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On p. 71 below eq. (6.12) in Schwartz book, they assume that $$\lim_{t \to \pm\infty}\partial_0 a_p(t)=0.\tag{1}$$ But I thought that this is just an 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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Schwartz's treatment of the LSZ construction is somewhat misleading:

  1. Schwartz fails to mention that the Fourier expansion $$ \hat{\phi}(x)~=~\int\! \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}} \left[\hat{a}_p(t)e^{-ip\cdot x}+\hat{a}^{\dagger}_p(t)e^{ip\cdot x} \right] \tag{6.7}$$ only applies to a free real field.

  2. It is not properly explained that $$\sqrt{2\omega_p} \hat{a}_p(t) = i \int \!d^3x~ e^{ipx}\stackrel{\leftrightarrow}{\partial}_0 \hat{\phi}(x)\tag{6.13}$$ is the actual definition of a (time-dependent) asymptotic annihilation operator $\hat{a}_p(t)$. A standard calculation then yields that $$\sqrt{2\omega_p} \partial_0\hat{a}_p(t) = i \int \!d^3x~ e^{ipx}(\Box+m^2) \hat{\phi}(x).$$ This vanishes for free fields where $\hat{a}_p(t)$ belongs to the Schrödinger picture and is therefore time-independent.

    OP's main question about $\partial_0\hat{a}_p(\pm \infty)$ therefore directly ties to the validity of the basic adiabatic assumption of the LSZ formalism. See also this & this related Phys.SE posts.

  3. Below eq. (6.7) it is wrongly asserted that $\hat{a}_p(t)$ belongs to the Heisenberg picture.

  4. Below eq. (6.8) it seems that Schwartz assumes that the theory is free for $|t|>T$. In fundamental QFT the interactions are not turned off at asymptotic regions: the Lagrangian does not depend explicitly on spacetime. In particular, $\epsilon=0$ in the Gell-Mann & Low theorem. For any finite $\epsilon>0$, not surprisingly $\partial_0\hat{a}_p(\pm \infty)=0$.

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).
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