Why does this term transform in this way?

[MichaelJ12]

A very good book about all the group-theoretical aspects is

R. U. Sexl, H. K. Urbantke, Relativity, Groups, Particles,
Springer, Wien (2001).

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.

 

[samalkhaiat]

How did you go from the third line to the fourth line?

$$(A , B) \otimes (A^{\prime} , B^{\prime}) = (A + A^{\prime} , B + B^{\prime}) \oplus (A + A^{\prime} - 1 , B + B^{\prime}) \oplus \cdots \oplus (|A - A^{\prime}| , B + B^{\prime}) \oplus (A + A^{\prime} , B + B^{\prime} - 1) \oplus \cdots \oplus (|A - A^{\prime}| , |B - B^{\prime}|) .$$

Could you recommend a good book?

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.

 

[MichaelJ12]

$$(A , B) \otimes (A^{\prime} , B^{\prime}) = (A + A^{\prime} , B + B^{\prime}) \oplus (A + A^{\prime} - 1 , B + B^{\prime}) \oplus \cdots \oplus (|A - A^{\prime}| , B + B^{\prime}) \oplus (A + A^{\prime} , B + B^{\prime} - 1) \oplus \cdots \oplus (|A - A^{\prime}| , |B - B^{\prime}|) .$$

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 meant where did you learn the physical meaning and significance of the gamma matrices projecting out certain spins.

 

[samalkhaiat]

I meant where did you learn the physical meaning and significance of the gamma matrices projecting out certain spins.

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.

 

[vanhees71]

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.

Then all textbooks by Weinberg are right for you :-)).