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Roger Penrose proposed a series of networks from which, fundamentally, space-time would emerge, called spin networks

(https://en.wikipedia.org/wiki/Spin_network)

In this article, it is said:

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 must be less than or equal to b + c, b less than or equal to a + c, and c less than or equal to 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.

But is this "norm" necessarily applied in spin networks? Are there spin networks that do not obey these rules?

I've read from Penrose that these rules are given from quantum mechanics (https://pdfs.semanticscholar.org/48d4/ee61c3361cefa678b99eecd8236fac53e147.pdf). But is it there any spin network model not based in quantum mechanics? Would then they "disobey" these rules?

I've read that there are models of classical spin networks. Are they not based in QM? Are norms not applied here? Can there be other other non-quantum spin networks?

Also, in the wikipedia article I quoted, it gives an alternative definition of spin networks apparently not based in QM.

And finally, I had a brief discussion with a user of this site with high reputation who told me that, while he was not 100% sure, he thought that there could be "non-quantum" spin networks.

So, in summary, are there any spin networks that do not obey norms? Are there any non-quantum spin networks?

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Are there any non-quantum spin networks?

As spin is intrinsically a quantum effect, I would not expect there to be such networks outside of a quantum context.

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  • $\begingroup$ but as I said, I read several articles working with classical spin networks. Aren't they non-quantum (since they are classical)? @MoziburUllah $\endgroup$ – user213961 Jan 23 at 19:35

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