An Introduction to Tensors and Group Theory for Physicists

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There are a few topics that I regret that we did not cover. A sketch of the solution to the twin paradox on a compact space. For our goals, I am not aware of any single textbook that introduces the topic at a pace and level appropriate for our class. I suggest using a combination of the following references. The lectures are meant to be connective tissue that gives an overarching narrative for what we are doing and why. Georgi, Lie Algebras in Particle Physics.

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It is convenient because it contains derivations of the main results. This is a bit more terse and goes beyond the scope of the course.

It is available free from the author. Zee, Group Theory in a Nutshell for Physicists. This is one of my favorite books for its conversational and pedagogical tone.

ISBN 13: 9780817647148

Introduces the geometry of Lie groups in careful detail. We will use this as a primary reference for tensor products. Familiarity with linear algebra at the level of Physics Quantum Mechanics. No background or primary interest in particle physics, field theory, or abstract algebra necessary. This is not only useful for general physics but it is what you have to master to fully understand quantum computing which goes on in the product space of qubits. This brings the first part of the book to a close.

I. General Information and Syllabus

It is short and not too theorem-proof bound to be impossible to understand. If you do read it then be prepared to think quite hard about some of the examples which tend to be more advanced than the main text. The second part of the book dives into groups without too much motivation. The whole aim is to get on to Lie groups as quickly as possible.

I can imagine that if this is your first introduction to groups it might be too fast. The second problem is that the important groups of classical and quantum physics are introduced equally quickly and the problem is that there is no complete list of symbols.

An introduction to tensors and group theory for physicists

What this means is that you inevitably end up having to go back and discover what SL or M is. Not too much of a problem, unless this is your first exposure to the nomenclature and you read the book in short bursts.

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  4. From here we move on to Lie algebra and its key ideas, such as the exponential map and so on. If you are prepared for it then the book will get you to understand that the Lie algebra is the infinitesimal form of the group, the tangent space of the group as a manifold and that the connection between the infinitesimal and the macro is the exponential map.

    However it is still possible to get lost in the detail. Once we have the basics over the next topic is representation theory. This is introduced with very little motivation. It basically doesn't tell you why we are interested in representations - from the physics point of view.