We shall derive the following expression for the quantum time correlation function
known as a Kubo transform relation. Since 
is given by the Heisenberg
equation:
it follows that
Evaluating the expression at 
gives
Thus,
By performing the trace in the basis of eigenvectors of 
, we obtain
But
Therefore,
which proves the relation. The classical limit can be deduced easily from the Kubo transform relation:
Note further, by using the cylic properties of the trace, that
