Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

This book is a collection of classic papers by V.A. Fock which were written in the period between the birth of quantum mechanics in the 1920s and the late 1960s. These papers include such fundamental notions of theoretical quantum physics as the Hartree-Fock method, Fock space, Fock symmetry of the hydrogen atom and the Fock functional method. Fock was a key contributor to one of the most exciting periods of development in 20th Century physics, and this book conveys the essence of that time. This collection of seminal works will be a useful reference for any undergraduate, graduate of researcher in theoretical and mathematical physics, especially those specialising in quantum mechanics and quantum field theory.

The last decade witnessed an increasing interest of mathematicians in prob lems originated in mathematical physics. As a result of this effort, the scope of traditional mathematical physics changed considerably. New problems es pecially those connected with quantum physics make use of new ideas and methods. Together with classical and functional analysis, methods from dif ferential geometry and Lie algebras, the theory of group representation, and even topology and algebraic geometry became efficient tools of mathematical physics. On the other hand, the problems tackled in mathematical physics helped to formulate new, purely mathematical, theorems. This important development must obviously influence the contemporary mathematical literature, especially the review articles and monographs. A considerable number of books and articles appeared, reflecting to some extend this trend. In our view, however, an adequate language and appropriate methodology has not been developed yet. Nowadays, the current literature includes either mathematical monographs occasionally using physical terms, or books on theoretical physics focused on the mathematical apparatus. We hold the opinion that the traditional mathematical language of lem mas and theorems is not appropriate for the contemporary writing on mathe matical physics. In such literature, in contrast to the standard approaches of theoretical physics, the mathematical ideology must be utmost emphasized and the reference to physical ideas must be supported by appropriate mathe matical statements. Of special importance are the results and methods that have been developed in this way for the first time."

In the period between the birth of quantum mechanics and the late 1950s, V.A. Fock wrote papers that are now deemed classics. In his works on theoretical physics, Fock not only skillfully applied advanced analytical and algebraic methods, but also systematically created new mathematical tools when existing approaches proved insufficient. This collection of Fock's papers published in various sources between 1923 and 1959 in Russian, German, French, and English. These papers explore some of the fundamental notions of theoretical quantum physics, such as the Hartree-Fock method, Fock space, the Fock symmetry of the hydrogen atom, and the Fock functional method. They also present Fock's views on the interpretation of quantum mechanics and the fundamental significance of approximate methods in theoretical physics. V.A. Fock was a key contributor to one of the most exciting periods of development in 20th-century physics, and this book conveys the essence of that time. The seminal works presented in this book are a helpful reference for any student or researcher in theoretical and mathematical physics, especially those specializing in quantum mechanics and quantum field theory.

During the past ten years, since the first edition of this book, gauge invariant models of elementary particle interactions were transformed from an attractive plausible hypothesis into a generally accepted theory confirmed by experiments. It was therefore natural that the development of the methods of gauge fields attracted the attention of the gr

Springer-Verlag has invited me to bring out my Selected Works. Being aware that Springer-Verlag enjoys high esteem in the scientific world as a reputed publisher, I have willingly accepted the offer. Immediately, I was faced with two problems. The first was that of acquaint ing the reader with the important stages in my scientific aetivities. For this purpose, I have included in the Selected Works eertain of my early works that have greatly influeneed my later studies. For the same reason, I have also in cluded in the book those works that contain the first, erude versions of the proofs for many of my basic theorems. The second problem was that of giving the reader the best possible opportunity to familiarize himself with the most important results and to learn to use my method. For this reason I have included the later improved versions of the proofs for my basic results, as weil as the monographs The Method of Trigo nometric Sums in Number Theory (Seeond Edition) and Special Variants of the Method of Trigonometric Sums."

Mathematics has a certain mystique, for it is pure and ex- act, yet demands remarkable creativity. This reputation is reinforced by its characteristic abstraction and its own in- dividual language, which often disguise its origins in and connections with the physical world. Publishing mathematics, therefore, requires special effort and talent. Heinz G-tze, who has dedicated his life to scientific pu- blishing, took up this challenge with his typical enthusi- asm. This Festschrift celebrates his invaluable contribu- tions to the mathematical community, many of whose leading members he counts among his personal friends. The articles, written by mathematicians from around the world and coming from diverse fields, portray the important role of mathematics in our culture. Here, the reflections of important mathematicians, often focused on the history of mathematics, are collected, in recognition of Heinz G-tze's life-longsupport of mathematics.

The last decade witnessed an increasing interest of mathematicians in prob lems originated in mathematical physics. As a result of this effort, the scope of traditional mathematical physics changed considerably. New problems es pecially those connected with quantum physics make use of new ideas and methods. Together with classical and functional analysis, methods from dif ferential geometry and Lie algebras, the theory of group representation, and even topology and algebraic geometry became efficient tools of mathematical physics. On the other hand, the problems tackled in mathematical physics helped to formulate new, purely mathematical, theorems. This important development must obviously influence the contemporary mathematical literature, especially the review articles and monographs. A considerable number of books and articles appeared, reflecting to some extend this trend. In our view, however, an adequate language and appropriate methodology has not been developed yet. Nowadays, the current literature includes either mathematical monographs occasionally using physical terms, or books on theoretical physics focused on the mathematical apparatus. We hold the opinion that the traditional mathematical language of lem mas and theorems is not appropriate for the contemporary writing on mathe matical physics. In such literature, in contrast to the standard approaches of theoretical physics, the mathematical ideology must be utmost emphasized and the reference to physical ideas must be supported by appropriate mathe matical statements. Of special importance are the results and methods that have been developed in this way for the first time."

This book focuses on the methods applied in the field theory of relativistic strings, which represent a direct generalization of the methods of gauge field theory. It deals with the geometrical aspects of the gauge field theory and illuminates the quantum theory of the Yang-Mills fields.

This book is based on notes from the course developed and taught for more than 30 years at the Department of Mathematics of Leningrad University. The goal of the course was to present the basics of quantum mechanics and its mathematical content to students in mathematics. This book differs from the majority of other textbooks on the subject in that much more attention is paid to general principles of quantum mechanics. In particular, the authors describe in detail the relation between classical and quantum mechanics. When selecting particular topics, the authors emphasise those that are related to interesting mathematical theories. In particular, the book contains a discussion of problems related to group representation theory and to scattering theory. This book is rather elementary and concise, and it does not require prerequisites beyond the standard undergraduate mathematical curriculum. It is aimed at giving a mathematically oriented student the opportunity to grasp the main points of quantum theory in a mathematical framework.