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 [itex]\mathbf p[/itex] is along the three direction. Below that equation the phase factor [itex]\exp([-a + b - \tilde{a} + \tilde{b}]\theta)=\exp([2\tilde b-2a]\theta)[/itex].

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