Quantum Analysis and Nonequilibrium Response

The quantum derivatives of $e^{-A}, A^{-1}$ and $\log A$, which play a basicrole in quantum statistical physics, are derived and their convergence isproven for an unbounded positive operator $A$ in a Hilbert space. Using thequantum analysis based on these quantum derivatives, a basic equation for theentropy operator in nonequilibrium systems is derived, and Zubarev's theory isextended to infinite order with respect to a perturbation. Using thefirst-order term of this general perturbational expansion of the entropyoperator, Kubo's linear response is rederived and expressed in terms of theinner derivation $\delta_{{\cal H}}$ for the relevant Hamiltonian ${\cal H}$.Some remarks on the conductivity $\sigma (\omega)$ are given.Comment: Latex, 16 pages, no figures, to be published in Prog. Theor. Phys. (1998