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I am currently reading Coleman's lecture note on QFT.(https://arxiv.org/abs/1110.5013) I have several questions regarding the scattering theory. Let $\phi$ be a real scalar field, and consider the interaction Hamiltonian $H_I(x)=g\phi(x)\rho(\vec{x})f(t)$, where $f$ represents the adiabatic turning on/off function.

The following is pp.91-92 of the lecture note:

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Questions:

  1. What is the difference between the vacuum with respect to $H_0$, and the vacuum with respect to $H_0 + H_I$? In the lecture note, these two vacuums are denoted by different notations $|0\rangle$ and $|0\rangle_P$. What is their difference?

  2. Somewhat ad hoc counterterm $a=E_0$ was added to $H_I$ to fix the divergent phase in $\langle0|S|0\rangle $. Adding $a$ will fix the divergent phase, but I think it does not necessarily means that $\langle0|S|0\rangle =1$. (As described in the lecture note, the correct expression should be $\langle0|S|0\rangle=e^{-i(\gamma_- +\gamma_+)}$.

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  1. The states $$|0\rangle~\in~{\cal H}_0 \quad\text{and}\quad |0\rangle_P~\in~{\cal H}$$ are the ground states for the Hamiltonians $\hat{H}_0$ and $\hat{H}$, respectively. More explicitly, using Coleman's notation p. 72-77, we have in the Schrödinger picture
    $$ |0 (t)\rangle_S~=~|0 (t)\rangle_S^{\rm in} ~=~e^{-i\gamma_-}|0 (t)\rangle_{P,S}\quad\text{for}\quad t\lesssim -\frac{T}{2}$$ and
    $$ |0 (t)\rangle_S~=~|0 (t)\rangle_S^{\rm out} ~=~e^{i\gamma_+}|0 (t)\rangle_{P,S}\quad\text{for}\quad t\gtrsim \frac{T}{2}.$$

  2. The phase $e^{-i(\gamma_- +\gamma_+)}$ is apparently absorbed via the $O(T/\Delta)$ counterterm.

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