Pauli Algebra from Cubic Point-Group Symmetry

Abstract:We show that the Pauli algebra, defined by the anticommutation relation {σ^a, σ^b} = 2δ^{ab} I, emerges necessarily from the O_h point-group symmetry of a simple cubic lattice. The T_{1g} subspace of the tensor product T_{1u} ⊗ T_{1u} contains three antisymmetric generators satisfying the Lie algebra of so(3) ≅ su(2). The two-dimensional irreducible representation of this algebra is precisely the Pauli algebra. The derivation uses only standard group theory and requires no assumptions about dynamics or quantization.