Intended for beginning graduate students, this text takes the reader from the familiar coordinate representation of quantum mechanics to the modern algebraic approach, emphsizing symmetry principles throughout. After an introduction of the basic postulates and techniques, the book discusses time-independent perturbation theory, angular momentum, identical particles, scattering theory, and time-dependent perturbation theory. It concludes with several lectures on relativistic quantum mechanics and on many-body theory

The mathematical study of the Bose gas goes back to the ?rst quarter of the twentieth century, with the invention of quantum mechanics. The name refers to the Indian physicist S.N. Bose who realized in 1924 that the statistics governing photons(essentiallyinventedbyMaxPlanckin1900)isdetermined(usingmodern terminology) by restricting the physical Hilbert space to be the symmetric tensor product of single photon states. Shortly afterwards, Einstein applied this idea to massive particles, such as a gas of atoms, and discovered the phenomenon that we now call Bose-Einstein condensation. At that time this was viewed as a mathematical curiosity with little experimental interest, however. The peculiar properties of liquid Helium (?rst lique?ed by Kammerlingh Onnes in 1908) were eventually viewed as an experimental realization of Bose- Einstein statistics applied to Helium atoms. The unresolved mathematical pr- lem was that the atoms in liquid Helium are far from the kind of non-interacting particles envisaged in Einstein’s theory, and the question that needed to be - solved was whether Bose-Einstein condensation really takes place in a strongly interacting system — or even in a weakly interacting system. That question is still with us, three quarters of a century later! The ?rst systematic and semi-rigorous mathematical treatment of the pr- lem was due to Bogoliubov in 1947, but that theory, while intuitively appealing and undoubtedly correct in many aspects, has major gaps and some ?aws. The 1950’s and 1960’s brought a renewed ?urry of interest in the question, but while theoreticalintuitionbene?tedhugelyfromthisactivitythemathematicalstructure did not signi?cantly improve.

Statistical Methods in Quantum Optics 2 - Non-Classical Fields continues the development of the methods used in quantum optics to treat open quantum systems and their fluctuations. Its early chapters build upon the phase-space methods introduced in the first volume Statistical Methods in Quantum Optics 1 - Matter Equations and Fokker-Planck Equations: the difficulties these methods face in treating non-classical light are exposed, where the regime of large fluctuations – failure of the system size expansion – is shown to be particularly problematic. Cavity QED is adopted as a natural vehicle for extending quantum noise theory into this regime. In response to the issues raised, the theory of quantum trajectories is presented as a universal approach to the treatment of fluctuations in open quantum systems.

When this publisher offered me the opportunity to \\Tite a book, some six years ago, I did not hesitate to say yes. I had just spent the last four years of graduate school struggling to understand the physics of strained quantum well lasers, and it seemed to me the whole experience was much more difficult that it should have been. For although many of the results I needed were easy to locate, the underlying physical premises and intervening steps were not. If only I had a book providing the derivations, I could have absorbed them and gone on my way. Such a book lies before you. It provides a unified and self-contained descrip­ tion of the essential physics of strained quantum well lasers, starting from first principles whenever feasible. The presentation I have chosen requires only the standard introductory background in quantum mechanics, solid state physics, and electromagnetics expected of entering graduate students in physics or elec­ trical engineering. A single undergraduate course in each of these subjects should be more than sufficient to follow the text. :'Iore advanced material on quantum mechanics is developed and collected in the first chapter. \Vhen pos­ sible, I have presented the results in a general setting and have later applied them to specific cases of interest. I find this the most satisfying way to ap­ proach the subject, and it has the additional benefit of solving many problems once and for all.

Another book on Quantum Optics? or Quantum Information? Well, not - actly. A more descriptive title might be: “A guided tour through basic qu- tum mechanics, quantum optics and quantum information”. Even better, a few words on its origin and our motivation for undertaking the task might be useful to the potential reader in deciding whether to turn the pages beyond this preface. For more than ten years now, a graduate course on quantum optics has been taught in the physics department of the University of Crete. Spanning two semesters, it originally consisted of a collection of topics representative of what can be found in the numerous by now excellent books on quantum optics. Over the last four years or so,however, the course acquired a gradually increasing segment of what is broadly referred to as quantum information, whichatthispointisapproximatelyhalfofthematerial.Inevitably,thetopics on standard quantum optics had to be reduced or compressed accordingly.

This course-based primer offers readers a concise introduction to the description of quantum mechanical systems with infinitely many degrees of freedom – and quantum spin systems in particular – using the operator algebraic approach. Here, the observables are modeled using elements of some operator algebra, usually a C*-algebra. This text introduces readers to the framework and the necessary mathematical tools without assuming much mathematical background, making it more accessible than advanced monographs.

Studies based on a meeting held at the Centro de Estudios Cientificos de Santiago, Dec. 17-20, 1987, review new developments in the field. Areas covered include: anomalous Jacobians and the vector anomaly; string phenomenology; quantum groups, integrable theories, and conformed models, small handles.

This book has been written twice. After having written and published it in German in 1990, I started allover again and rewrote the whole story for an English speaking audience. During the first round I received encouraging words and critical remarks from students and colleagues alike which have helped to sustain me the second time around. In the preface the author usually states that his or her book resulted from a course that he or she gave at some university. I cannot claim that the present book is any exception to the rule. But I expanded and remodelled the original material which circulated as a manuscript so that the printed version would follow a more stringent and coherent architectural plan. In doing so I have concentrated on the conceptual problems inherent in the path integral formalism rather than on certain highly specialized techniques used in applications. Nevertheless, I have also included those methods that are of fundamental interest and have treated specific problems mainly to illustrate them.

The first fifteen chapters of these lectures (omitting four to six chapters each year) cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically, the mathematics students have some background in advanced anal­ ysis, while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience, we decided to select material which is interesting from the viewpoint of modern theoretical physics, and which illustrates an interplay of ideas from various fields of mathematics such as operator theory, probability, differential equations, and differential geometry.

Imagine that two people have been chosen to be observers in a scien­ tific experiment involving two photons, or quanta of light. These pho­ tons originate from a single source and travel in opposite directions an equal distance halfway across the known universe to points where each will be measured or observed. Now suppose that before the pho­ tons are released, one observer is magically transported to a point of observation halfway across the known universe and the second ob­ server is magically transported to another point an equal distance in the opposite direction.

In light of the barrage of popular books on physics and cosmology, one may question the need for another. Here, two books especially come to mind: Steven Weinberg's The First Three Minutes, written 12 years ago, and the recent best-seller ABriefHistory of Time by Stephen Hawking. The two books are complementary. Weinberg-Nobel prize winner/physicist-wrote from the standpoint of an elementary particle physicist with emphasis on the contents of the universe, whereas Hawking wrote more as a general relativist with emphasis on gravity and the geometry of the universe. Neither one, however, presented the complete story.

Intended for beginning graduate students, this text takes the reader from the familiar coordinate representation of quantum mechanics to the modern algebraic approach, emphsizing symmetry principles throughout. After an introduction of the basic postulates and techniques, the book discusses time-independent perturbation theory, angular momentum, identical particles, scattering theory, and time-dependent perturbation theory. It concludes with several lectures on relativistic quantum mechanics and on many-body theory.

Foundational research focuses on the theory, but theories are to be related also to other theories, experiments, facts in their domains, data, and to their uses in applications, whether of prediction, control, or explanation. A theory is to be identified through its class of models, but not so narrowly as to disallow these roles. The language of science is to be studied separately, with special reference to the relations listed above, and to the consequent need for resources other than for theoretical description. Peculiar to the foundational level are questions of completeness (specifically in the representation of measurement), and of interpretation (a topic beset with confusions of truth and evidence, and with inappropriate metalinguistic abstraction).

Statistical Methods in Quantum Optics 2 - Non-Classical Fields continues the development of the methods used in quantum optics to treat open quantum systems and their fluctuations. Its early chapters build upon the phase-space methods introduced in the first volume Statistical Methods in Quantum Optics 1 - Matter Equations and Fokker-Planck Equations: the difficulties these methods face in treating non-classical light are exposed, where the regime of large fluctuations – failure of the system size expansion – is shown to be particularly problematic. Cavity QED is adopted as a natural vehicle for extending quantum noise theory into this regime. In response to the issues raised, the theory of quantum trajectories is presented as a universal approach to the treatment of fluctuations in open quantum systems.

Special relativity and quantum mechanics, formulated early in the twentieth century, are the two most important scientific languages and are likely to remain so for many years to come. In the 1920's, when quantum mechanics was developed, the most pressing theoretical problem was how to make it consistent with special relativity. In the 1980's, this is still the most pressing problem.