density.

The Keldysh technique provides an unambiguous way of extending this separation between noise and response to higher orders. A full exposition of this method is beyond the scope of this paper; we refer the reader to Ref. [@Kamenev2011book]. We sketch the main ideas needed here. In the path-integral formulation of the Keldysh technique, each operator corresponds to two different fields, the classical field $\xi_{\rm cl}(t)$ and the quantum field $\xi_{\rm q}(t)$. Averages of these fields (weighted by the appropriate Keldysh action describing the system) then correspond to operator averages with a particular time ordering. One finds that:

-   Averages only involving quantum fields are necessarily zero.

-   Averages involving at least one classical field $\xi_{\rm cl}(t)$ and one or more quantum fields $\xi_{\rm q}(t)$ can always be interpreted as response coefficients to an external perturbation of the form $\hat{H}_{\rm ext}(t)$.

-   Averages [*only*]{} involving classical fields $\xi_{\rm cl}(t)$ do not correspond to any kind of response function. Instead, they describe the intrinsic fluctuation properties of the system

Formally, this dichotomy arises because the perturbation $\hat{H}_{\rm ext}(t)$ enters the action of the system as a term that [*only*]{} involves the quantum field, i.e. $S_{\rm ext} = \int dt V(t) \xi_{\rm 