Angular momentum: An approach to combinatorial space-time by Roger Penrose

This week I have also been reading about Penrose’s “spin networks“. In this paper Penrose attempts to build a purely combinatorial description of spacetime starting from the mathematics of spin-1/2 particles. The appraoch he describes in this paper leads into twistor theory. The spin networks he describe give an interesting theory of space. Penrose’s spin networks are purely combinatorial structures: graphs with edges labelled by numbers j = 0, 1/2, 1, 3/2,… These numbers represent the total angular momentum or “spin”. In the spin networks described in this paper it is required that three edges meet at each vertex, with the corresponding spins j1, j2, j3 adding up to an integer and satisfying the triangle inequalities

|j1 – j2| ≤ j3 ≤ j1 + j2

These rules are motivated by the quantum mechanics of angular momentum in that if we combine a system with spin j1 and a system with spin j2, the spin j3 of the combined system satisfies exactly these constraints. In this paper a spin network represents a quantum state of the geometry of space and this interpretation is justified by computations using a special rule for computing the norm of any spin network.


A spin network


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