Relationship between lesser Green's function and greater Green's function in Keldysh formalism

Question

I wonder if there is any general relationship between lesser Green's function $G^<(t,t')$ and $G^>(t,t')$ in the non equilibrium case, which means they not only depend on the relative time but also the average time. The time evolution kernel becomes a Dyson Series.

Answer 1

TL;DR In general, no.

A longer but possibly irrelevant discussion follows. Consulting the classic review RevModPhys.58.323 by Rammer and Smith, the quantities you are considering are defined as (Eq. 2.5):

$$G^{<}(\boldsymbol x_1,t_1,\boldsymbol x_{1'},t_{1'})=\mp i\langle \psi^\dagger_{\mathcal H}(\boldsymbol x_{1'},t_{1'}) \psi_{\mathcal H}(\boldsymbol x_1,t_1)\rangle, $$

$$G^{>}(\boldsymbol x_1,t_1,\boldsymbol x_{1'},t_{1'})=- i\langle \psi_{\mathcal H}(\boldsymbol x_1,t_1) \psi^\dagger_{\mathcal H}(\boldsymbol x_{1'},t_{1'}) \rangle, $$

where $\mathcal H$ implies the Heisenberg picture, while $(\boldsymbol x_1,t_1)$ and $(\boldsymbol x_{1'},t_{1'})$ are at this point completely general.

In thermal equilibrium these functions depend only on the relative variables, i.e., $t_1 - t_{1'}$ and $\boldsymbol x_1 - \boldsymbol x_{1'}$. A well known consequence of this is the relationship concerning the Fourier transforms of the lesser and greater Green's functions, Eq. 2.65, $$\tilde G^{<}(E) = e^{-\beta E}\tilde G^{>}(E).$$ This relationship holds basically since the Hamiltonian at different times commutes with itself in an equilibrium state (also known as the Kubo-Martin-Schwinger boundary condition).

However, if the Hamiltonian does not commute with itself, which depends on the kind of perturbation considered, this relation is obviously not valid any more.

Depending on the perturbation, it should be possible to find similar relations (which now should depend on the average variables $t_1 + t_{1'}$ etc.), even though I've failed to find a reference to illustrate this point. In any case, such relations would involve a perturbative expansion, and no simple general relation exists as far as I know.

Answer 2

For the record, there is a "general relationship" between $G^{<}(t,t')|_{t'=t}$ and $G^{>}(t,t')|_{t'=t}$, that is, when they are evaluated "at equal times". It reads $$ G^{R}(t,t) - G^{A}(t,t) = G^{>}(t,t) - G^{<}(t,t) = -i,$$ and is essentially a consequence of the non-commutating operators, since $G^{>}(t,t) - G^{<}(t,t) = -i\langle a a^{\dagger} - a^{\dagger}a \rangle$. It is quite important to bear this relation in mind when deriving equations of motion for the Green functions. Note that $G^{R}(t,t') - G^{A}(t,t') = G^{>}(t,t') - G^{<}(t,t')$ always holds.