Part II
Formalism of Diagrammatic Perturbation
Theory
How can the electron propagator or other propagators actually be computed? And
will the eventual computational schemes offer distinct advantages over the conventional computational methodology based on wave functions and the Schrödinger
equation? A prominent route to the computation of propagators is provided by the
formalism of diagrammatic perturbation theory, in which the contributions to the PT
expansions of the propagator matrix elements are represented in the form of graphical schemes, the famous Feynman diagrams and variations thereof. The following
Chaps. 4–6 establish the diagrammatic perturbation theory specifically for the electron propagator. Three basic theorems, namely the Gell-Mann and Low theorem,
Wick’s theorem, and the linked-cluster theorem constitute the core of the formalism.
They are discussed in Chaps. 4 and 5. Here our presentation owes much to Chap. 3
in the exemplary textbook by Fetter and Walecka [1]. Based on those theorems,
the rules to draw and evaluate Feynman diagrams are derived and demonstrated in
Chap. 6. The final Chap. 7 deals with the time-ordered or Goldstone diagrams, which
allow for a direct diagrammatical access to the results of the various time or energy
integrations required in the evaluation of the Feynman diagrams.
It should be noted that there are non-diagrammatic approaches as well, such as
the algebraic propagator methods reviewed in Chap. 16, the hierarchy of coupled
time-dependent equations of motion for many-body Green’s functions [2], and the
method of functional derivatives [3].
References
1. Fetter AL, Walecka JD (1971) Quantum theory of many-particle systems. Mc Graw-Hill,
New York
2. Martin PC, Schwinger J (1959) Phys Rev A 115:1342
3. Kadanoff LP, Baym G (1962) Quantum statistical mechanics. Benjamin, Reading
Formalism of Diagrammatic Perturbation
Theory
How can the electron propagator or other propagators actually be computed? And
will the eventual computational schemes offer distinct advantages over the conventional computational methodology based on wave functions and the Schrödinger
equation? A prominent route to the computation of propagators is provided by the
formalism of diagrammatic perturbation theory, in which the contributions to the PT
expansions of the propagator matrix elements are represented in the form of graphical schemes, the famous Feynman diagrams and variations thereof. The following
Chaps. 4–6 establish the diagrammatic perturbation theory specifically for the electron propagator. Three basic theorems, namely the Gell-Mann and Low theorem,
Wick’s theorem, and the linked-cluster theorem constitute the core of the formalism.
They are discussed in Chaps. 4 and 5. Here our presentation owes much to Chap. 3
in the exemplary textbook by Fetter and Walecka [1]. Based on those theorems,
the rules to draw and evaluate Feynman diagrams are derived and demonstrated in
Chap. 6. The final Chap. 7 deals with the time-ordered or Goldstone diagrams, which
allow for a direct diagrammatical access to the results of the various time or energy
integrations required in the evaluation of the Feynman diagrams.
It should be noted that there are non-diagrammatic approaches as well, such as
the algebraic propagator methods reviewed in Chap. 16, the hierarchy of coupled
time-dependent equations of motion for many-body Green’s functions [2], and the
method of functional derivatives [3].
References
1. Fetter AL, Walecka JD (1971) Quantum theory of many-particle systems. Mc Graw-Hill,
New York
2. Martin PC, Schwinger J (1959) Phys Rev A 115:1342
3. Kadanoff LP, Baym G (1962) Quantum statistical mechanics. Benjamin, Reading