In the four chapters of Part II, the formalism of diagrammatic perturbation theory
is developed, based on three central theorems, the Gell-Mann and Low theorem,
Wick’s theorem, and the linked-cluster theorem. At the end of that part, the reader
should be able to draw and evaluate Feynman diagrams.
However, the diagrammatic arts do not yet establish a procedure to compute the
electron propagator or the physical information conveyed therein. So with Chap. 8
in Part III, the focus shifts to the issue of developing computational schemes. Here,
the prominent starting point is the Dyson equation, relating the electron propagator
to the so-called self-energy. The latter quantity is itself subject to a diagrammatic
perturbation expansion, where the diagrams are simpler than those for the electron
propagator. The subject of Chap. 9 is the algebraic–diagrammatic construction
(ADC), a general procedure to generate systematic higher-order approximations
(ADC(n) schemes) to the self-energy, being consistent through order n, and, crucially, reproducing the correct analytical structure of the self-energy. The ADC
procedure is quite versatile and can directly be applied to the electron propagator,
or, more accurately, to its ðN Æ 1Þ-electron parts, as is demonstrated in Chap. 10.
Then, in Chaps. 11 and 12, our tour takes a remarkable turn: The direct ADC
approximations can be derived via a radically different route, namely a
wave-function-based approach referred to as intermediate state representation
(ISR). (An impetuous reader, already familiar with the topics of Chaps. 1 and 2,
might take a shortcut directly to Chaps. 11 and 12). The ISR concept bridges the
gap between propagator and wave-function methods, lifts certain limitations
inherent to the diagrammatic propagator approach, and allows for a rigorous
foundation (Chap. 12) of the defining many-body features.
In Part IV, we turn toward the physics of N-electron excitations and the
polarization propagator relevant here. Chapter 13 discusses how diagrammatic
perturbation theory can be adapted to the polarization propagator. The ADC and
ISR concepts for N-electron excitations are presented in Chap. 14, while Chap. 15
reviews the prominent random-phase approximation (RPA), being a paradigmatic
model in many-body theory. The final part V takes a look at two related approaches,
which may be seen as ISR variants: The equation-of-motion (EOM) methods
(Chap. 16) and methods based on the coupled-cluster (CC) ansatz (Chap. 17).
Altogether 9 appendices supplement the main text: Appendix A.1 reviews
many-body perturbation theory and recollects some useful algebraic techniques;
some more lengthy proofs are deferred to Appendices A.2, A.3, A.4, and A.6;
extensions to Chaps. 8, 13, and 16 are given in Appendices A.5, A.7, and A.8,
respectively; the final Appendix A.9 compiles various explicit ADC expressions.
As may be permissible in a textbook, perhaps even advisable, the bibliography
has been kept relatively short and selective. In topics that are well documented in
the literature, only a few key papers or books are quoted. More comprehensive
reference is made to subjects or issues that are less familiar or amenable. And, of
course, I have tried to indicate the sources wherever the text draws upon exemplary
previous presentations.
Preface
vii
is developed, based on three central theorems, the Gell-Mann and Low theorem,
Wick’s theorem, and the linked-cluster theorem. At the end of that part, the reader
should be able to draw and evaluate Feynman diagrams.
However, the diagrammatic arts do not yet establish a procedure to compute the
electron propagator or the physical information conveyed therein. So with Chap. 8
in Part III, the focus shifts to the issue of developing computational schemes. Here,
the prominent starting point is the Dyson equation, relating the electron propagator
to the so-called self-energy. The latter quantity is itself subject to a diagrammatic
perturbation expansion, where the diagrams are simpler than those for the electron
propagator. The subject of Chap. 9 is the algebraic–diagrammatic construction
(ADC), a general procedure to generate systematic higher-order approximations
(ADC(n) schemes) to the self-energy, being consistent through order n, and, crucially, reproducing the correct analytical structure of the self-energy. The ADC
procedure is quite versatile and can directly be applied to the electron propagator,
or, more accurately, to its ðN Æ 1Þ-electron parts, as is demonstrated in Chap. 10.
Then, in Chaps. 11 and 12, our tour takes a remarkable turn: The direct ADC
approximations can be derived via a radically different route, namely a
wave-function-based approach referred to as intermediate state representation
(ISR). (An impetuous reader, already familiar with the topics of Chaps. 1 and 2,
might take a shortcut directly to Chaps. 11 and 12). The ISR concept bridges the
gap between propagator and wave-function methods, lifts certain limitations
inherent to the diagrammatic propagator approach, and allows for a rigorous
foundation (Chap. 12) of the defining many-body features.
In Part IV, we turn toward the physics of N-electron excitations and the
polarization propagator relevant here. Chapter 13 discusses how diagrammatic
perturbation theory can be adapted to the polarization propagator. The ADC and
ISR concepts for N-electron excitations are presented in Chap. 14, while Chap. 15
reviews the prominent random-phase approximation (RPA), being a paradigmatic
model in many-body theory. The final part V takes a look at two related approaches,
which may be seen as ISR variants: The equation-of-motion (EOM) methods
(Chap. 16) and methods based on the coupled-cluster (CC) ansatz (Chap. 17).
Altogether 9 appendices supplement the main text: Appendix A.1 reviews
many-body perturbation theory and recollects some useful algebraic techniques;
some more lengthy proofs are deferred to Appendices A.2, A.3, A.4, and A.6;
extensions to Chaps. 8, 13, and 16 are given in Appendices A.5, A.7, and A.8,
respectively; the final Appendix A.9 compiles various explicit ADC expressions.
As may be permissible in a textbook, perhaps even advisable, the bibliography
has been kept relatively short and selective. In topics that are well documented in
the literature, only a few key papers or books are quoted. More comprehensive
reference is made to subjects or issues that are less familiar or amenable. And, of
course, I have tried to indicate the sources wherever the text draws upon exemplary
previous presentations.
Preface
vii