 Jacobi sign test for supercharge components

ID: jacobi-sign-test-for-supercharge-components

Let

[J_{i}, q_{a}] = (R_{i})_{ab} q_{b}

, with numerical coefficient matrices and

[J_{i}, J_{j}] = i ϵ_{ijk} J_{k}

. The Jacobi identity then gives

(R_{j} R_{i} - R_{i} R_{j}) q = i ϵ_{ijk} R_{k} q

, so component coefficient matrices obey the opposite bracket to state-representation matrices. Thus

R_{i} = - σ_{i} /2

is consistent for an upper-column spinor, whereas

+ σ_{i} /2

in that same component prescription is not. Inverse adjoint actions and dual row conventions must be compared with their index contractions intact.

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