Special properties of spin network functions

My work on quantum geometric operators makes heavy use of special properties of the spin network functions in particular the fact that the action of the flux operators on spin network basis states can be mapped to the problem of evaluating angular momentum operators on angular momentum eigenstates, which is familiar from ordinary quantum mechanics. This results in a great simplification and provides a convenient way to work in the SU(2)gauge-invariant regime using powerful techniques from the recoupling theory of angular momenta.

This post provide details on the conventions used in the construction of recoupling schemes. The post is organized as follows:

  • Basic properties of matrix representations of SU(2), whose matrix elements are used for the definition of spin network functions.
  • The theory of angular momentum from quantum mechanics and the notion of recoupling of an arbitrary number of angular momenta in terms of recoupling schemes.
  • The definition of 6j-symbols andtheir basic properties. This provides an explicit notion of recoupling schemes in terms of polynomials of quantum numbers.

Representations of SU(2)
Irreducible matrix representations of SU(2) can be constructed in (2j+1)-dimensional linear vector spaces, where j ≥ 0 is a half integer number; j = 0 denotes the trivial representation.

General Conventions — Defining Representation j = 1/2
Generators
Generators of SU(2) use the τ-matrices given by τk := − iσk, with σk being the Pauli-matrices:

a1
Additionally use

a2

SU(2) Representations
In the defining representation of SU(2), for a group element h ∈ SU(2)

 

a3

also

a4

and we have the additional properties that

a5

Use the following convention for the matrix elements

a6

For the τk’s we additionally have

a7

General Conventions for (2j + 1)-dimensional SU(2) Representation Matrices
General Formula for SU(2) Matrix Element

The (2j +1)-dimensional representation matrix of h ∈ SU(2), given in terms of the parameters of the defining representation can be written as

a8

where ℓ takes all integer values such that none of the factorials in the denominator gets a negative argument. By
construction every representation of SU(2) consists of special unitary matrices and

a9

Generators and ε-Metric
Applying the representation matrix element formula and the ansatz,

a10

where in the defining two dimensional representation the exponential can be explicitly evaluated, one obtains
a11

and finds that

a12

Angular Momentum Theory
Basic Definitions
The angular momentum orthonormal basis u(j,m; n) =|j m ; n〉 of a general (2j + 1) dimensional representation of SU(2). The index n stands for additional quantum numbers, not affected by the action of the angular momentum operators J fulfilling the commutation
relations

b1

Formulate ladder operators as

b2

The |j m ; n〉  simultaneously diagonalize the two operators: the squared total angular momentum (J)² and the
magnetic quantum number J³ :

b3

That is, |j m ; n〉 is a maximal set of simultaneous eigenvectors of (J)² and J³ find the following commutation relations

b4

such that for the (2j + 1)-dimensional matrix representation with arbitrary weight j

b5

 

Fundamental Recoupling

Clebsch -Gordan theorem on tensorized representations of SU(2):
Theorem b1

If we couple two angular momenta j1, j2, we can get resulting angular momenta j12 varying in the range |j1 − j2| ≤ j12 ≤ j1 + j2. The tensor product space of two representations of SU(2) decomposes into a direct sum of representation spaces, with one space for every possible value of recoupling j12 with the according dimension
2j12 + 1.

b6

Recoupling of n Angular Momenta — 3nj-Symbols
The successive coupling of three angular momenta to a resulting j can be generalized to an n-fold tensor product of representations            πj1 ⊗ πj2 ⊗ . . . ⊗ πjn  by reducing out step by step every pair of representations.

This procedure is carried out until all tensor products are reduced out. One then ends up with a direct sum of representations, each of which has a weight corresponding to an allowed value of the total angular momentum to which the n single angular momenta j1, j2, . . . , jn can couple. However, there is an arbitrariness in how one couples the n angular momenta together, that is the order in which πj1 ⊗πj2 ⊗. . .⊗πjn is reduced out matters.

Consider a system of n angular momenta. First we fix a labelling of these momenta, such that we have j1, j2, . . . , jn. Again the first choice would be a tensor basis |j m〉 of all single angular momentum states |jk mk〉, k = 1 . . . n defined by:

b7

Recoupling Scheme
recoupling b1

Standard Basis

Theorem b2

3nj-Symbol

definitionb3

Properties of Recoupling Schemes

A general standard recoupling scheme is defined as follows:

b8
For the scalar product of two recoupling schemes we have:

b9

Properties of the 6j-Symbols
6j-symbols are the basic structure in recoupling calculations, as every coupling of n angular momenta can be expressed in terms of them.
Definition
The 6j-symbol is defined in

c1

The factors in the summation are Clebsch-Gordon coefficients.

Explicit Evaluation of the 6j-Symbols
A general formula for the numerical value of the 6j-symbols has been derived by Racah

c2

Symmetry Properties
The 6j-symbols are invariant under:

  • any permutation of the columns:

c5

• simultaneous interchange of the upper and lower arguments of two columns
c6

Orthogonality and Sum Rules
Orthogonality Relations

c7

Composition Relation

c8

Sum Rule of Elliot and Biedenharn

c9

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s