For a two-dimensional complex vector space, the spin matrices can be
calculated directly from the angular momentum commutator definition. The 3
Pauli matrices are retrieved and 23 other triplet solutions are found. In the
three-dimensional space, we show that no matrix fulfills the spin equations and
preserves the norm of the vectors. By using a Clifford geometric algebra it is
possible in the four-dimensional spacetime (STA) to retrieve the 24 different
spins 1/2. In this framework, spins 1/2 are rotations characterized by multivectors
composed of 3 vectors and 3 bivectors. Spins 1 can be defined as rotations
characterized by 4 vectors, 6 bivectors and 4 trivectors which result in unit
multivectors which preserve the norm. Let us note that this simple derivation
retrieves the main spin properties of particle physics.
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