Spin network

Source: Wikipedia, the free encyclopedia.
Spin network diagram, after Penrose

In

matrix groups
. The diagrammatic notation can thus greatly simplify calculations.

Roger Penrose described spin networks in 1971.[1] Spin networks have since been applied to the theory of quantum gravity by Carlo Rovelli, Lee Smolin, Jorge Pullin, Rodolfo Gambini and others.

Spin networks can also be used to construct a particular

gauge transformations
.

Definition

Penrose's definition

A spin network, as described in Penrose (1971),[1] is a kind of diagram in which each line segment represents the world line of a "unit" (either an elementary particle or a compound system of particles). Three line segments join at each vertex. A vertex may be interpreted as an event in which either a single unit splits into two or two units collide and join into a single unit. Diagrams whose line segments are all joined at vertices are called closed spin networks. Time may be viewed as going in one direction, such as from the bottom to the top of the diagram, but for closed spin networks the direction of time is irrelevant to calculations.

Each line segment is labelled with an integer called a

spin number. A unit with spin number n is called an n-unit and has angular momentum nħ/2, where ħ is the reduced Planck constant. For bosons, such as photons and gluons, n is an even number. For fermions, such as electrons and quarks
, n is odd.

Given any closed spin network, a non-negative integer can be calculated which is called the norm of the spin network. Norms can be used to calculate the

probabilities
of various spin values. A network whose norm is zero has zero probability of occurrence. The rules for calculating norms and probabilities are beyond the scope of this article. However, they imply that for a spin network to have nonzero norm, two requirements must be met at each vertex. Suppose a vertex joins three units with spin numbers a, b, and c. Then, these requirements are stated as:

  • Triangle inequality: a ≤ b + c and b ≤ a + c and c ≤ a + b.
  • Fermion conservation: a + b + c must be an even number.

For example, a = 3, b = 4, c = 6 is impossible since 3 + 4 + 6 = 13 is odd, and a = 3, b = 4, c = 9 is impossible since 9 > 3 + 4. However, a = 3, b = 4, c = 5 is possible since 3 + 4 + 5 = 12 is even, and the triangle inequality is satisfied. Some conventions use labellings by half-integers, with the condition that the sum a + b + c must be a whole number.

Formal approach to definition

Formally, a spin network may be defined as a (directed)

intertwiners
of the edge representations adjacent to it.

Properties

A spin network, immersed into a manifold, can be used to define a

gauge transformations
.

Usage in physics

In the context of loop quantum gravity

In

countable; it constitutes a basis of LQG Hilbert space
.

One of the key results of loop quantum gravity is

eigenstate
of each such operator, and the area eigenvalue equals

where the sum goes over all intersections i of Σ with the spin network. In this formula,

According to this formula, the lowest possible non-zero eigenvalue of the area operator corresponds to a link that carries spin 1/2 representation. Assuming an Immirzi parameter on the order of 1, this gives the smallest possible measurable area of ~10−66 cm2.

The formula for area eigenvalues becomes somewhat more complicated if the surface is allowed to pass through the vertices, as with anomalous diffusion models. Also, the eigenvalues of the area operator A are constrained by ladder symmetry.

Similar quantization applies to the volume operator. The volume of a 3D submanifold that contains part of a spin network is given by a sum of contributions from each node inside it. One can think that every node in a spin network is an elementary "quantum of volume" and every link is a "quantum of area" surrounding this volume.

More general gauge theories

Similar constructions can be made for general gauge theories with a compact Lie group G and a

diffeomorphism invariance are needed to make the duality exact (smearing Wilson loops is tricky). Later, it was generalized by Robert Oeckl to representations of quantum groups in 2 and 3 dimensions using the Tannaka–Krein duality
.

String-net condensation produces topologically ordered
states in condensed matter.

Usage in mathematics

In mathematics, spin networks have been used to study

skein modules and character varieties, which correspond to spaces of connections
.

See also

Wikimedia Commons has media related to Spin networks.

References

  1. ^
    Proc. Conf., Oxford, 1969), Academic Press, pp. 221–244, esp. p. 241 (the latter paper was presented in 1969 but published in 1971 according to Roger Penrose, "On the Origins of Twistor Theory" (Archived June 23, 2021) in: Gravitation and Geometry, a Volume in Honour of I. Robinson
    , Biblipolis, Naples 1987).

Further reading

Early papers

Modern papers

Books
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