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.
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