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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.

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  • $\begingroup$ Are you following a reference? Which pages? $\endgroup$ – Qmechanic Jul 12 at 16:05
  • $\begingroup$ Yes I red and red a reference $\endgroup$ – 정재훈 Jul 13 at 6:38
  • $\begingroup$ I'm following Peskin,Schwartz book, and Wiki $\endgroup$ – 정재훈 Jul 13 at 9:16
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  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.

References:

  1. M.D. Schwartz, QFT & the standard model, 2014; p. 22 eq. (2.78); p. 23 eq. (2.81); p. 71 below eqs. (6.11) & (6.12).

  2. C. Itzykson & J.B. Zuber, QFT, 1985, eq. (3-44); Section 5-1-3, p. 204-208.

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