Finite Temperature Green Function Formalism

Developing a Green function formalism similar to the \(T=0\) FDPT is much more complicated. Kubo realized that the Boltzmann factor appearing in the evaluation of the Green function at finite-T can be treated as a time evolution operator by introducing imaginary times. \( e^{-\beta H} = e^{-i H (-i \beta ) t}=U(-i \beta ,0)\) where \(\beta = 1/k_B T\) . Recall that the time evolution operator in Schrödingers picture is: \(U(t,0)= e^{-i H t}\) . Hence, based on the above we can interpret the Boltzmann factor as evolution in imaginary times. This is a Wick rotation from real to imaginary time: \(t \rightarrow -i \tau \) where \(\tau \) is a real number in the interval \(0 \le \tau \le \beta \) as we will discuss below. Lets now study how this substitution \(t \rightarrow -i \tau \) is implemented in practical terms.