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On page 112, he tries to show why "+ie" term is the correct choice to add in the denominator of the Lippmann-Schwinger equations for the scattering "in" states. The proof is based on the residue theorem of a complex variable. The idea is to show the integral associated with the Lippmann-Schwinger equations approaches zero when the time tends to -infinity.

So for this purpose and with the energy as the variable of integration, Weinberg uses the residue theorem and extends its integration domain to a semi-circle contour on the upper complex half-plane. Then he argues that integral is indeed zero when the energy goes to +/- infinity and time goes to -infinity.

Here is the part that I don't understand. Physically, energy should have a lower bound, so the domain of integration of the integral associated with the Lippmann-Schwinger equations should be from the finite energy lower bound to +infinity. But the integral that Weinberg proves zero is an integral with energy running from -infinity to +infinity.

On page 114, Weinberg uses similar argument.

So my question is why Weinberg sets the energy integration range from -infinity to +infinity but not from a finite lower bound to +infinity?