Editing Spin network (section)

== Definition ==

=== Penrose's definition ===
A spin network, as described in Penrose (1971),<ref name=Penrose71>R. Penrose (1971a), "Angular momentum: an approach to combinatorial spacetime," in T. Bastin (ed.), ''Quantum Theory and Beyond'', Cambridge University Press (this paper can be found online on [[John C. Baez]]'s [http://math.ucr.edu/home/baez/penrose/ website]); and R. Penrose (1971b), "Applications of negative dimensional tensors," in D. J. A. Welsh (ed.), ''Combinatorial Mathematics and its Applications'' ([[Proceedings|Proc.]] [[Academic conference|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, [https://web.archive.org/web/20210623190333/http://users.ox.ac.uk/~tweb/00001/ "On the Origins of Twistor Theory" (Archived June 23, 2021)] in: ''Gravitation and Geometry, a Volume in Honour of [[Ivor Robinson (physicist)|I. Robinson]]'', Biblipolis, Naples 1987).</ref> 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 [[boson]]s, such as [[photon]]s and [[gluon]]s, ''n'' is an even number. For [[fermion]]s, such as [[electron]]s and [[quark]]s, ''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) [[graph theory|graph]] whose [[graph theory|edges]] are associated with [[Irreducible representation|irreducible]] [[Representations of Lie groups/algebras|representations]] of a [[Compact group|compact]] [[Lie group]] and whose [[vertex (graph theory)|vertices]] are associated with [[intertwiner#Representation theory|intertwiner]]s of the edge representations adjacent to it.

=== Properties ===
A spin network, immersed into a manifold, can be used to define a [[functional (mathematics)|functional]] on the space of [[Connection (mathematics)|connections]] on this manifold. One computes [[holonomy|holonomies]] of the connection along every link (closed path) of the graph, determines representation matrices corresponding to every link, multiplies all matrices and intertwiners together, and contracts indices in a prescribed way. A remarkable feature of the resulting functional is that it is invariant under local [[gauge transformation]]s.