Derivation of a general expression in quantum algebra

F.P. Laussy - fabrice.laussy@gmail.com
26 January 2011

Clear["Global`*"]

Pauli matrices:

σd=(({{0, 0}, {1, 0}}));σ=(({{0, 1}, {0, 0}}));

Commute[μ,ν,η,θ] computes [(σ^(†μ))(σ^ν),(σ^(†η))(σ^θ)] through the formula that is spelled down:

Commute[μ_,ν_,η_,θ_]:=σ ν θ (η-μ)+σd μ η(ν-θ)+(IdentityMatrix[2]-2σd.σ)((1-μ)ν η(1-θ)+μ(1-ν)(1-η)θ)

All the possible cases:

ind=Flatten[Table[{i,j,k,l},{i,0,1},{j,0,1},{k,0,1},{l,0,1}],3]

A particular case:

Commute[0,μ,η,0]//MatrixForm

So [(σ^μ),(σ^(†η))]=η μ (1-2(SuperDagger[σ])σ)  [obviously].

All the cases:

{#[[1]],MatrixForm[#[[2]]]}&/@Transpose[{ind,Commute@@#&/@ind}]//Column

How the formula was obtained?

I computed it by brute force:

bruteCommute[μ_,ν_,η_,λ_]:=MatrixPower[σd,μ].MatrixPower[σ,ν].MatrixPower[σd,η].MatrixPower[σ,λ]-MatrixPower[σd,η].MatrixPower[σ,λ].MatrixPower[σd,μ].MatrixPower[σ,ν]

{#[[1]],MatrixForm[#[[2]]]}&/@Transpose[{ind,bruteCommute@@#&/@ind}]//Column

Extracted the nonzero elements:

({{{{0,1,1,0},1-2(SuperDagger[σ])σ}}, {{{0,1,1,1},σ}}, {{{1,0,0,1},-1+2(SuperDagger[σ])σ}}, {{{1,0,1,1},-(SuperDagger[σ])}}, {{{1,1,0,1},-σ}}, {{{1,1,1,0},(SuperDagger[σ])}}})

encoding the associated list as coefficients, for instance, {0,1,1,0}→μ(1-ν)(1-η)θ.

Working out the algebra by hand brings me to the formua.