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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."
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