Chapter 5
Introducing Diagrams
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
where
ˆ
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.org/10.1007/978-3-319-93602-4_5
61