This exposition is devoted to a consistent treatment of quantization problems, based on appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert spaces. The authors begin by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes to the naive treatment. It then builds the necessary mathematical background following it by the theory of self-adjoint extensions. By considering several problems such as the one-dimensional Calogero problem, the Aharonov-Bohm problem, the problem of delta-like potentials and relativistic Coulomb problemIt then shows how quantization problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems. In the end, related problems in quantum field theory are briefly introduced. This well-organized text is most suitable for students and post graduates interested in deepening their understanding of mathematical problems in quantum mechanics. However, scientists in mathematical and theoretical physics and mathematicians will also find it useful.

This book contains a systematic analysis of the formalisms of quantum electro dynamics in the presence of an intense external field able to create pairs from the vacuum, and thereby violate the stability of the latter. The approach developed is not specific to quantum electrodynamics, and can equally well be applied to any quantum field theory with an unstable vacuum. It should be noted that only macroscopic external fields are considered, whereas problems associated with the superstrong Coulomb (micro) field are not treated. As a rule, the discussion is confined to those details of the formalism and calculations that are specific to the instability property. For instance, renormalization is not discussed here since, in practical calculations, it is carried out according to standard methods. The presentation is based mainly on original research undertaken by the authors. Chapter 1 contains a general introduction to the problem. It also presents some standard information on quantum electrodynamics, which will be used later in the text. In addition, an interpretation of the concept of an external field is given, and the problems that arise when one tries to keep the interaction with the external field exactly are discussed. In Chapter 2, the perturbation expansion in powers of the radiative interac tion is developed for the matrix elements of transition processes, taking the arbitrary external field into account exactly."