The discussion centers on the transformation properties of the Dirac equation, specifically the term ##\gamma^\mu \partial_\mu## and its behavior under Lorentz transformations. Participants clarify that while the derivative transforms as ##\partial_\mu \to (\Lambda^{-1})^\nu_{\; \mu} \partial_\nu##, the gamma matrices ##\gamma^\mu## remain constant matrices and do not undergo transformation. The key takeaway is that the expression ##\gamma^\mu \partial_\mu \psi## is Lorentz invariant due to the appropriate transformation of the spinor ##\psi##, not the operator itself. Misunderstandings regarding the notation and implications of transformations are also addressed.
PREREQUISITESPhysicists, particularly those specializing in quantum mechanics and quantum field theory, as well as students studying special relativity and particle physics.
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.