Abstract
The purpose of the present paper is to formulate the quantum analysis of multivariate operator functions f({A_j}) by introducing auxiliary operators {H_j} satisfying the conditions that [H_j, H_κ]=0 , [H_j, A_κ ]=0 for j≠κ, and [H_j, [H_κ, A_κ]]=0. Then we have d^nf=Σ_<j1>...<jn> δ_H_<J1>・・・δ_H_<jn>f using the inner derivation δ_A : δ_AQ=[A, Q]=AQ-QA. The operator Taylor expansion formula is given in the form : f({A_j+x_jdA_j})=e^Σ_j^x_j^d_jf({A_j})=exp(Σ_jx_jδ_H_j)f({A_j})=Σ_nΣ_<j1>...<jn>x_<j1>・・・x_<jn>δ<j1>..., _<jn>f≡Σ_nΣ_<j1>..., _<jn>x_<j1>・・・x_<jn>f^<(n)>_<j1>..., _<jn> : dA_<J1>・・・dA_jn with dA_j≡[H_j, A_j]=δ_H_jA_j, and with the partial derivatives {d_j} with respect to {A_j}. Here, δ_<j1>..., _<jn> denotes an ordered partial inner derivation, and n!f^<(n)>_<j1>...<jn> denotes the n th derivative of f({A_j}).
