Quantum Theory A Wide Spectrum

This book is based on lectures given in quantum theory over the years at
various levels culminating into graduate level courses given to the students
in physics. It is a modern self-contained textbook and covers most aspects of
the theory and important recent developments with fairly detailed presentations. In addition to traditional or so-called standard topics, it emphasizes on
modern ones and on theoretical techniques which have become indispensable
in the theory. I have included topics which I believe every serious graduate
student in physics should know. This volume is also a useful source of information and provides background for research in this discipline and related
ones as well. As such, the book should be valuable to the graduate student,
the instructor, the researcher and to all those concerned with the intricacies
of this subject. To make this work accessible to a wider audience, some of
the technical details occurring in the presentations have been relegated to
appendices. A glance at the Contents will reveal that although the book is
fairly advanced, it develops the entire formalism afresh. As for prerequisites,
a familiarity with general concepts and methods of quantum physics as well
as with basic mathematical techniques which most students entering graduate school seem to have is, however, required. The evident interest of my
students in my quantum theory courses has led me quite often to expand and
refine my notes that eventually became the book. I often witness many of my
earlier students, who have already taken my courses, coming back to sit in
my lectures and continue to do so. Some of these learners are A-students. In
developing the formalism, at the very early stages, and of the rules for computations, I have followed a method based on Schwinger’s (1970, 1991, 2001)
elegant and incisive approach of direct analyses of selective measurements,
rather than of the historical one, as well as in the introduction of quantum
generators and the development of transformation theory. The selective measurement approach has its roots in Dirac’s abstract presentation in terms of
projection operators and provides tremendous insight into the physics behind
the formalism. Other authors who have also shown interest in this approach
include Kaempffer (1965), Gottfried (1989) and Sakurai (1994).
Some of the highlights of the book are: 1) Selective measurements. Direct
analyses of such measurements and extensions thereof lead to the development of the underlying rules of the theory in the most natural and elegant
way. 2) Wigner’s Theorem on symmetry transformations. This theorem is of
central importance in quantum theory and provides the nature of symmetry
transformations and is the starting point on how to implement them. 3) Continuous transformations as well as supersymmetry and discrete transformations. 4) Hilbert space concepts and self-adjoint operators. 5) General study
of the spectra of Hamiltonians. 6) Localizability, uncertainties and stability
of quantum systems, such as of the H-atom, and their relations to boundedness of the corresponding Hamiltonians from below. 7) Decay of quantum
systems and the Paley-Wiener Theorem. 8) Harmonic oscillators at finite temperatures, with external sources and coherent states. Bose-Fermi oscillators.
9) Hyperfine splitting of the H-atom for arbitrary angular momentum states.
10) The non-relativistic Lamb shift. 11) The anomalous magnetic moment of
the electron. 12) Measurement, interference and the role of the environment.
13) Schrödinger’s cat and quantum decoherence. 14) Bell’s test. 15) Quantum teleportation and quantum cryptography. 16) Geometric phases under
non-adiabatic and non-cyclic conditions. The AB effect. Rotation of a spinor
by 2π radians. Neutron interferometry. 17) Analytical quantum dynamical
treatment of the Stern-Gerlach effect. 18) Ramsey oscillatory fields method
and applications. 19) Green functions, and how they provide information on
different aspects of the theory in a unified manner. 20) Path integrals and
constrained dynamics. 21) The quantum dynamical principle as a powerful,
simple and most elegant way of doing quantum physics. This approach has
not yet been sufficiently stressed in the literature and it is expected to play
a very special role in the near future not only as a practical way for computations but also as a technically rigorous method. 22) The stability of matter
in this monumental theory. This problem is undoubtedly one of the most important and serious problems that quantum physics has resolved. The Pauli
exclusion principle is not only sufficient for stability but it is also necessary.
23) The intriguing problem of “bosonic matter” and the collapse of matter if
the Pauli exclusion principle were abolished with the energy released upon
the collapse of two such macroscopic objects in contact being comparable to
that of an atomic bomb. 24) Systematics of quantum scattering including a
detailed treatment of the Coulomb problem. Emphasis is also put on the connection between configuration and momentum spaces in a scattering process.
25) Spinors, quantum description of relativistic particles, helicity and relativistic equations for any mass and any spin. As the energy and momentum
of a particle become large enough, the Schrödinger equation, with a nonrelativistic kinetic energy, becomes inapplicable. One is then confronted with
the development of a formalism to describe quantum particles in the relativis
tic regime. The chapter in question emerging from this endeavor provides the
precursor of relativistic quantum theory of fields. 26) Spin & Statistics, as
probably one of the most important results not only in physics but in all of
the sciences, in general. The spin and statistics connection is responsible for
the stability of matter, without it the universe would collapse. 27) Detailed
mathematical appendices, with proofs, tailored to our needs which may be
otherwise not easy to read in the mathematics literature.
The above are some of the topics covered in addition to the more standard
ones. I have made much effort in providing a pedagogical approach to some
of the more difficult ones just mentioned. These relatively involved topics are
treated in a more simplified manner than that in a technical journal without, however, sacrificing rigor, thus making them more accessible to a wider
audience and not only to the mathematically inclined reader. The problems
given at the end of each chapter form an integral part of the book and should
be attempted by every serious reader. Some of these problems are research
oriented. With the rapid progress in quantum physics, I hope that this work
will fill a gap, which I feel does exist, and will be useful, and also provides a
challenge, to all those concerned with our quantum world.