Many-body perturbation expansions without diagrams. I. Normal states

On the basis of an exact perturbational expression for the interacting one-particle Green function $G$ corresponding to bosons / fermions in terms of the bare interaction potential $v$ and permanents / determinants of the non-interacting one-particle Green function $G_0$, we deduce four recursive perturbation expansions for the self-energy $\Sigma$. With $W$ denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of $\Sigma$ in terms of (i) self-energy diagrams and $(v, G_0)$, (ii) $G$-skeleton self-energy diagrams and $(v, G)$, (iii) $W$-skeleton self-energy diagrams and $(W, G_0)$, and (iv) $G$- and $W$-skeleton self-energy diagrams and $(W,G)$. For the calculation of $W$, we rely on a similar exact perturbational expression for the interacting two-particle Green function $G_2$ as for $G$. From this expression, we deduce four recursive perturbation expansions for the polarization function $P$, necessary for the calculation of $W$, that are similar to those for $\Sigma$ specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-$\tfrac{1}{2}$ fermions on a lattice in arbitrary $d$ spatial dimensions. In two further appendices, we present methods and short programs for determining the $\nu$th-order diagrams corresponding to the perturbation expansions of $G$ in terms of $(v,G_0)$, and $\Sigma$ in terms of $(v,G_0)$ and $(v,G)$ on the basis of the cycle decompositions of the elements of the symmetric group $S_{2\nu}$.