The discussion revolves around the transformation properties of the Dirac equation and the associated gamma matrices under Lorentz transformations. Participants explore the implications of these transformations on the spinor and the differential operator, addressing questions of invariance and the correct application of transformation rules.
Participants express differing views on the transformation properties of the gamma matrices and the implications for Lorentz invariance. There is no consensus on the interpretation of the transformation symbol ##\to##, leading to further discussion and clarification.
Some participants highlight the potential for confusion regarding the transformation of the spinor and the implications of the Lorentz transformation being invertible. The discussion reflects varying levels of familiarity with the concepts involved, particularly among those self-studying.
vanhees71 said:A very good book about all the group-theoretical aspects is
R. U. Sexl, H. K. Urbantke, Relativity, Groups, Particles,
Springer, Wien (2001).
(A , B) \otimes (A^{\prime} , B^{\prime}) = (A + A^{\prime} , B + B^{\prime}) \oplus (A + A^{\prime} - 1 , B + B^{\prime}) \oplus \cdots \oplus (|A - A^{\prime}| , B + B^{\prime}) \oplus (A + A^{\prime} , B + B^{\prime} - 1) \oplus \cdots \oplus (|A - A^{\prime}| , |B - B^{\prime}|) .MichaelJ12 said:How did you go from the third line to the fourth line?
Almost all books on group representations in general and the representation theory of the Lorentz group \mbox{SL}(2 , \mathbb{C}) in particular teach you these stuff. The latter can be found in all textbooks on Supersymmetry.Could you recommend a good book?
samalkhaiat said:(A , B) \otimes (A^{\prime} , B^{\prime}) = (A + A^{\prime} , B + B^{\prime}) \oplus (A + A^{\prime} - 1 , B + B^{\prime}) \oplus \cdots \oplus (|A - A^{\prime}| , B + B^{\prime}) \oplus (A + A^{\prime} , B + B^{\prime} - 1) \oplus \cdots \oplus (|A - A^{\prime}| , |B - B^{\prime}|) .
Almost all books on group representations in general and the representation theory of the Lorentz group \mbox{SL}(2 , \mathbb{C}) in particular teach you these stuff. The latter can be found in all textbooks on Supersymmetry.
I have already said that you can learn about this stuff from all textbooks on supersymmetry. I just know this stuff because it is my business to know it.MichaelJ12 said:I meant where did you learn the physical meaning and significance of the gamma matrices projecting out certain spins.
Then all textbooks by Weinberg are right for you :-)).MichaelJ12 said:Thank you very much. I love those kinds of books. Have you read other books that belong to the same category as Relativity, Groups, Particles, meaning that they are meant to impart understanding? I only want books about "advanced" physics or mathematics like quantum field theory, general relativity, differential geometry, topology, and any other branches of pure mathematics.