# Numerical work with sagemath 11: Tetrahedron Dynamics

This week I’ve been following up some of the mathematical physics seen in the paper Background independence in a nutshell: the dynamics of a tetrahedron. I have have been looking at how sagemath and python handle polyhedra.

So I’ve written a sagemath program to simulate the dynamics of the classical tetrahedon:

which governed by the equations:

As a sagemath program this gives:

Typical results are tabulated below showing the behaviour long times T, as theta C tends to 90 degrees.

As part of this work of been developing some python functions in particular a function which will produce the equation of a plane through three points . This is especially useful  for finding the dihedral angle in polyhedra.

# Background independence in a nutshell: the dynamics of a tetrahedron by Rovelli et al

This week I’ve been looking at the dynamics of the quantum tetrahedron. So I’ve been reading a couple of papers and doing calcualtions in sagemath which I’ll post later.

The first paper I looked at was this paper, in which  the authors study how physical information can be extracted from a background independent quantum system. They use an extremely simple system that models a finite region of 3d euclidean quantum spacetime with a single equilateral tetrahedron. They show that the physical information can be expressed as a boundary amplitude and how the notions of ‘evolution’ in a boundary proper-time and ‘vacuum’ can be extracted from the background independent dynamics. The paper discusses the  classical theory, classical time evolution, the quantum theory and  the quantum time evolution.

Introduction
In a background independent field theory the distance and time separation must be extracted from the dynamical variables.An idea for solving this problem is to study the quantum propagator of a finite spacetime region, as a function of the boundary data.The key observation is that in gravity the boundary data include the gravitational field, the geometry of the boundary and so all relevant relative distances and time separations. The boundary formulation realizes very elegantly in the quantum context the complete
identification between spacetime geometry and dynamical fields.
Formally, the idea consists in extracting the physical information from a background independent quantum field theory in terms of the quantity;

The particle scattering amplitudes can be effectively computed from W[] in quantum gravity. This equation becomes a generalized Wheeler-DeWitt equation in the background independent context. Theboundary picture is appealing, but its implementation in the full 4d quantum gravity theory is difficult because of the technical complexity of the theory. It is useful to test and illustrate it in a simple context.That is is what is done in this paper. The authors consider riemannian general relativity in three dimensions.

To further simplify the context, the authors triangulate spacetime, reducing the field variables to a finite number. Taking a minimalist triangulation: a single tetrahedron with four equal edges. The number of variables dealt with is reduced to a bare minimum. The result is an extremely simple system, which is sufficient to realize the conceptual complexity of a background independent theory of spacetime geometry.

The authors show that this simple system has in fact a background independent classical and quantum dynamics. The classical dynamics is governed by the relativistic Hamilton function the quantum dynamics is governed by the relativistic propagator, both these
functions explicitly computed . The classical dynamics, which is equivalent to the Einstein equations, fixes relations between quantities that can be measured on the boundary of the tetrahedron. The quantum dynamics gives probability amplitudes for ensembles of boundary measurements.
The model and its interpretation are well-defined with no need of picking a particular
variable as a time variable. However, it is posssible to identify an elapsed proper time T among the boundary variables, and reinterpret the background independent theory as a theory describing evolution in the observable time T.

Two  interpretations of the model are described , in the classical as well in the quantum theory. The distinction between the nonperturbative vacuum state and the Minkowski vacuum that minimizes the energy associated with the evolution in T are illustrated , and it is shown that the usual  technique suggested for computing the Minkowski vacuum state from the nonperturbative vacuum state works in this context. This system captures the essence of background independent physics in a nutshell.

Elementary geometry of an equilateral tetrahedron

Consider a tetrahedron immersed in euclidean three-dimensional space. Let a be the length of the top edge and b the length of the bottom edge, assume that the other four side edges have equal length c. Such a tetrahedron is called an equilateral tetrahedron. There are “bottom”, “top” and “side” dihedral angles at the edges with length a, b, c. Elementary geometry gives;

Classical theory

Regge action
Consider the action of general relativity, in the case of a simply connected finite spacetime region R. In the presence of a boundary

have to add a boundary term to the Einstein–Hilbert action, in order to have well defined equations of motion. The full action reads;

Here g is the metric field, R is the Ricci scalar, n is the number of spacetime dimensions, while q is the metric, and k the trace of the extrinsic curvature.

In general, the Hamilton function of a finite dimensional dynamical system is the value of the action of a solution of the equations of motion, viewed as a function of the initial and final coordinates; the general solution of the equations of motion can be obtained from the Hamilton function. Since the bulk action vanishes on a vacuum solution of the equations of motion, the Hamilton function of general relativity reads

where the extrinsic curvature k[q] is a nonlocal function, determined by the Ricci-flat metric g bounded by q.

In the paper only the three-dimensional riemannian case is considered, where n = 3 and the signature of g is [+ + +]. The discretization of the theory is provided by a Regge triangulation. Let i be the index labelling the links of the triangulation and call li the length of the link i. In three dimensions, the bulk Regge action is;

where theta(i,t) the dihedral angle of the tetrahedron t at the link i, and the angle in the parenthesis is therefore the deficit angle at i. The boundary term is;

Notice that the angle in the parenthesis is the angle formed by the boundary, which can be seen as a discretization of the extrinsic curvature.

Choosing the minimalist triangulation formed by a single tetrahedron, and considering only the case in which the tetrahedron is equilateral. Then there are no internal links, the Regge action is the same as the Regge Hamilton function. The expression for the dihedral angles as functions of the edges length, for a flat interior geometry, gives the Hamilton function;

The dynamical model and its physical meaning
The Hamilton function (defines a simple relativistic dynamical model. The model has three variables, a, b and c, these are partial observables. That is, they include both the independent -‘time’ and the dependent-dynamical variables, all treated on equal footing.

The equations of motion are obtained following the general algorithm of the relativistic Hamilton– Jacobi theory: define the momenta;

These equations give the dynamics, namely the solution of the equations of motion. Explicitly, the calculation of the momenta is simplified by the observation that the action is a homogeneous function of degree one, so this gives;

The evolution equations are;

Time evolution
In the description given so far, no reference to evolution in a preferred time variable was considered. To introduce regard the direction of the axis of the equilateral tetrahedron as a temporal direction. In particular interpret b as an initial variable and a as a final variable. The length c of the side links can then be regarded as a proper length measured in the temporal direction, namely as the physical time elapsed from the measurement of a to the measurement of b.

Renaming c as T . The Hamilton function reads then S(a, b, T ) and can now be interpreted as the Hamilton function that determines the evolution in T of a variable a. The variable b is interpreted as measured at time T = 0 and the variable a at time T ; therefore b can be viewed as an integration constant for
the evolution of a in T .

In this system, the hamiltonian that evolves the system in the time T , which is called ‘proper-time hamiltonian’, can obtained from the energy

Notice that the angle theta can vary between 0 and pi/2, and therefore so does the arccos. Therefore the energy can vary between 2pi and 4pi. The fact that the domain of the energy is bounded has important consequences. For instance –  should expect time to become discrete in the quantum theory.

The relativistic background independent system can be reinterpreted as an evolution system, where the ‘proper time’ on the boundary of the region of interest is taken as the independent time variable. The Hamilton equation generated by the hamiltonian for a(T ) and pa(T ) are:

In the large T limit we have the behaviour;

Quantum equilateral tetrahedron

Specialize the formalism to the case of an equilateral tetrahedron. The simplest way to do so is to restrict attention to the states where four of the six edge lengths are equal. More precisely, put:

ja ≡ j13,
jb ≡ j24,
jc ≡ j12 = j23 = j34 = j41
and consider only the states

|ja, jb, jc> = |jc, ja, jc, jc, jb, jc>.

The states are restricted to the subset of (SU(2))^6. The boundary Hilbert state K is spanned by the states |ja, jb, jc>. The boundary observables a, b, c, pa, pb, pc that measure the length of the edges of the tetrahedron and the external angles are represented by Casimir and trace operators, and the dynamics is given by the propagator

which expresses the probability amplitude of measuring the lengths determined by ja, jb, jc. The predictions of the theory are given by the
quantization of the lengths and by the relative probability amplitude, W() above.

# Disappearance and emergence of space and time in quantum gravity by D. Oriti

This week I’ve been studying GFT condensates, Bose-Einstein condensates and the Gross-Pitaevskii Equation and the emergence of spacetime. I have posted on spacetime as a GFT condensate and numerical work with python on Bose-Einstein condensates. This post is based on D. Oriti’s paper, ‘Disappearance and emergence of space and time in quantum gravity’

In this paper he looks at the disappearance of continuum space and time at a microscopic scale. These include arguments for the discrete spacetimes and non-locality in a quantum theory of gravity. He also discusses how these ideas are realized in specific quantum gravity approaches. He then considers the emergence of continuum space and time from the collective behaviour of discrete, pre-geometric atoms of quantum space such as quantum tetrahedra, and for understanding spacetime as a kind of condensate and presents the case for this emergence process being the result of a phase transition, called ‘geometrogenesis’. Oriti then discusses some conceptual issues of this scenario and of the idea of emergent spacetime in general. A concrete example is given in the form of the GFT framework for quantum gravity, and he illustrates a procedure for the
emergence of spacetime in this framework.
GEOMETROGENESIS
An example of emergent spacetime in the context of the GFT framework is given in this paper. It aims at extracting cosmological dynamics directly from microscopic GFT models, using the idea of continuum spacetime as a condensate, possibly emerging from a big bang phase transition.

GFTs are defined usually in perturbative expansion around the Fock vacuum. In this approximation, they describe the interaction of quantized simplices and spin networks, in terms of spin foam models and simplicial gravity. The true ground state of the system, however, for non-zero couplings and for generic choices of the macroscopic parameters, will not be the Fock vacuum. The interacting system will organize itself around a new, non-trivial state, as in the case of standard Bose condensates. The relevant ground states for different values of the parameters will correspond to the different macroscopic, continuum phases of the theory, with the dynamical transitions
from one to the other being phase transitions of the physical system called spacetime.
The fact that the relevant ground state for a proper continuum geometric phase would probably not be the GFT Fock vacuum can be argued also on the basis of the pregeomet ”meaning of it: it is a quantum state in which no pregeometric excitations at all are present, no simplices, no spin networks. It is a no space state, the absolute void. It can be the full non-perturbative, diffeo-invariant quantum state around which one defines the theory – in fact, it is analogous to the diffeoinvariant vacuum state of loop quantum gravity, but it is not where to look for effective continuum physics. Hence the need to change vacuum and study the effective geometry and dynamics of a different
one.

As described in my last post it is possible to define an approximation procedure that associates an approximate continuum geometry to the set of data encoded in a generic GFT state. This applies to GFT models whose group and Lie algebra variables admit an interpretation in terms of discrete geometries, i.e. in which the group chosen is SO(3, 1) in the Lorentzian setting or SO(4) in the Riemannian setting and additional simplicity conditions are imposed, in the model, to reduce generic group and Lie algebra elements to discrete counterparts of a discrete tetrad and a discrete gravity connection.

A generic GFT state with a fixed number N of GFT quanta will be associated to a set of 4N Lie algebra elements: BI(m) , with m = 1, …,N running over the set of tetrahedra/vertices, I = 1, …, 4 indicating the four triangles of each tetrahedron. In turn, the geometricity conditions imply that only three elements are independent for each tetrehadron.

If the tetrahedra are embedded in a spatial 3-manifold M with a transitive
group action. The embedding is defined by specifying a location of the tetrahedra, i.e. associating for example one of their vertices with a point on the manifold, and three tangent vectors defining a local frame and specifying the directions of the three edges incident at that vertex. The vectors eA canbe interpreted  as continuum tetrad vectors integrated over paths in M corresponding to the edges of the tetrahedron. Then, the variables gij(m) can be used to define the coefficients of continuum metric at a finite number N of points, as: gij(m) = g(xm)(vi(m), vj(m)), invariant under the action of the group SO(4).

The reconstruction of metric coefficients gij(m), at a finite number of points, from the variables associated to a state of N GFT quanta – such as quantum tetrahedra, depends only on the topology of the assumed symmetric manifold M and on the choice of group action H. The approximate metric will be homogenous if it has the same coefficients gij(m) at any m. This captures the notion of the metric being the same at every point. It also implies that the same metric would be also isotropic if H = R3 or H = SU(2).

The quantum GFT states obtained using the above procedure can be interpreted as continuum homogeneous quantum geometries. In such a second quantized setting, the definition of states involving varying and even infinite numbers of discrete degrees of freedom is straightforward, and field theory formalism is well adapted to dealing with their dynamics.

The crucial point, from the point of view of emergent spacetime and of the idea of spacetime as a condensate of quantum pregeometric and not spatio-temporal building blocks is that quantum states corresponding to homogeneous continuum geometries are exactly GFT condensate states. The hypothesis of spacetime as a condensate, as a quantum fluid, is realized in a literal way. The simplest state of this type -one-particle GFT condensate, for which we assume a bosonic quantum statistics, is

This describes a coherent superposition of quantum states of arbitrary number of GFT quanta, all of them described by the same distribution ϕo of pregeometric variables. The function ϕo is a collective variable characterizing such continuum geometry, and it depends only on invariant homogeneous geometric data.  It is a second quantized state characterized by the fact that the mean value of the fundamental quantum operator φ is non-zero:

<ϕo|φ(gi)|ϕo> = ϕo(gi),

contrary to what happens in the Fock vacuum.

The effective dynamical equations for the condensate can be extracted directly from the fundamental GFT quantum dynamics.The generic form of the dynamics for the condensate ϕo is, schematically:

where Keff and Veff are modified versions of the kinetic and interaction kernels entering the fundamental GFT dynamics, reflecting the approximations needed to interpret ϕo as a cosmological condensate.This is a non-linear and non-local, Gross-Pitaevskii-like equation for the spacetime condensate function ϕo. In the simple case in which:

and we assume that the function ϕo depends on four such SU(2) variables. The final equation one gets for non-degenerate geometries) is:

that is the Friedmann equation for a homogeneous universe with constant curvature k.