# How exact are the Kadanoff-Baym equations?

In a General nonequilibrium Quantum chemical System, the Kadanoff-Baym equations have the form:

$(i \partial_t -h_2-\Sigma_{H,2})G_<(1,2) = \delta_C(1, 2) + \int d3 \Sigma (1,3)G_<(3,2)$

where a number stands to a configurational state, e.g. $1=(x_1,t_1,\dots)$, $G_<(1,2) = <a^\dagger(1)a(2)>$ is the 1-particle Green function on the Schwinger-Keldysh contour (the Delta Distribution on contour is $\delta_C$), $h_2$ is the 1-particle part of the Hamiltonian actong on configuration 2 and $\Sigma(1,2)$ is the self-energy term.

This self-energy term is linked to other equations (Martin-Schwinger hierarchy) and contains contributions of Exchange and correlation in Quantum chemistry. The Special term $\Sigma_{H}$ is the Hartree term, which contributes to mean-field Coulomb Repulsion (local term, therefore not in integral on right Hand side). From the term $\Sigma$ I gain an Exchange term (nonlocal) that can also be found in Hartree-Fock equations and a Boltzmann-equation like binary collision term; These I aobtain by one Iteration from the Martin-Schwinger hierarchy (MSH). If I further iterate the MSH, I get also correlations mediated by other particles (Terms beyond binary "Boltzmann" collsion).

Questions:

• If I would try to simulate chemical reaction Dynamics, how necessary is the binary Boltzmann term to incorporate? When I can truncate Terms beyond Hartree-Fock?

• If electronic correlations are present, is it mostly sufficient to describe these correlations only by the binary "Boltzmann" term? Or should I incorporate higher order collision integrals?

• In a high correllated System, highly nonlocal Terms will arise in the Kadanoff-Baym equations. Are These related to Quantum entanglement phenomena? Would These theoretically predict that two molecules are on two different places at one time due to ist wave nature (can even large objects that are super-high correlated be on two places simulataneously)?

I have used some statistical mechanics terminology for better understanding; I know that Kadanoff-Baym equations are a lot more General.

Many chemical reactions are difficult to describe with many-electron Systems. Thus, it might be more useful to reduce Degree of freedom that are not so important for a chemical reaction. For example, the reaction

$A + B \rightarrow C$

can be modelled with Annihilation Operators for species $c_A,c_B,c_C$ (their conjugate are the corresponding creation operators) that go into the interaction Hamiltonian as follows:

$H_{int} = \int Dx_{A}Dx_BDx_C V_{ABC}(x_A,x_B,x_C)c_C^\dagger c_Ac_B$.

Here, $x_A,x_B,x_C$ are specific physical Degree of freedom for the species and $V_{ABC}$ the interaction potential. One can derive by e.g. using Ehrenfest's Theorem Quantum kinetic equations (Kadanoff-Baym equations).

In General, These equations are a hierarchy of infinitely many n-Point Green's functions. However, these can be truncated at some order of $\hbar$. One can perform an Expansion of the Wigner function $f(x,k,t)=\frac{1}{(2 \pi)^4}\int d^4(x-y)G_<(x,y)e^{ik(x-y)}$ (x,y and k are relativistic 4-vectors, e.g. $x=(t,\vec{x})$, where scalar product is taken with Minkowski metric) in Terms of $\hbar$. External potentials are expanded in Terms of gradients to do this (therefore, this is called the Gradient expansion).

To Leading order $\hbar \mapsto 0$, a Boltzmann equation can be derived. It has correlation terms obtained by the leading order self-energy graphs and in case of electrons (in this question) addittionally Hartree-Fock terms for the electrostatic interaction.

Hence, the leading correlation term must be respected in any out-of-Equilibrium Situation.

On the next higher order of $\hbar$, correction of the Self-Energy graph (this is a vertex correction) is incorporated. Here, Quantum effects like non-local correlations are described. The Quantum terms look similar to the semiclassical Boltzmann collision term with the main difference that the collision rate $w((A,x_A;B,x_B)\mapsto(C,x_C))$ depends on the Propagators $G_<,G_>$. Collisions are mediated by a correlation with dynamic objects that might be even far apart. The probability Amplitude will also Change during such an entangled collision; Maybe beyond the scope of what the semiclassical "Boltzmann terms" are doing. Hence, the quantum entanglement terms may allow that some objects could have a probability to be found on a place, where it is not expected to be.