# Numerical work using python 1: Bose-Einsten condensation

In my last post ‘Constructing spacetime from the quantum tetrahedron: Spacetime as a Bose-Einstein Condensate’ I started to explore how spacetime could be formed as a Bose-Einstein condensate of quantum tetrahedra within the framework of Group Field Theory (GFT).

This week I have been adapting some of the algorithms found in Statistical Mechanics: Algorithms and Computations by Werner Krauth  to allow me to begin exploring Spacetime as a Group Field Theory Bose-Einstein Condensate.

First I used a python program to look at how bosons occupy energy levels as the temperature decreases: Basically they all end up occuping the lowest energy level.

The data obtained is plotted in a graph of occupancy number against Energy – notice how the lower energy levels are more highly occupied that the higher ones.

Following this I used a python program with a more sophisticated algorithm to explore how the degree of condensation varies with temperature as used this to produce an animation showing the formation of the Bose-Einstein condensate.

The animation of the data obtained from the python program shows how the Bose-Einstein condensate forms as the temperature is lowered.

(Click picture to see animation)

Given the crudeness of the algorithms I am using so far I think this models an actual Bose-Einstein condensate formation quite well.

# 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.

A quantum of the GFT field –  a quantum  tetrahedron

A single field “quantum” maybe a spin network vertex or quantum tetrahedro.n

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.

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

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.

Summary

• 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

# Review of the Quantum Tetrahedron

This week I have been reviewing the state of play with regard to the quantum tetrahedron, from next week I will be looking at how a Bose-Einstein condensate of quantum tetrahedra could from spacetime! This review is based on E. Bianchia’ s lectures.

# Numerical work with Sagemath 11: Length eigenvalues in LQG

Whist working on A length operator for canonical quantum gravity by T. Thiemann I calculated the spectrum for the length operator using J1, J2 and J3 values for tri-, di- and j=1/2 values. I’ll be reworking this program and adding  interactive capability – then I’ll post it  at wiki.sagemath.org

As can be seen the length eigenvalues are of the order of the Planck length lp=1.61619926*10^-35

# A length operator for canonical quantum gravity by T. Thiemann

This week I have been exploring the length operator in LQG. One of the papers I’ve been reading is this one by Thiemann. Like all of Tiemann’s work it is mathematically quite sophisticated so I’ll precis it in a slightly less mathematical manner.

In this paper the author constructs an operator that measures the length of  a curve for four dimensional Lorentzian vacuum quantum gravity. He works in a representation in which a SU(2) connection is diagonal and finds that the operator obtained  after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization.

In the paper it is shown that the length operator admits self-adjoint extensions. Part of the length operator’s spectrum is computed and found like the volume and area operators to be purely discrete and roughly is quantized in units of the Planck length. It also contains full and direct  information about all the components of the metric tensor which aids the construction of a new type of weave states which approximate a given  classical 3-geometry.

Spectrum of length operator

The full expression is so complicated that I’ll just give some special cases:

• Trivalent Vertex

For  triavalent vertex with J1, J2, J3 we have eigenvalues :

• Divalent Vertex

For divalent vertex with J1 = J2 = Jo we have eigenvalues:

• The Quantum of Length

The quantum of length is achieved for Jo = 1/2 and given by:

The length can change only in packets of:

which approaches zero for large spin so that for high spin the length looks like a continuous operator.

# Space of the vertices of relativistic spin networks by A. Barbieri

This week I been reading quite a short paper about three and four dimensional vertices in LQG.

In this paper the general solution to the constraints that define relativistic spin neworks vertices is given and their relations with 3-dimensional quantum tetrahedra are discussed.

Quantum  Tetrahedra

The author’s basic idea is to use as fundamental variables the bivectors defined by 2-dimensional faces of the tetrahedron rather than the edges. Since the set of bivectors in n dimensions is isomorphic to the dual of the Lie algebra of SO(n), the Hilbert space of a quantum bivector is defned as the orthogonal sum of the  irreducible representations of the universal covering of SO(n);

If n = 3 the covering group is SU(2),

while

if n = 4 it is Spin(4) ≃ SU(2) x SU(2).

The author focuses on these two subcases i.e n=3 and n=4. The next step  is to take the tensor product of four such spaces (one for each face) and impose some constraints which so that the faces have suitable properties, which in terms of bivectors. These constraints are that

• The bivectors defined by the faces must add up to zero.
• The bivectors are all simple i.e. they can be written as wedge products of two vectors.
• The sum of two bivectors is also simple.

In three dimensions all bivectors are simple and the only constraint is that the faces close: the resulting space is that of 4-valent SU(2)-spin network vertices. Whilst in n four dimensions there are non-simple bivectors

# Numerical work with Sagemath 10: Angle, area and volume of a quantum tetrahedron

This week I have been consolidating the work I’ve been doing on the angle, area and volume operators in LQG. I have taken the numerical work  done in posts:

Then combined it to be able to find the angle, area and volume in a quantum tetrahedron.

The numerical values are in good accord with those suggested by Lee Smolin as shown below.