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 2pi and 4pi. 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.

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