About eq. (5.7.23) in Weinberg's The quantum theory of fields vol. I

[diraq]

On page 237, Weinberg checked eq. (5.7.23) with an example when $\mathbf p$ is along the three direction. Below that equation the phase factor $\exp([-a + b - \tilde{a} + \tilde{b}]\theta)=\exp([2\tilde b-2a]\theta)$.

Under the transformation
$$p^0\rightarrow -p^0;\mathbf p\rightarrow -\mathbf p,$$
the phase factor becomes $(-1)^{2\tilde b-2a}\exp([2\tilde b-2a](-\theta))$. The major difference is that $\exp(\pm\theta d)$ should be transformed into $(-1)^d\exp(\mp\theta d)$ for any integer $d$. This cannot lead to the conclusion of eq. (5.7.23). Please enlighten me on this issue. Thanks.

 

[azntoon]

The transformation p^0\rightarrow -p^0;\mathbf p\rightarrow -\mathbf p does not change the phase factor \exp([-a + b - \tilde{a} + \tilde{b}]\theta)=\exp([2\tilde b-2a]\theta). This is because under this transformation, the four terms in the exponential [-a + b - \tilde{a} + \tilde{b}] remain unchanged. However, under the transformation p^0\rightarrow -p^0;\mathbf p\rightarrow -\mathbf p, the phase factor becomes (-1)^{2\tilde b-2a}\exp([2\tilde b-2a](-\theta)). The major difference is that \exp(\pm\theta d) should be transformed into (-1)^d\exp(\mp\theta d) for any integer d, which leads to the conclusion of eq. (5.7.23).