Chapter 7
Time-Ordered or Goldstone Diagrams
The analytical expressions deriving from the Feynman or Abrikosov diagrams are
not yet entirely explicit, since, at nth order, they involve an n-fold time integral with
regard to the time arguments of the inner vertices. The evaluation of these integrals,
involving products of time-dependent free Green’s functions, is rather cumbersome.
In the so-called energy (or ω-) representation, the time variables and time integrations are replaced by energy variables and integrations as the result of appropriate
Fourier transformations. As will be discussed in Sect. 7.1, the ω-representation is
somewhat simpler than the original time representation, but does not solve the integration problem, as one is still left with n ω-integrations. Fortunately, the diagrammatic formulation can be extended as to allow one to obtain the result of the inner
time or ω-integrations in an explicit, albeit fragmented form. This is accomplished
by inspecting the set of (n + 2)! time-ordered or Goldstone diagrams associated
with a given nth-order Feynman (or Abrikosov) diagram. The rules for drawing and
evaluating the time-ordered diagrams are presented in Sect. 7.2. A derivation of the
Goldstone rules is given in Appendix A.4.
7.1 Energy Representation of Diagrams
As discussed in Chap. 3, one can switch back and forth between the time and energy
(or ω-) representations of the electron propagator via Fourier transformations,
G pq (ω) =
∞
−∞
d(t − t
) e
iω(t−t
) G pq (t, t
)
© 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_7
95
Time-Ordered or Goldstone Diagrams
The analytical expressions deriving from the Feynman or Abrikosov diagrams are
not yet entirely explicit, since, at nth order, they involve an n-fold time integral with
regard to the time arguments of the inner vertices. The evaluation of these integrals,
involving products of time-dependent free Green’s functions, is rather cumbersome.
In the so-called energy (or ω-) representation, the time variables and time integrations are replaced by energy variables and integrations as the result of appropriate
Fourier transformations. As will be discussed in Sect. 7.1, the ω-representation is
somewhat simpler than the original time representation, but does not solve the integration problem, as one is still left with n ω-integrations. Fortunately, the diagrammatic formulation can be extended as to allow one to obtain the result of the inner
time or ω-integrations in an explicit, albeit fragmented form. This is accomplished
by inspecting the set of (n + 2)! time-ordered or Goldstone diagrams associated
with a given nth-order Feynman (or Abrikosov) diagram. The rules for drawing and
evaluating the time-ordered diagrams are presented in Sect. 7.2. A derivation of the
Goldstone rules is given in Appendix A.4.
7.1 Energy Representation of Diagrams
As discussed in Chap. 3, one can switch back and forth between the time and energy
(or ω-) representations of the electron propagator via Fourier transformations,
G pq (ω) =
∞
−∞
d(t − t
) e
iω(t−t
) G pq (t, t
)
© 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_7
95