Differentiating semi-classical Kubo trans. correlation functions wrt t

[jelathome]

I do not fully understand how non local correlation functions are derived from the differentiation of local ones. Specifically how in the following paper equation 23 is derived from 21 since the paper implies H and dpdq are invariant under time.

http://www.its.caltech.edu/~ch10/Papers/Miller_2.pdf

 

[Avodyne]

On the RHS of eq.(21), the only place $t$ appears is in the factor of $\bar q_t$; this is the Heisenberg-picture operator $\bar q$ evaluated at time $t$. The time derivative of $\bar q_t$ is $\bar v_t$. So $(d/dt)\tilde c_{qq}(t)$ is given by the RHS of eq.(21), but with $\bar q_t$ replaced with $\bar v_t$.

Next, we use time-translation invariance of $H$ and the integration measure to replace $\bar q_0\bar v_t$ with $\bar q_{-t}\bar v_0$.

Now we differentiate with respect to $t$ again; this gives $(d/dt)^2 c_{qq}(t)$. The factor of $\bar q_{-t}$ becomes $-\bar v_{-t}$.

Finally, time translate again to replace $\bar v_{-t}\bar v_0$ with $\bar v_0\bar v_t$.

This proves eq.(20).