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I am trying (unsuccessfully) to verify this relation regarding the helicity of Dirac spinors: $$ { \sigma }_{ \vec { p } }u_{ r }\left( \vec { p } \right) =\frac { \vec { \Sigma } \cdot \vec { p } }{ \left| \vec { p } \right| } u_{ r }\left( \vec { p } \right) =\left( -1 \right) ^{ r+1 }u_{ r }\left( \vec { p } \right) $$ Further details can be found in the book by F.Mandl and G.Shaw "Quantum Field Theory" at page 60 equation 4.35. I am using the Dirac representation for the Gamma Matrices but I am starting wondering if this relation could be representation dependent (I have tried also to compute it in the Weyl representation but again unsuccessfully). I calculated it both by hand and using Mathematica with the same negative outcome.

I noticed that this equation is valid if one assumes to be in the reference frame in which the impulse p is directed along the z axis p = (0,0,p3).

My questions are: is that relation dependent of the gamma matrices representation? is that relation dependent of the reference frame?

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  • $\begingroup$ Welcome to Physics! Please note that "derive this equation for me" type questions are considered off-topic. Could you rephrase your question to be more about the concept that's troubling you, rather than about the mathematics involved? $\endgroup$
    – Kyle Kanos
    Jun 19, 2015 at 12:17
  • $\begingroup$ I am sorry, this is the first time I am asking something. I will take more care in future. $\endgroup$ Jun 19, 2015 at 16:10

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