## About Physics 729

This course will present the foundations of theoretical physics with the use of Lie algebras and Lie Groups. Generally, this course assumes a maturity in mathematics and physics rather than a particular body of knowledge. Although the formal requirements are only advanced calculus, matrix theory, and a core of undergraduate physics; one will not appreciate the course as much without a good framework to which to attach the work. It would be helpful if the student also had completed a year of graduate work in classical mechanics (including relativity), quantum mechanics, and electromagnetic theory but these are not absolutely necessary. We will review fundamental concepts both in mathematics and in physics to make sure that the student understands the notation and starting points of the course. It is my objective to make this course as easy as is humanly possible – to create the course that I wish that I had had.

It is my intent to continue to expand and to hyperlink new material, problems, seminars, special lectures, research articles, and publications to form an evolving and dynamic graduate course in theoretical physics. Some extra topics will present more advanced and specialized material. Some topics will present material that is remedial in nature to support the main thrust of the course. Some material will extend the course at the same level with more examples, problems, and discussion. My eventual objective is to have a coherent body of material that will serve as a foundation for core areas of theoretical physics and at the same time to encapsulate concepts that a graduate student would need in order to work in research in any area of theoretical physics.

I will begin with an introduction and a foundational orientation. Core concepts in mathematics, especially modern algebra, classical mechanics, relativity, and electromagnetic theory will be reviewed. Next, we will cover an elementary presentation of group theory and then launch immediately into the most important Lie groups and Lie algebras. Thus I will first give just the essential concepts of Lie groups and algebras, and then immediately apply these concepts to the most important Lie Algebras and thus teach by doing. I will thereby derive the foundations of quantum theory, angular momentum, relativity, relativistic quantum theory, second quantization, and basic concepts in internal symmetries. We will explicitly study the Heisenberg, rotation, Lorentz, Poincare, unitary, general linear, affine, Markov and other related groups. Then later in the course, I will derive the more formal and technical foundations of semisimple and non-semisimple Lie groups and algebras. Finally I will explore some extensions of current concepts with the general linear group, and the Markov subgroup, and then apply these ideas to a theory of uncertainty in areas of logic, number systems, information, network (graph) topology and quantum theory. A number of problems will be described that can serve as research areas, some straight forward and some conceptually difficult.

### How It Works

This course will consist of a collection of lectures on DVDs and then placed on the Web server, in 25 minute lengths or less. I have found that I can cover a 50 minute lecture in 25 minutes if I can proceed with the knowledge that the student can back up the tape and review the material as required. All of the resulting material will be printed and bound in a manual for distribution to students as well as available on the associated web site. This has the consequence that the student does not spend time copying equations and making elaborate notes but can use this book to add comments and remarks for their individual use. The student will soon realize that I truly cover the content of a 50 minute lecture in 25 minutes and thus a 25 minute lecture constitutes a full class meeting and is highly compressed. By using video tape, I do not have to spend time waiting for the student to copy material (they already have it), nor to repeat (one can simply back up the disk), nor to ask questions, nor to wait on students to all arrive in class, nor to discuss the time of the next test, which days class will be held and which are holidays. In other words, this is the essence of the information – highly concentrated and at high speed. Questions and discussions can be held with me at other times each week. At that time, students can request other modules, both remedial and advanced. They can also point out deficiencies and mistakes which they find. This modular form will allow me to revise and edit tapes: As time passes, I may wish to redo a lecture and it can then be substituted for the original. Lectures may have mistakes of different forms and I will have to live with that for a while if I am to ever finish. These tapes will not have the polish of commercial presentations but a more informal ‘one-on-one’ atmosphere of presentation. I will strive for a core completeness first and then gradually improve the quality and also expand the scope. A hyperlinked type structure will allow later additions of more extensive topics and discussions.

Lie Algebra Notes pt1 Lie Algebra Notes pt2 Lie Algebra Notes pt3