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In Srednicki QFT page 37. In the derivation of LSZ reduction formula, he introduces the time-order operator $T$, so no time-dependent creation/annihilation operators are left in the transition amplitude. How can this be justified mathematically? And if I understand this correctly, if time ordering is not used, then a term like $$\langle 0\mid a_{1}\left ( -\infty \right )a_{2}\left ( -\infty \right )a_{1^{\tilde{}}}^{\dagger}\left ( \infty \right )a_{2^{\tilde{}}}^{\dagger}\left ( \infty \right )\mid 0\rangle $$ this mean that there's a contribution that depend on the amplitude of transition from the final momenta to the initial momenta but quantum mechanically the final momenta are not known in advance so such a term can't contribute to the process. Does this explain the use of time-ordering physically?

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What do you mean by "justified mathematically"? –  Frederic Brünner Jan 21 '13 at 17:10
    
He, probably, means "obtained mathematically as a result of mathematical transformations". It might be there is an implicit deception there. –  Vladimir Kalitvianski Jan 21 '13 at 17:23
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