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I'm currently trying to learn more about the Lorentz- and Poincare Lie-algebras and the representation theory about them. But I'm really struggling with the material that we were given and I'm also having a hard time finding something suitable on the internet. Until now I've looked into:

  1. Notes on Quantum Mechanics, K. Shulten, Chapter 10.1
  2. Introduction to relativistic quantum field theory, R. Soldati, Chapters 1.2.6-1.2.8
  3. The Representation Theory of the Lorentz Group, Jackson Burzynski

Number [2] covers basically everything I need, but I'm having a hard time following his conclusions -- at the end of each section he just starts stating things as facts without giving any kind of explanation (for example, he mentions two dimensional left spinor Weyl representations, but doesn't explain anything about them). Number [1] was just to short, it didn't cover enough of the material needed and number [3] had a nice approach on deriving the Lie-algebra of the Lorentz-group but was, for me at least, not clear enough when talking about the representation theory (although a lot better than [1]). It obviously lacks the part about the Poincare group.

So what I'm looking for: Books, papers, etc. that explain how on can find the Lie-algebra of the above mentioned groups and show how we can find the representations which are used in QFT. I don't mind if the recommendation is repetitive or long, etc, as long as it explains these concepts well.

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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

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    $\begingroup$ I only want to let you know that there is an adjourned version (2019) of Soldati's notes on his site, where probably some mistakes are corrected. It's this one: chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/robertosoldati.com/archivio/news/107/Campi1.pdf $\endgroup$ – Run like hell Feb 2 at 23:43
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    $\begingroup$ Ryder’s QFT book has an excellent treatment of spinor reps of the Lorentz group. $\endgroup$ – bapowell Feb 3 at 15:58
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    $\begingroup$ An excellent account of the spinor reps of the Lorentz group is given in the first few subsections of "ideas and methods of supersymmetry and supergravity" by Buchbinder and Kuzenko. Weinbergs chapter 2.7 of his first volume of QFT is very good for the Unitary irreps of the Poincare group. You can also use subsection 1.5 and 1.8 of the first book I mentioned for this. $\endgroup$ – NormalsNotFar Feb 3 at 21:24
  • $\begingroup$ @NormalsNotFar Thanks for the recommendations, but could you maybe be a bit more specific on which book you mean by "Weinberg's"? I only found "Classical Solutions in Quantum Field Theory", from Erick J. Weinberg, in which chapter 2.7 hasn't really any connection to the above mentioned topics... $\endgroup$ – Sito Feb 3 at 21:50
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    $\begingroup$ It's called "The Quantum Theory of Fields: Volume 1" by Steven Weinberg. $\endgroup$ – NormalsNotFar Feb 3 at 22:06

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