Module 2 : Conceptual overview and Linear response theory
Lecture 5 : Linear response theory-II     (Properties of response kernel)
 
At this point, one must remember that this formula is deceptively simple looking--that this is the case becomes clear upon remembering that $A_{H} (t)$ does not commute with $A_{H} (t')$ unless $A$ is a conserved quantity and itself commutes with the system Hamiltonian $H$.

Before we move on to discuss the properties of the linear response function in more detail, it is perhaps important to first highlight the following subtlety: When $A=B$ is a conserved quantity, the linear response is predicted to be zero by the foregoing formula, since the commutator is zero. However, we know that a uniform external field, that couples to the total magnetization (which is a conserved quantity), does change the magnetization of the system by an amount proportional to the field, i.e. within linear response. So what goes wrong with the formalism developed here? The answer has to do with the fact that we have assumed that the initial state of the system at time $t_0$ far in the past is drawn from an ensemble governed by the Gibbs probability for the system with Hamiltonian $H$. And since the quantity to which the external perturbation couples is conserved, there is no way for the system to change this "incorrect'' probability distribution and replace it by a slightly perturbed probability distribution that takes into account the linear effect of the uniform field that couples to the conserved magnetization. This makes sense: If the magnetization were truly conserved on the time scale of the experiment, and if the external field was truly uniform, then it would indeed be impossible for the external field to change the magnetisation. A better description of the real experimental conditions is therefore to take a slightly non-uniform external field varying in time at some low frequency $\omega$, and measure the linear response at the corresponding small wavevector $q$ and frequency $\omega$. Then, the linear response formalism developed here will give a sensible result, which should be extrapolated to the uniform d.c. limit by first sending the frequency to zero keeping the wavevector non-zero, and then send q to zero.