*(English)*Zbl 1155.81005

This book is the second volume of an impressive monograph that provides introductory accounts of important topics in mathematical physics for graduate students and research workers. It combines theories and applications to demonstrate how the rigorous mathematical point of view helps to clarify and answer questions arising in quantum field theory with a special emphasis on QED. It has been written by someone at the frontier of research in applied functional analysis. Together with Volume I, this unique exposition will become a guide to the rudiments of quantum field theory beyond the more phenomenological aspects. Volume II contains the following material. Part I: Introduction, Part II: Basic Ideas in Classical Mechanics, Part III: Basic Ideas in Quantum Mechanics, Part IV: Quantum Electrodynamics, Part V: Renormalization. Zeidler emphasizes that it is crucial to understand the procedure called *renormalization* by physicists. He therefore studies different aspects of renormalization in each of the six volumes of his monograph, in particular the regularization of divergent series and integrals. Detailed hints are given for using the BPHZ renormalization procedure and Bogoliubov’s iterative R-method in perturbation theory. Supplemented with an extensive bibliography and historical remarks and citations, this is in my perspective a perfect book for mathematicians and physicists alike.

For the first part, cf. “Quantum field theory. I: Basics in mathematics and physics. A bridge between mathematicians and physicists.” [Berlin: Springer (2006; Zbl 1124.81002)].

##### MSC:

81-01 | Textbooks (quantum theory) |

81T15 | Perturbative methods of renormalization (quantum theory) |

81V10 | Electromagnetic interaction; quantum electrodynamics |

81U20 | $S$-matrix theory, etc. (quantum theory) |

81S05 | Commutation relations (quantum theory) |