One stupid question about Weinberg's Volume 1

[ap_nhp]

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

I don't agree with the $$-\sigma$$ in the result of space reversal transformation.

Can anyone explain it for me?

Thanks

 

[meopemuk]

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

I don't agree with the $$-\sigma$$ in the result of space reversal transformation.

Can anyone explain it for me?

Thanks

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

Eugene.

 

[ap_nhp]

$$\sigma$$ is an eigenvalue of the helicity operator $$(\mathbf{J} /cdot \mathbf{P})P^{-1}$$. This operator changes its sign under the space reversal transformation. Therefore, $$\sigma$$ 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 $$\sigma$$ in $${\Psi _{k,\sigma }}$$ as eigenvalue of $$J_{3}$$ but in $${\Psi _{p,\sigma }}$$ is helicity. That makes me confuse.

 

[ap_nhp]

$$\sigma$$ is an eigenvalue of the helicity operator $$(\mathbf{J} /cdot \mathbf{P})P^{-1}$$. This operator changes its sign under the space reversal transformation. Therefore, $$\sigma$$ 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 $$\sigma$$ change in process. Before, he used it as eigenvalue of $${J _ {3}}$$. 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.