Mattuck - Guide to Feynman Diagrams in the Many-Body Problem

I was delighted by the extreme reactions to the first edition of this book. One reviewer called it a pedagogical jewel ... useful as a crutch for poorly-prepared students, while another felt that it was primarily for people who were well-prepared and courageous. The preface to the Russian edition referred to the pinball game analogy on p. 29 as some sort of worlds record in popularization, but an English critic complained that the pinball picture was highly offensive and had no place in a serious work of science. A student told me that at his university, the book was known as Feynman Diagrams for Idiots, while other students felt that it was only for people exceptionally well-grounded in quantum mechanics. One critic stated that the possibilities for classroom use should be rather wide, but others claimed that the book was useless, since no detailed calculations were carried out in it.

In short, the first edition is too elementary and too advanced. Therefore, the purpose of the second edition is to make the book more advanced and more elementary. Toward this end, on the elementary side, a zeroth and first chapter have been added which are on the pre-kindergarten or nursery school level. This gives a view of the entire field based almost purely on pictures, cartoons, and virtual movies, with essentially no mathematics.

On the more advanced side, I have added to many chapters a new section showing in mathematical detail how typical many-body calculations with Feynman diagrams are carried out. For example, chapter 3 contains the detailed calculation of the energy and lifetime of an electron in an impure metal. In chapter 9, the single pair-bubble approximation is used to compute the quasi particle lifetime diagrammatically. The pair-bubble integrations are done in detail in chapter 10 and the results employed to obtain the form of the effective interaction in an electron gas, and the plasmon dispersion law in chapter 13. Chapter 14 contains the calculation of the finite temperature pair-bubble.

A number of new exercises have been added, some of which give the student the opportunity to carry out simpler many-body calculations himself. For example, Ex. 10.7 requires solving the K -matrix equation in ladder approximation, computing the integrals and showing that the hole lines give a negligible contribution in the low density case.

A new chapter on the quantum field theory of phase transitions has been added. It includes, on the kindergarten level, an analysis of the staring crowd transition (see p. 290) and on the more advanced level, the diagrammatic calculation of the magnetization and transition point for the ferromagnetic phase. There are also new chapters on the Kondo problem and on the renormalization group.

I have also written several new appendices. Appendix L reviews the analytic properties of propagators, which I make considerable use of at various points in the text. Appendix M shows the relation between the equation of motion and Feynman diagram methods for calculating the propagator. Appendix N gives the basic ideas of the reduced graph method, used in connection with the Kondo problem.

In preparing the second edition, special thanks are due to Stud. Scient. Nikolai Nissen for pointing out better methods for carrying out many of the calculations, and for carefully reading and criticizing the new material.

I am also very grateful to my colleague Dr Ulf Larsen for the many fruitful and stimulating discussions of many-body theory we have had during the last five years, for his help in working out the chapter on the Kondo problem, and for weeding out many of the inaccuracies which had crept into the book.

I would like to thank Professor P. W. Atkins of Lincoln College, Oxford, for pointing out how the book could be modified to make it of more value to chemists.

I am much indebted to my cousins son, David Lustbader, B.A., for his aid in improving chapter Zero, and to my own son, Allan, for help in pasting together the thousands of pieces of paper which were the raw material for the second edition. And I want to express my gratitude to my students, whose unending stream of questions forced me to replace fuzziness by clarity throughout the book.

And finally, a word of thanks to the many people who, by telling me how much they enjoyed the first edition (one wrote: Please allow me to express my gratitude for a ray of sunshine that you have cast into the windowless office of a second year graduate student in the form of your book on Feynman diagrams), gave me the inspiration and fortitude to sweat my way through the production of the second edition.

Copenhagen, 1974