A simple proof for a basic linked cluster theorem

Moritz Helias, David Dahmen

Donald H. Kobe

The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.

Thomas H. Schucan, Hans A. Weidenmüller

The ground-state energy of nuclear matter is expanded in such a way that each term of the expansion is characterized by the number of hole lines. This procedure reverses the usual sequence of steps, which consists in the use of the linked-cluster theorem, the introduction of $G$ matrices, and the partial summation of $G$-matrix diagrams leading to the hole-line expansion. In this way, we find an algebraic method which yields automatically all diagrams with a fixed number of independent hole lines. As a by-product, a new proof for the linked-cluster theorem is obtained. We have applied the method to diagrams with four independent hole lines and find complete agreement with results published previously by B. D. Day.

M. E. Arons

Daniel D. Spiegel

This expository note presents a constructive proof of Wigner's theorem using only a few basic facts about Hilbert spaces, such as the existence of orthonormal bases and the Fourier decomposition of a vector. Our proof is based on a proof by Steven Weinberg found in the first volume of his series of textbooks on quantum field theory, but differs in a few places for the sake of greater simplicity and rigor.

Baoming Tang, Ehsan Khatami, Marcos Rigol