# Why is a spin network a spinfoam's boundary?

i read in Wikipedia that spinfoams boudaries are spin networks, but nothing is said about what is a boundary in this case. i know that a spin foam is a spin network where other colours and intertwiners are added to faces and edges. are boundaries of a spinfoam a part or them where these colors are neglected?

## 1 Answer

Sorry this answer is a bit late. This is pretty trivial, actually.

A spin network is a graph with the following data associated to its nodes and links (note the terminology here - we've reserved terms vertex and edge for the components of spinfoams):

• To a link we associate an irreducible unitary representation of the gauge group $$G$$ (in LQG this is $$G=SU(2)$$).
• To a node we associate an intertwining tensor between the representations of incident links.

A spinfoam is a 2-complex with the following data associated to its vertices, edges and faces:

• To a face we associate an irreducible unitary representation of $$G$$.
• To an edge we associate an intertwining operator between the representations of incident faces.
• To a vertex we don't associate any data. Instead, each vertex contributes with the spinfoam amplitude evaluated at the vertex which depends on the data associated to incident edges and faces.

Now imagine a spinfoam to be "cut" by a sphere which never passes through vertices. Each face will be cut in two halves separated by a link, and each edge will be cut in two halfs separated by a node. Moreover, the incidence relation between edges and faces becomes the incidence relation between nodes and links.

Thus, the boundary of a spinfoam is a spin network.

• I see how a spin network can be built from a spinfoam. it is a choice. a spinfoam has no naturel boundary. suppose that the spinfoam is not embedded in a manyfold you have no sphere cutting rhe edges. what are the rules for a good choice of faces? (only one face is not a goof choice) Commented Jun 5, 2018 at 15:47
• @Naima I'm not sure I agree with "spinfoam has no natural boundary". Maybe not in the mathematical sense, but in quantum physics we are always looking to find quantum states associated to the boundary. Try reading about the boundary formalism in any of the LQG textbooks or seminal papers e.g. arxiv.org/abs/gr-qc/0604044. It's probably one of the deepest and most general insights of QFT and especially TQFT – that Hilbert spaces are associated to boundaries, and evolution operators are associated to sums over the bulk objects of some sort. Commented Jun 6, 2018 at 18:41
• thanks. i understand now that not having a unique boundary is not a problem. you give a great tool to get spin networks of a spinfoam! Commented Dec 24, 2020 at 10:14