= Pauli matrix completeness identity
{c}
{title2=$\sum_a(\sigma^a)_{ij}(\sigma^a)_{kl}=2\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl}$}
The identity follows because the identity matrix and the three <Pauli matrices> form an orthogonal basis of two-by-two matrices under the trace inner product. It rewrites a spin-vector bilinear square as a scalar four-fermion expression, with the Grassmann signs supplied when fields are reordered. This is the algebra behind a <spin-vector Hubbard–Stratonovich decoupling>.
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