3.5 DYSON Equation

The DYSON equation can be achieved by classifying the various contributions in arbitrary FEYNMAN diagrams. DYSON's equation summarizes the FEYNMAN-DYSON perturbation theory in a particularly compact form. The exact GREEN's function can be written as the non-interacting GREEN's function plus all connected terms with a non-interacting GREEN's function at each end, see (3.40). This structure is shown in Fig. 3.5, where the double line denotes $G$ and the single line $G_{0}$.

Figure 3.5: The GREEN's function expanded in terms of connected diagrams.

By introducing the concept of self-energy $\Sigma$ the structure in Fig. 3.5 takes the form shown Fig. 3.6.

Figure 3.6: FEYNMAN diagrams showing the general structure of $G$.

The corresponding analytic expression is given by

$$\begin{array}{l}\displaystyle \displaystyle G({\bf {r}},t;{\... ...r}},t;1) \ \Sigma(12) \ G_{0}(2;{\bf {r'}},t') \ , \end{array}$$ (3.41)

where the abbreviation $1\equiv({\bf r_1},t_1)$ and $\int d1 \equiv \int d{\bf r_1}\int dt_1$ is used. The self-energy $\Sigma$ describes the renormalization of single-particle states due to the interaction with the surrounding many-particle system and the DYSON equation determines the renormalized GREEN's function.

Another important concept is the proper self-energy insertion which is a self-energy insertion that can not be separated into two pieces by cutting a single-particle line. By definition, the proper self-energy is the sum of all proper self-energy insertions, and will be denoted by $\Sigma^{*}$. Using the perturbation expansion, one can define the proper self-energy $\Sigma^*$ as an irreducible part of the GREEN's function. Based on this definition first-order proper self-energies, which are resulted from the first-order expansion of the GREEN's function (see Section 3.4.2), are shown in Fig. 3.7. These diagrams are irreducible parts of Fig. 3.4-b and Fig. 3.4-c and are referred to as the HARTREE ( $\Sigma^\mathrm{H}$) and the FOCK ( $\Sigma^\mathrm{F}$) self-energies.

Figure 3.7: FEYNMAN diagrams of the first-order proper self-energies.

The self-energy can also in principle be introduced variationally [203]. A variational derivation of the self-energies for the electron-electron and electron-phonon interactions is presented in Appendices F.1 and F.2, respectively. It follows from these definitions that the self-energy consists of a sum of all possible repetitions of the proper self-energy

$$\begin{array}{l}\displaystyle \displaystyle \Sigma({\bf {r}}... ...2) \ \Sigma^{*}(2;{\bf {r'}},t') \ + \ \ldots \ . \end{array}$$ (3.42)

Correspondingly, the GREEN's function in (3.41) can be rewritten as

$$\begin{array}{l} \displaystyle \displaystyle G({\bf {r}},t;{... ...{*}(12) \ G_{0}(2;{\bf {r'}},t') \ + \ \ldots \ , \end{array}$$ (3.43)

which can be summed formally to yield an integral equation (DYSON equation) for the exact GREEN's function which is shown in Fig. 3.8.

Figure 3.8: FEYNMAN diagrams representing DYSON's equation.

The corresponding analytic expression is given by

$$\begin{array}{l} \displaystyle \displaystyle G({\bf {r}},t;{... ...{r}},t;1) \ \Sigma^{*}(12) \ G(2;{\bf {r'}},t')\ . \end{array}$$ (3.44)

The validity of (3.44) can be verified by iterating the right hand-side, which reproduces (3.43) term by term. In a similar manner, one can show that the DYSON equation can be also written as

$$\begin{array}{l} \displaystyle G({\bf {r}},t;{\bf {r'}},t') ... ...}},t;1) \ \Sigma^{*}(12) \ G_0(2;{\bf {r'}},t')\ . \end{array}$$ (3.45)

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