= Kubo formula
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The <Kubo formula> expresses a linear response through an <equilibrium> correlation <function>. For a perturbation $H'(t)=-f(t)B$ and observable $A$, one convention gives $\delta\langle A(t)\rangle=\int_{-\infty}^t\chi_{AB}(t-t')f(t')\,dt'$, with
$$
\chi_{AB}(t)=\frac{i}{\hbar}\Theta(t)\langle[A(t),B(0)]\rangle_{\mathrm{eq}}.
$$
The step <function> enforces causality, and the commutator records the quantum response.
The sign depends on the declared perturbing-Hamiltonian convention. Applying the framework to currents or polarization relates transport coefficients to fluctuations in an unperturbed system, forming a foundation of <linear response theory> and the <fluctuation-dissipation theorem>.<Statistical-Mechanical Theory of Irreversible Processes. I by Ryogo Kubo>[1957 general response, relaxation and correlation framework]{ref}
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