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- I am having a little confusion regarding the limit taken in Peskin and Schroeders quantum field theory book in ch. 4.

In P&S, it is shown that $$e^{-iHT}\ket{0}=e^{-iH_{0}T}\ket{\Omega}\bra{\Omega}\ket{0}+\sum_{n\neq 0}e^{-iE_nT}\ket{n}\bra{n}\ket{0}$$.

It is then claimed that by letting $$T\to (\infty(1-i\epsilon)) $$ that the other terms die off much quicker than $$e^{-iE_0T}$$, but my question is why is this the case? For example, why wouldn't the other terms also die off quicker if we simply sent $$T\to \infty$$ instead? Perhaps there is something about the limit of complex numbers I'm not understanding. Any insight would be appreciated. Thanks.

It is then claimed that by letting $$T\to (\infty(1-i\epsilon)) $$ that the other terms die off much quicker than $$e^{-iE_0T}$$, but my question is why is this the case? For example, why wouldn't the other terms also die off quicker if we simply sent $$T\to \infty$$ instead? Perhaps there is something about the limit of complex numbers I'm not understanding. Any insight would be appreciated. Thanks.

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