I really wonder about the time derivative of creation and annihilation operators in the derivation of LSZ

In Schwartz book, they assume that $$\lim_{t \to \pm\infty}\partial_0 a_p(t)=0.\tag{1}$$ But I thought that is just assumption. so we have to construct the mathematical description. I found the Gell-Mann and Low theorem. In that theorem, full Hamiltonian is $$H(t)=H_0+e^{-\epsilon|t|}V(0)\tag{2}$$
and time evolution operator is $$U(t,t')=T\exp\bigg[-i\int^{t}_{t'}dt~H(t)\bigg]\tag{3}$$ and $$U_I(t,t')=T\exp\bigg[-i\int^{t}_{t'}dt~e^{-\epsilon|t|}V_I(t)\bigg].\tag{4}$$ So I thought that $$\lim_{t \to \pm\infty}\partial_0 a_p(t)= iU_I^\dagger(\pm\infty,0)e^{-\epsilon|\pm\infty|}[V(\infty),a_p]U_I(\pm\infty,0).\tag{5}$$ and since $$t$$ is going to infinity, that is 0. by the way, I fell in another problem. If I deal with that time evolution operator as the above, when we use the LSZ formula and $$n$$-point function, I'll meet the $$e^{-\epsilon|t|}$$. the function will interrupt what delta function about energy is made. what is my fault? please tell me about my incorrect logic. I spend the time a lot due to this problem.

• Are you following a reference? Which pages? – Qmechanic Jul 12 at 16:05
• I'm following Peskin,Schwartz book, and Wiki – 정재훈 Jul 13 at 9:16

1. The annihilation operator $$\hat{a}_p(t)$$ are defined as Fourier coefficient of the Heisenberg picture field $$\hat{\phi}(x)$$, $$\hat{\phi}(x)~=~\int\! \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}} \left[\hat{a}_p(t)e^{-ip\cdot x}+\hat{a}^{\dagger}_p(t)e^{ip\cdot x} \right],\tag{6.7}$$ where $$p\cdot x=\omega_p t- \vec{p}\cdot \vec{x}$$. Ref. 1 is assuming that interactions are truncated in temporal asymptotic regions. There the annihilation operator $$\hat{a}_p(t)$$ belongs to the Schrödinger picture and is therefore time-independent.
2. A related approach can be found in Ref. 2 where $$\hat{a}_p(t)~=~i\int\! d^3x ~e^{ip\cdot x}\stackrel{\leftrightarrow}{\partial_0}\hat{\phi}(x) \tag{3-44}$$ is a free in (or out) annihilation operator by definition, and therefore time-independent. Further steps are standard: Match in/out operators with interacting operators at temporal asymptotic regions. See Ref. 2 for details.