Introducing Diagrams

Abstract

The step from the Gell-Mann and Low (GML) formulation of the PT expansion of the electron propagator toward a diagramatic representation is enabled by Wick's theorem. According to this theorem, the expectation values of time-ordered fermion operator products arising in the GML expression can be evaluated in terms of contractions of operator pairs (Sect. 5.1). The contractions are related to free electron propagators and can be represented graphically by a directed line between the respective fermion operators. Together with the wiggly lines as graphical symbols for the interaction integrals, this allows one to replace the original analytic terms with diagrams (Sect. 5.2). The contraction concept allows one to distinguish linked and unlinked contributions (or diagrams) to the propagator PT expansion. The linked-cluster theorem, discussed in Sect. 5.3, states that the denominator in the GML expressions exactly cancels all unlinked parts so that only linked contributions (or diagrams) need to be considered in the PT expansion of the electron propagator. 5.1 Wick's Theorem The perturbation expansion (4.58) for the electron propagator established in the preceding Chap. 4 involves expectation values of time-ordered fermion operator products, being of the form 0 | ˆ T T T ˆ H I (t 1). .. ˆ H I (t n)c p (t)c † q (t) | 0 wherêwherê H I (t) = w rs c † r (t)c s (t) + 1 2 V uvrs c † u (t)c † v (t)c s (t)c r (t) (5.1) © Springer Nature Switzerland AG 2018 J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture Notes in Chemistry 94, https://doi.