# Finite temperature QFT: Can vertex correction function/ initial correlations induce short-time tunneling?

The collision term in the Kadanoff-Baym equation has the structure

$$I(\tau_1,\tau_2) = \int_C d \tau'\Sigma(\tau_1,\tau') G(\tau',\tau_2)$$

where the contour $$C$$ is along the time-Forward $$C_+$$, time-backward $$C_-$$ and along the (imaginary time) Matsubara $$C_M$$ branch and $$\tau_1,\tau_2$$ are both NOT on Matsubara branch. I want to incorporate initial correlations with a bath of temperature $$T$$ and chemical potential $$\mu$$. Now I consider the interaction with a particle having Green function $$G$$ and an Exchange particle having Green function $$G^*$$, such that for the self-energy

$$\Sigma(\tau_1,\tau') = \int_C d \eta_1 \int_C d \eta_2 \Lambda(\tau',\eta_1,\eta_2) G(\tau_1,\eta_1) G^*(\tau_1,\eta_2).$$

In the case, where the vertex function $$\Lambda(\tau_1,\eta_1,\eta_2)$$ is local (i.e. can be expressed in Terms of Delta functions on equal contours), I will get the initial correlation term

$$I_{IC}(\tau_1,\tau_2) = \int_{C_M} d \tau'\Sigma_\downarrow(\tau_1,\tau') G_\uparrow(\tau',\tau_2)$$

that mimicks contact with the (initial) bath ($$\uparrow$$ = Propagator from Matsubara contour to non-Matsubara contour, $$\downarrow$$ = Propagator in the other direction).

My Question is, if the term $$\Lambda(\tau_1,\eta_1,\eta_2)$$ is nonlocal and has Arguments on different branches. One calls this "vertex correction". There will be Quantum correlation Terms like

$$I_{aQCterm}(\tau_1,\tau_2) = \int_{C_M} d \tau'\Sigma_{aQCterm}(\tau_1,\tau') G^\uparrow(\tau',\tau_2)$$, while

$$\Sigma_{aQCterm}(\tau_1,\tau') = \int_{-\infty}^\infty d \eta_1 \int_{C_M} d \eta_2 \Lambda_\downarrow(\tau',\eta_1,\eta_2) G_{ret}(\tau_1,\eta_1) G_{ret}^*(\tau_1,\eta_2).$$

Intuitively These describe a reaction

particle + exchange particle -> particle in state of initial correlation

and due to nonlocality, the outgoing particle will be located on a different Point than the Incoming reactants.

Questions:

a) Can vertex corrections, if also an additional external potential barriers are included in the equations, describe dynamical (Maybe only temporal) Quantum Tunneling? Maybe yes, the outgoing particle may appear outside the barrier, while Incoming colliding particles are inside barrier. Is my thought correct?

b) Can initial states of a particle in the past be "revived" due toprocess described above? I think also yes, I think that initial correlations make that the particle appears similar to its initial state at some later state due to Quantum collisions.

## 1 Answer

In this paper expansions of nonequilibrium Green's functions, the Green's function formalism has been generalized to treat arbitrary time-dependent systems initially prepared with an arbitrary density matrix.

For the recent developments about initial correlation in the framework of NEGF, you can follow the publications of Gianluca Stefanucci and Robert van Leeuwen.