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MBPT

Green's function theory, also called (improperly) Many-Body Perturbation Theory (MBPT), is a Quantum Field Theory based on a formalism of second quantization of operators. The fundamental degree of freedom is the Green's function or propagator, $G(r_1,t_1,r_2,t_2)$, which represents the probability amplitude for the propagation of an electron from $r_1,t_1$ to $r_2,t_2$. The main advantages of this theory are that:

  1. it avoids having indices running over many particles;
  2. fermionic antisymmetrization is automatically imposed;
  3. systems with varying number of particles can be treated;
  4. and most importantly, all the physics of the system is condensed inside the Green's function.

As in any other quantum field theory (for example QED), the many-body system can be expanded in perturbation theory, with the coupling being the many-body interaction term. The Green's function (as well as any other quantity of the theory, such as the self-energy or the polarization) can be calculated at a given order of perturbation theory. A Feynman diagrammatic analysis is hence possible. The theory at the first order is equivalent to Hartree-Fock theory.

hedin's equations

However the coupling is not small (compare to, for example, the electron-ion interaction) and the expansion does not converge. The second order is not necessarily smaller than the first. Hence one needs to resort to more complicated methods to solve the theory, such as partial resummations of diagrams at all orders, or better, iterative methods.

In iterative schemes one introduces new quantities into the theory but relating them to the old, in the hope that at the end one can succeed in closing the equations. Indeed, MBPT can be solved thanks to a set of five integro-differential equations, called the Hedin equations, that have to be solved iteratively until self-consistency is achieved.

So far, nobody has solved the Hedin equations for a real system. Approximations are required to simplify the problem. Among the most widely used approximate schemes are the GW approximation for the self-energy and the Bethe-Salpeter Equation approach and its related approximations.