I was reviewing the first few chapters of Weinberg VolI and found a hole in my understanding in page 112, where he tried to show in the asymptotic past [itex]t=-\infty[/itex], the in states coincide with a free state. In particular, he argued the integral [tex]\int d\alpha\frac{e^{-iE_{\alpha}t}g(\alpha)T_{\beta\alpha}^+\Phi_\beta}{E_\alpha-E_\beta+i\epsilon}\ldots(1)[/tex] would vanish, where [itex]d\alpha=d^3\mathbf{p}[/itex](also involves discrete indices like spin, but of no relevance here). In his argument, he used a contour integration in the complex [itex]E_\alpha[/itex] plane, in which the integral of central interest is the integration along real line [tex]\int_{-\infty}^\infty dE_\alpha\frac{e^{-iE_{\alpha}t}g(\alpha)T_{\beta\alpha}^+\Phi_\beta}{E_\alpha-E_\beta+i\epsilon}\ldots(2)[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

I don't see how to obtain (2) from (1), since the lower bound of energy is the rest mass, in the best case I could get something like [itex]\int_{m}^\infty dE_\alpha\cdots[/itex], but how could one extend this onto the whole real line.

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# Confusion (8) from Weinberg's QFT.(Lippmann-Schwinger eqn)

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