GFT Condensates and Cosmology

This week I have been studying some papers and a seminar by Lorenzo Sindoni and Daniele Oriti on Spacetime as a Bose-Einstein Condensate. I have also been reading a great book ‘The Universe in a Helium Droplet’ by Volovik and a really good PhD Thesis, ‘Appearing Out of Nowhere: The Emergence of Spacetime in Quantum Gravity‘ by Karen Crowther – I be posting about these next time.

Spacetime as a Bose-Einstein Condensate has been discussed in a number of other posts including:

Simple condensates

Within the context of  Group Field Theory (GFT), which is a field theory on an auxiliary group manifold. It incorporates many ideas and structures from LQG and spinfoam models in a second quantized language. Spacetime should emerge from the collective dynamics of the microscopic degrees of freedom. Within Condensates all the quanta are in the same state. These simple quantum states of the full theory, can be put in correspondence with Bianchi cosmologies via symmetry reduction at the quantum level. This leads to an effective dynamics for cosmology which makes  contact with LQC and Friedmann  equations.

Group Field Theories the second quantization language for discrete geometry

Group field theories are quantum field theories over a group manifold. The basic defintiion of a GFT is

gftequ1

which can denoted as:

gftequ2

The theory is formulated in terms of a Fock space and Bosonic statistics is used.

gftequ3

Gauge invariance on the right is required, that is:

gftequ4

GFT quanta: spin network vertices  and quantum tetrahedra

Considering D=4 with group G=SU(2). These quanta have a natural interpretation in terms of 4-valent spin-network vertices.

 

gftfig1

Via a noncommutative Fourier transform it can be formulated in group variables. Considering  SU(2), we have:

gftequ5

gftfig2

We now  have a second quantized theory that creates quantum tetrahedra

gftfig3 represented as gftequ6.

 Correlation functions of GFT and spinfoams

When computing the correlation functions between boundary states the Feynman rules glue tetrahedra into 4-simplices. This is controlled by the combinatorics of the interaction term. This amplitude is designed to match spinfoam amplitudes. For example,
the interaction kernel can be chosen to be the EPRL vertex in a group representation.

gftfig5

The dynamics can be designed to give rise to the transition amplitudes with sum over 4d geometries included using a discrete path integral for gravity.

By  proceeding as in condensed matter physics and we can design
trial states, parametrised by relatively few variables, and deduce from the dynamics of the fundamental model the optimal induced dynamics.

Now we select some trial states to getthe  effective continuum dynamics. We choose trial states that contain the relevant information about the regime that we want to explore. Fock space suggests several interesting possibilities such as field coherent states;

gftequ7

This is a simple state, but not a state with an exact finite number of particles. It is  inspired by the idea that spacetime is a sort of condensate and can be generalized to other states  such assqueezed, and multimode.

The condensates can be naturally interpreted as homogeneous cosmologies:

gftequ8

Elementary quanta possessing the same wavefunction so that  the metric tensor in the frame of the tetrahedron is the same everywhere. This Vertex or wavefunction homogeneity can be interpreted in terms of homogeneous cosmologies, once a
reconstruction procedure into a 3D group manifold has been specified.The reconstruction procedure is based on the idea that each of these tetrahedra is embedded into a background manifold: the edges are aligned with a basis of left invariant vector fields.