Query in Zeidler's Volume II QFT

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Discussion Overview

The discussion revolves around a query regarding a specific claim made in Zeidler's Quantum Field Theory Volume II, particularly on the application of spin operators to a wave function. Participants are exploring the implications of the operator notation and its effects on the wave function, focusing on the contributions of different components of the spin operator.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions why the terms S^1 and S^2 do not contribute when applying the spin operator S to the wave function Ψ^+_{p,s}, despite the S^3k term yielding a contribution of skΨ^+_{p,s}.
  • Another participant suggests that defining the terms involved in the equality could facilitate better understanding and responses from others knowledgeable in QFT.
  • One participant mentions uploading images of relevant pages from the book to clarify the notation, indicating the complexity of the material.
  • There is a discussion about the nature of the S^1 operator, with one participant asserting it is a 4x4 matrix with specific 2x2 matrices as diagonal blocks, and questioning its effect on the wave function.
  • Another participant reiterates the understanding of S^1 and raises a question about its application to an eigenstate of gamma^3, seeking clarification on the implications of this application.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the application of the spin operators, with some seeking clarification and others providing insights. There is no consensus on the contributions of the different terms of the spin operator to the wave function.

Contextual Notes

Participants note the complexity of the notation involved, which may hinder clear communication. The discussion reflects uncertainty regarding the mathematical operations and their implications on the wave function.

Avogadro Number
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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!
 
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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 said:
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!
 

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Avogadro Number said:
@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)
 
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!
 
Avogadro Number said:
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 ?
 

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