Proof of quantum correlation functions

[victorvmotti]

Reading through David Tong lecture notes on QFT.On pages 76, he gives a proof on correlation functions . See below link:

[QFT notes by Tong][1] [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdfI am following the proof steps to obtain equation (3.95). But several intermediate steps of the proof are not clear.

**First question**

why we can write:

$$T\phi_{1I} \dots \phi_{nI}S=U_{I}(+\infty, t_{1})\phi_{1I}U(t_{1},t_{2})\phi_{2I}\dots \phi_{nI}U_{I}(t_{n},-\infty)$$

I mean after dropping the $$T$$ shouldn't we have?:

$$=\phi_{1I}\phi_{2I}\dots \phi_{nI}S$$$$=\phi_{1I}\phi_{2I}\dots \phi_{nI}U_{I}(+\infty,-\infty)$$

Does $$T$$ operate on the $$\phi_{1}\dots\phi_{n}$$ only or the $$\phi_{1}\dots \phi_{nI}S$$ and $$U_{I}(+\infty,-\infty)=U_{I}(+\infty, t_{1})U_{I}(t_{1},t_{2})\dots U_{I}(t_{n},-\infty)$$?**Second question**

How we convert each of the $$\phi_{I}$$ into $$\phi_{H}$$ using $$U_{I}(t_{k},t_{k+1})=Texp(-i\int_{t_{k}}^{t_{k+1}}H_{I})$$ to arrive at

$$T\phi_{1I} \dots \phi_{nI}S=U_{I}(+\infty, t_{0})\phi_{1H}\dots \phi_{nH}U_{I}(t_{0},-\infty)$$

**Third question**

Why we have

$$U_{I}(t, -\infty)=U(t,-\infty)$$

 

[eljose79]

and

$$U_{I}(+\infty, t)=U(+\infty, t)$$

The reason we can write:

$$T\phi_{1I} \dots \phi_{nI}S=U_{I}(+\infty, t_{1})\phi_{1I}U(t_{1},t_{2})\phi_{2I}\dots \phi_{nI}U_{I}(t_{n},-\infty)$$

is because of the time ordering operator $$T$$, which ensures that the operators are arranged in the correct chronological order. In this case, the time ordering operator acts on the operators $$\phi_{1I}\dots \phi_{nI}S$$, which includes both the interaction picture operators ($$\phi_{1I}\dots \phi_{nI}$$) and the scattering operator $$S$$. Therefore, the time ordering operator $$T$$ operates on the entire expression, not just on the interaction picture operators.

To convert each of the interaction picture operators $$\phi_{I}$$ into the corresponding Heisenberg picture operators $$\phi_{H}$$, we use the relation $$\phi_{H}=U_{I}(t, t_{0})\phi_{I}U_{I}(t_{0}, t)^{\dagger}$$, where $$t_{0}$$ is an arbitrary initial time. By using the expression for $$U_{I}(t_{k},t_{k+1})$$ given in the proof, we can rewrite the expression as $$\phi_{H}=U(t_{k},t_{k+1})\phi_{I}U(t_{k+1},t_{k})^{\dagger}$$, and by taking the limit as $$t_{k+1}\rightarrow +\infty$$ and $$t_{k}\rightarrow -\infty$$, we get $$U_{I}(+\infty, t_{0})\phi_{H}U_{I}(t_{0},-\infty)$$. Since this is true for each of the $$\phi_{I}$$, we can rewrite the entire expression as $$U_{I}(+\infty, t_{0})\phi_{1H}\dots \phi_{nH}U_{I}(t_{0},-\infty)$$.

Finally, we have $$U