One stupid question about Weinberg's Volume 1

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When I read quantum field theory in Weinberg's Volume 1. In equation 2.6.22 :
[tex]P{\Psi _{p,\sigma }} = {\eta _\sigma }\exp ( \mp i\pi \sigma ){\Psi _{{\cal P}p,{-\sigma} }}[/tex]

I don't agree with the [tex]-\sigma[/tex] in the result of space reversal transformation.

Can any one explain it for me?

Thanks
 
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When I read quantum field theory in Weinberg's Volume 1. In equation 2.6.22 :
[tex]P{\Psi _{p,\sigma }} = {\eta _\sigma }\exp ( \mp i\pi \sigma ){\Psi _{{\cal P}p,{-\sigma} }}[/tex]

I don't agree with the [tex]-\sigma[/tex] in the result of space reversal transformation.

Can any one explain it for me?

Thanks
[tex]\sigma[/tex] is an eigenvalue of the helicity operator [tex](\mathbf{J} /cdot \mathbf{P})P^{-1}[/tex]. This operator changes its sign under the space reversal transformation. Therefore, [tex]\sigma[/tex] also changes its sign. Don't you agree with that?

Eugene.
 
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[tex]\sigma[/tex] is an eigenvalue of the helicity operator [tex](\mathbf{J} /cdot \mathbf{P})P^{-1}[/tex]. This operator changes its sign under the space reversal transformation. Therefore, [tex]\sigma[/tex] also changes its sign. Don't you agree with that?

Eugene.
Thank meopemuck. I 've read again. And I see that the definition of Weinberg is little bit change in the process. He used [tex]\sigma[/tex] in [tex]{\Psi _{k,\sigma }}[/tex] as eigenvalue of [tex] J_{3}[/tex] but in [tex]{\Psi _{p,\sigma }}[/tex] is helicity. That makes me confuse.
 
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[tex]\sigma[/tex] is an eigenvalue of the helicity operator [tex](\mathbf{J} /cdot \mathbf{P})P^{-1}[/tex]. This operator changes its sign under the space reversal transformation. Therefore, [tex]\sigma[/tex] also changes its sign. Don't you agree with that?

Eugene.
Thank meopemuck. I've read Weinberg again. And I see that the definition of [tex]\sigma[/tex] change in process. Before, he used it as eigenvalue of [tex]{J _ {3}}[/tex]. And then he used it as helicity. That makes me confuse. But now I understand the reason of his definition.

Thank you one more time.
 

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