Lectures On Quantum Theory Mathematical And Structural Foundations

This book is based on material taught to final-year physics undergraduates as part of the theoretical physics option at Imperial College. After a self-contained introduction to the essential ideas of vector spaces and linear operators, a bridge is built between the concepts and mathematics of classical physics, and the new mathematical framework employed in quantum mechanics. The axioms of nonrelativistic quantum theory are introduced, and shown to lead to a variety of new conceptual problems. Subjects discussed include state-vector reduction, the problem of measurement, quantum entanglement, the Kochen-Specker theorem, and the Bell inequalities. The book includes twenty-five problems with worked solutions.

These notes are the contents of a lecture course given to third year physics undergraduates at the Imperial College who are taking the theoretical physics option. The subject of ?Algebra and Groups? is of considerable importance in a number of branches of modern theoretical physics, and therefore one major objective of the course is to introduce the students to the basic ideas on the subject, bearing in mind the potential applications to quantum theory. However, another equally important aim of the course is to introduce the student to the art of genuine ?mathematical? thinking. The notes are therefore written in a more precise mathematical style than is usually the case in courses aimed at physics students. Quite apart from the general educational value of such an exposure to abstract thinking, it is also the case that much modern theoretical physics draws on sophisticated ideas from pure mathematics and therefore it is most important that a perspective graduate student can approach these subjects without experiencing a total culture shock! The course is divided into three parts. The first is a short introduction to general group theory, with particular emphasis being placed on the matrix Lie groups that play such a crucial role in modern theoretical physics. The second part deals with the theory of vector spaces, with particular attention being paid to the theory of Hilbert spaces and the basic analytical techniques that are needed to handle the infinite dimensional situation. The final part of the course is a short introduction to the theory of group representations and the associated theory of characters.

Quantum gravity is the name given to a theory that unites general relativity - Einstein's theory of gravitation and spacetime - with quantum field theory, our framework for describing non-gravitational forces. The Structural Foundations of Quantum Gravity brings together philosophers and physicists to discuss a range of conceptual issues that surface in the effort to unite these theories, focusing in particular on the ontological nature of the spacetime that results. Although there has been a great deal written about quantum gravity from the perspective of physicists and mathematicians, very little attention has been paid to the philosophical aspects. This volume closes that gap, with essays written by some of the leading researchers in the field. Individual papers defend or attack a structuralist perspective on the fundamental ontologies of our physical theories, which offers the possibility of shedding new light on a number of foundational problems. It is a book that will be of interest not only to physicists and philosophers of physics but to anyone concerned with foundational issues and curious to explore new directions in our understanding of spacetime and quantum physics.

The Advanced School on Quantum Foundations and Open Quantum Systems was an exceptional combination of lectures. These comprise lectures in standard physics and investigations on the foundations of quantum physics.On the one hand it included lectures on quantum information, quantum open systems, quantum transport and quantum solid state. On the other hand it included lectures on quantum measurement, models for elementary particles, sub-quantum structures and aspects on the philosophy and principles of quantum physics.The special program of this school offered a broad outlook on the current and near future fundamental research in theoretical physics.The lectures are at the level of PhD students.

In many ways the last decade has witnessed a surge of interest in the interplay between theoretical physics and some traditional areas of pure mathematics. This book contains the lectures delivered at the NATO-ASI Summer School on `Recent Problems in Mathematical Physics' held at Salamanca, Spain (1992), offering a pedagogical and updated approach to some of the problems that have been at the heart of these events. Among them, we should mention the new mathematical structures related to integrability and quantum field theories, such as quantum groups, conformal field theories, integrable statistical models, and topological quantum field theories, that are discussed at length by some of the leading experts on the areas in several of the lectures contained in the book. Apart from these, traditional and new problems in quantum gravity are reviewed. Other contributions to the School included in the book range from symmetries in partial differential equations to geometrical phases in quantum physics. The book is addressed to researchers in the fields covered, PhD students and any scientist interested in obtaining an updated view of the subjects.

Four concise, brilliant lectures on mathematical methods in quantum mechanics from Nobel Prize–winning quantum pioneer build on idea of visualizing quantum theory through the use of classical mechanics.

A leisurely but mathematically honest presentation of quantum mechanics for graduate students in mathematics with an interest in physics.