Contour ordering in the Keldysh Formalism

Question

I am currently working on some transport problems using the non-equilibrium Green's function techniques.

I am trying to understand the contour ordered intgeral that the Keldysh Formalism uses to define Green's functions.

I am quite sure I understand how the contour ordering works for a single operator: $$\tilde{T}[e^{i\int_{t_0}^{t}dt'H_{h}'(t')}]O_{h}(t)T[e^{-i\int_{t_0}^{t}dt'H_{h}'(t')}]=T_{C_{t}}[e^{-i\int_{t_0}^{t}dt'H_{h}'(t')}O_{h}(t)]$$ where $\tilde{T}$ is the anti-time ordering operator, $T$ is the time ordering operator and $T_{C_{t}}$ is the contour ordering operator which is defined as placing operators at times that come later in the conour $C_{t}$ left to that of operators at times that come before.

Now, the way I understand it is that the contour $C_{t}$ provides only a way to order the operators so that the terms on the L.H.S and R.H.S are equal. It's not that we are integrating in the complex plane over the contour itself.

Usually, $t_0\rightarrow -\infty$.

Moving on to two operators at different times, we arrive at the definition of the Green's function given here:

$$G_{ij}(t,t')=\frac{\langle \Psi_{H}|T_{C_{t,t'}}[c_i(t)c^{\dagger}_{j}(t')]|\Psi_{H}\rangle}{\langle \Psi_{H}|\Psi_{H}\rangle}$$

What I don't understand is how would this give rise to the four different Green's functions,$G^{++}, G^{-+}, G^{+-}, G^{--}$ given in the reference. To my understanding,the thing that we actually measure in other places, the expectation value - $\langle \Psi_{H}|T[c_{iH}(t)c^{\dagger}_{jH}(t')]|\Psi_{H}\rangle$, will upon reducing the two operators to the interaction picture, give two separate contours for the two operators.

I am obviously not getting the motivation behind making four things out of one entity.So, finally, my question is what is going on when I am defining the Green's function? How am I getting four distinct possibilities or rather why am I using four different entities?

Answer 1

The expectation value in the question, $$\langle\Psi_{H}|T[c_{iH}c_{jH}^{\dagger}]||\Psi_{H}\rangle$$ is actually the causal Green function $G^{--}_{ij}$.

Let us assume the Hamiltonian: $H=h+H'(t)$

After transforming the operators to the interaction picture and doing some gymnastics, $G^{--}_{ij}$ can be written by choosing the reference time at $t_0=-\infty$ as: $$G^{--}(1,2)=\langle U_{I}(-\infty,+\infty)U_{I}(\infty, t_1)c_{iI}(t_1)U_{I}(t_1,t_2)c_{jI}^{\dagger}(t_2)U_{I}(t_2,-\infty)\rangle\theta(t_1-t_2)-\langle U_{I}(-\infty,+\infty)U_{I}(\infty, t_2)c^{\dagger}_{jI}(t_2)U_{I}(t_2,t_1)c_{iI}(t_1)U_{I}(t_1,-\infty)\rangle\theta(t_2-t_1)$$

I have left out the bra and the ket because the average could also be taken over an ensemble.

The entity defined above can be conveniently represented by the expression: $$G^{--}(1,2)=\langle T_{C}[e^{-i\int_{C}H_{I}'(\tau)d\tau}c_{iI}(t_1)c_{jI}^{\dagger}(t_2)]\rangle$$

As in the question, the contour ordering rearranges the operators in a way that times that come later on the contour are to the left of times that come earlier.

In the image shown here, we are assuming that $t_1 >t_2$, the main point is that the two time variables occur in the upper branch(represented by $(-)$) of the contour, and the contour ordering acts on them like a simple time ordering operator.

A way to prove this would be by noticing that $\int_{C}$can be broken up into four pieces and then expanding the exponential.

Similarly,$G^{++}$ is the anti-causal Green's function with the usual definition using the anti-time ordering operator.

A similar contour can be constructed for this case except now,contrary to the previous case,both the times of the operators would have to be placed on the lower branch $(+)$ of a contour that would look like this:

We have assumed here that $t_2>t_1$. In this case, the contour ordering acts like an anti-time ordering operator on the lower branch.

In the same way, the lesser$(G^{-+})$ and the greater$(G^{+-})$ Green's functions are defined using a similar contour with different placements of the operators on the two branches as indicated by their indices.

This is an exhaustive review of the whole thing.