Two point function or Feyman propagator of K-G field
G_2=D_f (x)=\theta(t)D(x)+\theta(-t)D(-x)
describe the amplitude of particle propagating from 0 to x on spacetime,
which is just to say for the observation we can't distinguish the particle move from 0 to x and the antipaticle move from x to 0 along -t and so we
have to take the sum as final result.

On how to write G_2 in a single integral which contains a i\epsilon (\epsilon>0 and is infinitesimal), it's merely a mathematical trick.
The simplest way is to check D_f(x) on 4-momentum space whether equal to
the form in terms of D(x) by calculating the energy integral in the D_f(x), which is essentially the treatment in Peskin's QFT book (here you have to
make the choice of countor in order to identify
the two expressions, and note that \epsilon^2=0, 2\epsilon E_p shall be
taken as \epsilon due to infinitesimal, so
p^2-m^2+i\epsilon=(p^0-E_p+i\epsilon)(p^0+E_p-i\epsilon)).

You can also take an integral representation of \theta(t), here obviouse we have \theta(t)=\int d\omega e^{i\omega t}/[2\pi i(\omega-i\epsilon)],
where the integral contour c=infinite semicircle on upper half plane, t>0
or lower half plane, t<0. This integral indeed equal 1 if t>0 and 0 if t<0.
Substituting above integral, it's directly to get the usual propagator
as an integral on 4-momentum space (take D(x) the form of (2.50) in
Peskin's QFT book, and move the pole by seting \omega=p^0-E_p, p^0+E_p).

In a word, the only physical point is the expression of Feyman propagator at first, and the choice of countor is just a way to represent \theta(t) or to identify two forms of Feyman propagator.

楼主写的算得上是正解

另一方面正如西门所说，例如计算推迟或超前传播子，取的围道与求feynman传播子时是不同的。

楼主提醒\epsilon^2=0时，可能说一句“把高阶无穷小取为零”更让人好理解。事实上，在讨论场的拉格朗日密度在某无穷小连续变换下不变时，常常就包含了默认取高阶无穷小为零。

我曾经有篇文章，涉及讨论场的拉格朗日密度在某无穷小连续变换下不变，结果审稿人认为拉格朗日密度的变分并不是恰好为零，因此不具备我说的那种对称性，把稿子枪毙了，审稿人就是犯了低级错误：拉格朗日密度的变分虽然不是恰好为零，但却是无穷小变换参数的高阶无穷小！因此我对此印象深刻。

大概楼主这篇文章也是针对在下的一个问题的回复吧。
对我来说，重要的就是要知道不同形式的传播子背后的物理含义；
知道物理含义之后，剩下的就是纯粹的技术性的数学问题了，比较好说。
楼主说得很清楚，多谢！

重要的就是要知道不同形式的传播子背后的物理含义；
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
这个全在于各Green函数的原始定义形式，如楼主所说。

另外，针对楼主而言，\theta(t)的积分形式最好用e^{-i\omega t}（多个负号）表示。
这是因为，无论如何，四动量积分中所出现的e指数必须是平面波，
而平面波公认的约定是e^{ik.x-i\omega t}（k和x是三维矢量）。
我所见过的所有资料概莫能外。
这种选择与度规选择无关，但在不同的度规选择下有不同的表现形式。
平面波的这一约定其实直接跟量子力学的以下基本对应直接相关：
-ið_x→p_x,ið_t→E（用了一个不恰当的符号来表示偏导数）

The integral form of the \theta(t) is determined by how
to move the pole. As in my note, if you want to sum two
integrals, you should turn \omega->-\omega, then move pole
\omega=p_0-E_p for \theta(t). For \theta(-t)=\int e^{-i\omega t}
/[2\pi i(\omega +i\epsilon)] with same choice of contour,
we should move pole \omega=p_0+E_p.

Plane wave has positive and negative frenquency components,
only in quantuam physics, we always use its positive components,
e^{-iEt} due to the identification of energy and frenquency
and E is positive as assumption.

我的前一回复有些贸然。实际上，由于Feynman传播子的定义中含两项，而每项分别含\theta(t)和\theta(-t)，故我的“最好”一说没有多大意义。但Feynman传播子的积分表达式中须出现Exp[-iEt]一点是对的。