Schild’s Ladder

Spin Networks

Mathematical Details

The effect of parallel-transporting vectors along a path, viewed as a linear map between the tangent spaces at the beginning and end of the path, is known as the holonomy for that path, and will always take the form of some rotation, R. The family of geometries for which the applet evaluates each spin network is characterised by the following simple rule: parallel transport along a straight line from the point (x0,y0,z0) to the point (x1,y1,z1) rotates vectors around the axis a, by an angle equal to the magnitude of a, where:

a = κ (y0 z1z0 y1, z0 x1x0 z1, x0 y1y0 x1) (1)

and κ is a curvature parameter. This in turn means that parallel transport around a square loop with side ε in one of the coordinate planes rotates vectors around the remaining coordinate axis by an angle:

θ(loop) = 2 ε2 κ (2)

The effect of each holonomy on a particle with total spin j is determined by a unitary matrix, Uj, which is found by using the appropriate representation of SU(2) — a homomorphism from the group SU(2) to the group of unitary linear operators on the Hilbert space that contains the particle’s spin state. The applet calculates these matrices with a combinatorial formula based on an embedding of the spin-j representation in the 2j-fold tensor product of the fundamental, spin-(1/2) representation. An amplitude can then be associated with each edge which depends on the m value at the start and end of the edge:

aedge(ms,me) = <jme| Uj(R) |jms> (3)

For any rotation R, given Uj(R) the corresponding effect on dual vectors is:

Uj*(R) <jm| = <jm| Uj(R)–1 (4)

The spin states at a node with incoming edges labelled by j1 and j2 and the outgoing edge labelled by j3 are compared by means of a linear map C between the tensor product of the Hilbert spaces for the incoming particles and that for the outgoing particle. This yields an amplitude at the node of:

anode(m1,m2,m3) = <j3m3| C(|j1m1> ⊗ |j2m2>) (5)

C is required to commute with the effects of any rotation, R:

C(Uj1(R) |j1m1> ⊗ Uj2(R) |j2m2>) = Uj3(R) C(|j1m1> ⊗ |j2m2>) (6)

in order for the amplitude to be unchanged if every state at the node is subject to the same rotation:

anode(m1,m2,m3) = (Uj3*(R) <j3m3|) C(Uj1(R) |j1m1> ⊗ Uj2(R) |j2m2>)  
  = <j3m3| Uj3–1(R) Uj3(R) C(|j1m1> ⊗ |j2m2>)  
  = <j3m3| C(|j1m1> ⊗ |j2m2>) (7)

The need to satisfy Equation (6) is enough to determine C up to an overall factor. Specifically, the coordinates of C are the Clebsch-Gordan coefficients, which give the amplitudes for a two-particle state being found to have various values of total spin.

The evaluation of a spin network is found by multiplying together all the node amplitudes given by Equation (5), and all the edge amplitudes given by Equation (3), and summing the product for every possible value of m at the beginning and end of each edge — while taking account of some constraints due to the fact that the Clebsch-Gordan coefficients will be zero unless the sum of m values on the incoming edges equals that on the outgoing edge. (My thanks to Dan Christensen, who worked out a more efficient approach to doing this than the method I was using originally.)

In the language of group theory, the map C is described as an intertwiner between the two representations of SU(2) that apply to the incoming and outgoing states. In more general spin networks — used in other theories of quantum gravity, or other field theories entirely — the edges of the network are labelled by irreducible representations of any group, G, the nodes are labelled by intertwiners between the representations, and the spin network evaluation is given as the trace of a big fat tensor that is formed by multiplying together the intertwiners and the linear maps the representations assign to the holonomies that the geometry (or other kind of field) dictates for each edge.

References: “Spin Networks in Nonperturbative Quantum Gravity” and “An Introduction to Spin Foam Models of Quantum Gravity and BF Theory” by John C. Baez; “The Future of Spin Networks” by Lee Smolin. Dan Christensen has a page of links to a number of other articles and web pages that deal with spin foams and spin networks, including many links to John Baez’s site.

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Schild’s Ladder / Spin Networks (Technical Notes) / created Saturday, 30 September 2000 / revised Sunday, 8 October 2000
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