Correlators depending on initial time

Question

Imagine a Hamiltonian of the form:- $$ H = H_{o} + V(t) $$ where, $H_{o}$ is a constant term, and $V(t)$ is a time dependent perturbation. In the most general case, $[H(t_{i}),H(t_{j})] \neq 0$ for $i \neq j$. Also, $[H_{o},V(t)] \neq 0$.

I am trying to work through the chapter on Imaginary time green's functions, in the textbook by Bruus and Flensberg. In it, they prove that the correlator depends solely on the difference between the initial and the final time. The proof is outlined below, for $\tau > \tau^{\prime}$. $$\mathcal{C}_{A B}\left(\tau, \tau^{\prime}\right) \equiv-\left\langle T_{\tau}\left(A(\tau) B\left(\tau^{\prime}\right)\right)\right\rangle$$, where $T_{\tau}$ is the time ordering operator. $$\begin{aligned} \mathcal{C}_{A B}\left(\tau, \tau^{\prime}\right) &=\frac{-1}{Z} \operatorname{Tr}\left[e^{-\beta H} e^{\tau H} A e^{-\tau H} e^{\tau^{\prime} H} B e^{-\tau^{\prime} H}\right] \\ &=\frac{-1}{Z} \operatorname{Tr}\left[e^{-\beta H} e^{-\tau^{\prime} H} e^{\tau H} A e^{-\tau H} e^{\tau^{\prime} H} B\right] \\ &=\frac{-1}{Z} \operatorname{Tr}\left[e^{-\beta H} e^{\left(\tau-\tau^{\prime}\right) H} A e^{-\left(\tau-\tau^{\prime}\right) H} B\right] \\ &=\mathcal{C}_{A B}\left(\tau-\tau^{\prime}\right) \end{aligned}$$

It is unclear to me as to why the correlator must depend only on the time difference between the initial and the final time, even for the most general hamiltonian. It sees to me that I could define a $V(t)$ that drastically changes the hamiltonian at a later time, so that the correlator also depends on the initial time. If so, where are we missing that in our proof above?

A guess on my part leads me to think that it is where we assume that the operators $e^{-\beta H} e^{\alpha H}$ commute. the expansions of the operator product is as follows:- $$e^{-\beta H}e^{\alpha H}=e^{-\beta H_{0}} T_{\tau} \exp \left(-\int_{0}^{\beta} d \tau_{1} \hat{V}\left(\tau_{1}\right)\right)e^{\alpha H_{0}} T_{\tau} \exp \left(\int_{0}^{\alpha} d \tau_{2} \hat{V}\left(\tau_{2}\right)\right)$$ If $H_{o}$ and V don't commute, neither should the second and third terms in the expression above. I am unable to find any sources to back this up, since all references I have seen have taken that $e^{-\beta H} e^{\alpha H}$ commute readily. I would be grateful if someone pointed out the fallacy in my thinking.

Answer 1

The developments here are for the system in thermodynamic equilibrium, i.e. there is an assumption that whatever the initial correlations existed between the variables have already decayed. Bruus&Flensberg is a good modern book covering many subjects, but it may be a bit sloppy on such details. Therefore for the basics I would recommend using one of the classics: Fetter &Walecka or Abrikosov, Gorkov &Dzyaloshinskii (more mathematically sophisticated). Mahan is good in terms of its coverage, but suffers from the same shortcomings as your book. Negele& Orland is a good alternative in terms of path integrals.

A couple more remarks:

• Abrupt changes in potential or the dependence on the initial conditions are correctly analyzed within the Keldysh formalism, which requires knowing the baduc equilibrium techniques.
• Operators $e^{-\beta H}$ and $e^{\alpha H}$ of course commute, since they are functions of the same operator.