Commutator

\([A,B] = AB-BA\)

• If \([A,B] = 0\) then \(A\) and \(B\) commute
• \([A,A] = 0\)
• \([A,a] = 0\)
• \([A,B] = -[B,A]\)
• Linearity: \([A,B+C+D] = [A,B] + [A,C] + [A,D]\)
• \([A,BC] = [A,B]C + B[A,C]\)
• Jacobi Identity: \([A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0\)
• If \(A,B\) are Hermitian and \(AB\) is Hermitian then \([A,B]=0\).

\(\{A,B\} = AB+BA\)