Query in Zeidler's Volume II QFT

[Avogadro Number]

Hello!

I am studying Zeidler's QFT Volume II, and I have a query on page 808:
It is claimed that
S Ψ^+_{p,s} = (sk)Ψ^+_{p,s} when p=p^3 k.
I tried my hand at deriving this, but when we write S=S^1i+S^2j+S^3k,
then the S^3k term acting on Ψ^+_{p,s} does give skΨ^+_{p,s},
but I don't see why the S^1 i and S^2 j terms don't give any contribution.
To those who are knowledgeable and happen to have access to the book,
could you please help me out? Many thanks!

 

[dextercioby]

I am sure there are far more people browsing the internet knowledgeable of QFT who do not have an electronic/a paper copy of this book than the ones who do, so if you can at least define the terms of the equality, it would increase your chances of receiving an answer.

 

[Avogadro Number]

I am sure there are far more people browsing the internet knowledgeable of QFT who do not have an electronic/a paper copy of this book than the ones who do, so if you can at least define the terms of the equality, it would increase your chances of receiving an answer.

@dextercioby: Yes, thanks for the suggestion. I should have done it the first time round, but the notation is rather heavy, and it is hard to type it all. So I have uploaded the images of 3 relevant pages. :) I wonder it it will help. Thanks!

 

[nrqed]

@dextercioby: Yes, thanks for the suggestion. I should have done it the first time round, but the notation is rather heavy, and it is hard to type it all. So I have uploaded the images of 3 relevant pages. :) I wonder it it will help. Thanks!

$S^1$ contains $\sigma^{23}$ ,right? What is the result of applying this to the wave function (which is an eigenstate of the spin in the z direction)

 

[Avogadro Number]

Yes, my understanding was that S^1 is the 4x4 matrix with the 2x2 matrices (1/2)*sigma^1 as its diagonal blocks.
Then if what Zeidler's claim is true, this S^1 ought to kill the 4x1 column vector u appearing in the wave function, but it does not.
What am I doing wrong? Thanks!

 

[nrqed]

Yes, my understanding was that S^1 is the 4x4 matrix with the 2x2 matrices (1/2)*sigma^1 as its diagonal blocks.
Then if what Zeidler's claim is true, this S^1 ought to kill the 4x1 column vector u appearing in the wave function, but it does not.
What am I doing wrong? Thanks!

S^1 is sigma^(23). What happens if we apply this to an eigenstate of gamma^3 ?