Linear response function

Remarks

The function exists and is unique, and $\chi_{AB}(t) = 0$ for $t < 0$; this is causality. Its Fourier transform $\tilde\chi_{AB}(\omega) = \int_0^\infty e^{i\omega t} \chi_{AB}(t)\, dt$ is the dynamical susceptibility, and it satisfies the Kramers–Kronig relations. Example: for a two-level system with $H = \tfrac{\hbar\Omega}{2}\sigma_z$, $A = B = \sigma_x$ and $\rho_0$ the ground state, one finds $\chi_{AB}(t) = \tfrac{2}{\hbar}\theta(t)\sin(\Omega t)$, which vanishes for $t < 0$ as required. Non-example: the function $\psi(t) = \tfrac{2}{\hbar}\sin(\Omega t)$ for all $t$, without the factor $\theta(t)$, is not the linear response function of that system. Take $f$ supported in $[1, 2]$ and $t = 0$: the left side is $0$ because the state has not yet been disturbed, while $\int \psi(-s) f(s)\, ds$ is nonzero for suitable $f$. In classical mechanics the same definition is used with a probability density on phase space and the Liouville equation in place of $\rho$ and the von Neumann equation.