Quantum Analysis of Linear and Nonlinear Responses, and Correlation Identities

The quantum derivatives of e, A and logA which play a basic role in quantum statistical physics are derived and their convergence is proven for an unbounded positive operator A in a Hilbert space. Using the quantum analysis based on these quantum derivatives, Kubo’s linear response theory is reformulated in terms of the inner derivation δH for the relevant Hamiltonian H, and Zubarev’s theory is extended up to infinite order with respect to a perturbation. Some remarks on the conductivity σ(ω) are given. Quantum correlation identities are presented as an example of the quantum analysis. They are applied to the Heisenberg and Ising models and locally separable quantum spin systems.