.

Firstly they  use the group field theory  formalism for quantum gravity which is related to loop quantum gravity and spin foam models.Having  identified criterion for discrete geometries to approximate continuum ones and to be compatible with spatial homogeneity, they propose a class of GFT states describing continuum macroscopic homogeneous geometries: GFT condensates which are superpositions of N-particle states satisfying the criterion for spatial homogeneity at each N, which are then spatially homogeneous to arbitrary accuracy.

The appearance of macroscopic geometries is described by a process similar to Bose-Einstein condensation of fundamental quanta, leading to the idea of spacetime as a condensate. Following the dynamics of such condensate states is explored directly from the fundamental quantum GFT dynamics, following the procedure used for quantum fluids. This effective dynamics is shown to have the form of a nonlinear and nonlocal extension of quantum cosmology. It is shown that any GFT model involving a Laplacian kinetic term, as suggested gives rise in a WKB approximation to an equation describing the classical dynamics of a homogeneous universe, and in the isotropic case to a modified Friedmann equation with corrections determined by the fundamental GFT dynamics.

The  procedure in this paper applies to any GFT model incorporating appropriate pregeometric data, It opens a new way of formulating effective equations for an emergent spacetime geometry from a pregeometric scenario and bolsters claims that such quantum gravity models correspond to general relativity in a semiclassical continuum approximation.

Group field theory
Group field theories are quantum field theories on group manifolds with a nontrivial combinatorial structure of quantum states and histories. Their quantum states are in fact four-valent graphs labeled by group or Lie algebra elements, which can be equivalently represented as 3D cellular complexes. The quantum dynamics, in a perturbative expansion around the vacuum, gives a sum of Feynman diagrams dual to 4D cellular complexes of arbitrary topology. The Feynman amplitudes for these discrete histories can be written either as spin foam models or as simplicial gravity path integrals.

GFTs can be defined in terms of a (complex) bosonic field  ϕ(g1, g2, g3, g4) on SO(4), A quantum of the GFT field, created by the operator ϕ(g1, . . . , g4), is interpreted as a tetrahedron whose geometry is given by the four parallel transports of the gravitational SO(4) connection along links dual to its faces.

BEC tetrahedron

A quantum of the GFT field –  a quantum  tetrahedron

A single field “quantum” maybe a spin network vertex or quantum tetrahedro.nBEC tetrahedron 2

In this picture, a superposition of N-particle states in the GFT corresponds to a spin network with N vertices or a complex with N tetrahedra.

BEC tetrahedra 3

A superposition of N-particle states         

One can use a noncommutative Fourier transform to define the field on conjugate Lie algebra variables ϕ(B1,B2,B3,B4). The variables BI so(4) are bivectors associated to the faces of the tetrahedron:

Where e is a cotetrad field encoding the simplicial geometry.

In order to ensure this interpretation, the variables BI must satisfy two types of conditions:

  • Simplicity constraints which impose a restriction on the domain of ϕ to be a submanifold of SO(4).
  • A second condition is invariance under gauge transformations which encodes a closure constraint: the bivectors BI must close to form a tetrahedron

Approximate geometries and homogeneity

The N-particle state BI is interpreted as a discrete geometry of N tetrahedra with bivectors BI(m) associated to the faces.

To approximate a continuum spatial geometry, we think of the tetrahedra as embedded into a three-dimensional topological manifold. An embedding of each tetrahedron is specified by the location of one of the vertices and three tangent vectors specifying the directions of the three edges emanating from this vertex. This requires assuming that the tetrahedra are associated to regions in the embedded manifold which are sufficiently flat, so that we can approximate the tetrad as constant.

We can say that a discrete geometry of N tetrahedra, specified by the data gij(m), is compatible with spatial homogeneity if

gij(m) = gij(k) ∀ k,m = 1, . . . ,N.

The correspondence between N-particle GFT states and continuum geometries can be viewed as the result of sampling the metric at N points.

GFT condensates as continuum homogeneous geometries

The quantum counterpart of the classical homogeneity condition becomes the requirement that the GFT N-particle state has a product structure in which the same wave function is assigned to each GFT quantum tetrahedron. Then, arbitrary superpositions of such N-particle states can be considered with N arbitrarily large. This means that  the reconstructed spatial geometry is homogeneous to arbitrary accuracy. This is then  a GFT quantum condensate state.

Two examples of GFT condensates

BE GFT condensate states

Effective cosmological dynamics

The effective dynamics for a homogeneous quantum space, i.e. for GFT condensates, can be extracted from a generic GFT for 4D quantum gravity, by following closely the standard procedures used in quantum fluids. Applying these to GFT condensates the authors obtain an effective equation for the
collective cosmological wave functions. This is the GFT analog of the Gross-Pitaevskii equation for BECs. It is a nonlinear and nonlocalequation for the collective cosmological wave function.


  • GFT is analogous to QFT for atoms in condensed matter system.
  • Continuum spacetime with GR-like dynamics emerges from collective behaviour of large numbers of GFT building blocks such as quantum tetrahedra.
  • A spacetime as a quantum condensate requires that there is a GFT analogue of the thermodynamic limit and in the macroscopic approximation an appropriate phase appears such that continuum spacetime is a GFT condensate
  • A phase transition leading to spacetime and geometry (GFT condensation) is what replaces Big Bang singularity – geometrogenesis

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