J v2wpi2tt [ui2pi pQ itV 2wpi

where, anticipating four-momentum conservation, any dependence of the function / (the remaining factors in the integral) on the momentum p\ has been expressed in terms of the other momenta, {p;}. Note that the contour must be closed in the lower half plane because t+r is always negative, and the integrand is damped only when pio has a negative imaginary part. Similarly, for the final state projections chose an initial time t' later than T, so that T < t' —> +oo and

because t' + t will always be positive, again forcing the contour to be closed in the lower half plane. Hence, the action of each projection operator puts an external particle on-mass-shell and turns the corresponding external propagator into a factor of i^/njuj. The emergence of these mass shell poles, which are the lowest energy states for the external particles, is a practical application of the discussion leading up to Fig. 14.4. Now, let T —* oo and reconstruct the delta function, giving

where

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