Peskin and Schroeder Equation 2.56 In Peskin and Schröder's QFT, equation 2.56, could anyone give a list of all the arguments necessary in order to make all the transitions mathematically rigorous?
I tried composing such a list myself and I came up with:


*

*$\frac{d}{dx}\theta(x)=\delta(x)$

*$\theta(x)\delta(x)\equiv0$?

*If this holds $\int d^4x(\partial_\mu\partial^\mu\theta(x^0-y^0))<0|[\phi(x),\phi(y)]|0>=-\int d^4x(\partial^\mu\theta(x^0-y^0))(\partial_\mu<0|[\phi(x),\phi(y)]|0>)$ then this holds: $(\partial_\mu\partial^\mu\theta(x^0-y^0))<0|[\phi(x),\phi(y)]|0>=-(\partial^\mu\theta(x^0-y^0))(\partial_\mu<0|[\phi(x),\phi(y)]|0>)$. But why? Is it because of the $e^{-ip_\alpha(x^\alpha-y^\alpha)}$ term in $<0|[\phi(x),\phi(y)]|0>$?

*Why is the KG operator applied on the Green's function equal to $-i\delta(x-y)$ and not $\delta(x-y)$? Is it a matter of convention?

 A: Probably not a complete answer.  However, your item 1. is correct but
1'. $\partial^\mu\theta(x^0)=g^{\mu 0} \delta(x^0)$ is better for this purpose. And then $\partial_\mu\partial^\mu\theta(x^0) = \partial_0\delta(x^0)=\delta'(x^0)$.  
Your item 2. is incorrect, the distribution on the l.h.s. is not defined as far as I know. But fortunately it is not needed.
Regarding 3., one can use that $f(x)\delta'(x)=-f'(x)\delta(x)$ if $f(0)=0$, which is obviously true.  Now, as a function of $x^0-y^0$ the operator $[\phi(x^0,\vec{x})\phi(y^0,\vec{y})]$ does vanish at $x^0-y^0=0$ because that makes $x-y$ spacelike. (Except at the origin $x-y=0$. Sorry, I don't have the time to deal with that now.  But the result is singular there anyways...).
That gives you (2.56).  
Regarding your item 4., you may redefine your Green function by a multiplicative factor to obtain the coefficient you want in front of your $\delta$.  But the modulus-squared of the coefficient in the tree-level propagator is the  coefficient in the kinetic term in the Lagrangian density, which you want correctly normalized. So you have only the freedom to redefine your Green's function by a phase factor.  And that is a matter of convention.  
