Angle and Volume Studies in Quantized Space by Michael Seifert

In this paper seifert reviews the angle and volume operators in Loop Quantum Gravity. I’m particularly interested at the moment in his treatment of the angle operator.

In LQG, quantum states are represented by spin networks –  graphs with weighted edges. Spatial observables such as area, volume, and angle are given by the eigenvalues of Hermitian operators on the spin network states. The author presents results on  the angle and volume operators which act on the vertices of spin networks. He finds that the minimum observable angle is inversely proportional to the square root of the total spin of the vertex, with a fairly slow decrease to  zero. He also presents numerical results indicating that the angle operator can reproduce the classical angle distribution. Noting that the volume operator is significantly harder to investigate analytically the author presents analytical and  numerical results indicating that the volume of a region scales as the 3/2 power of its bounding surface, a result which corresponds to the classical model of space.

Angular Momentum Operators

One of the aims of this paper is to describe the angles associated with a given vertex.  To define angles, need to be able to associate a direction with an edge or a group of edges. In Penrose’s original formulation of spin networks, the labels on the edges refer to angular momenta carried by the edges  so one of the natural directions to associate with an edge is its angular momentum vector. The angular momenta can be measured by the angular momentum operators, which act upon the edges. Angular momentum operators are expressed in terms of the Pauli spin matrices σi:

seirfeit pauli matrices

Using this, we can define angular momentum operators Ji = 1/2hσi which act on the edges. Can also construct the Jˆ2 operator this is equal to the sum of the squares of the three Ji operators.

The eigenvalues simplify to the  form:

Jˆ2 = 1/2hj(j +1)

The Area Operatorseirfeit areas

The total area of  is then given by

Discreteness of Area and Volume fig 3

where the summation of i is over all edges that intersect , and Ji is the edge label of the edge I.

The Angle Operator

seirfeit angle partition

The angle operator will measure the angle between the edges that traverse S1 and those that traverse S2. The  direction that is associated with the edges traversing S1, S2, and Sr is the angular momentum vectors associated with their internal edges J1,J2, and Jr.

Classically it is expected that

cos θ = J1.J2 / |J1||J2|

The angle operator is then defined as

seirfeit angle op

The eigenvalues of the angle operator:

seirfeit angle eigenvalue

where ji = ni/2.

From the discreteness of the angle operator Seifert derives three  results: will be referred to as

Small angle  property for angle e

seirfeit small angle

where the total value of core spin nT = n1 + n2 + nr

Angular resolution property

seirfeit angle resolution

where n is valence of the vertex.

Angular distribution property

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