# Peskin & Schroeder QFT，eq. (2.56) derivation

I'm trying to derive the eq.2.56 of P&S's QFT textbook step by step:

$$(\partial^2+m^2)D_R(x-y)=-i\delta^{(4)}(x-y). \tag{2.56}$$

I have no problem with the first step:

$$(\partial^2+m^2)D_R(x-y)=(\partial^2\theta(x^0-y^0))\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle+2(\partial_\mu\theta(x^0-y^0))(\partial^\mu\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle))+\theta(x^0-y^0)(\partial^2+m^2)\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle$$ (*)

And here is what my derivation, please point out the mistakes I've made, I would be appreciated for your help:

1.for the third term in the RHS of the eq.(*):

I think it used the Klein-Gordon equation so that $$(\partial^2+m^2)\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle=\langle0\rvert[(\partial^2+m^2)\phi(x),\phi(y)]\rvert0\rangle+\langle0\rvert[\phi(x),(\partial^2+m^2)\phi(y)]\rvert0\rangle=0$$

2.for the first term in the RHS of the eq.(*):

Since $$\theta(x^0-y^0)$$ only contains the time variable, then only partial derivative wrt. time remained, then $$\partial^2\theta(x^0-y^0)=\partial_t\delta(x^0-y^0)$$, then

$$\partial_t\delta(x^0-y^0)\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle=-\delta(x^0-y^0)\partial_t\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle$$

in which the property of $$\delta$$ function $$f(x)\delta^\prime(x)=-f^\prime(x)\delta(x)$$ has been used.

3.for the second term in the RHS of the eq.(*):

it equals to $$2\delta(x^0-y^0)\partial_t\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle$$

Finally, combined the three terms we have obtained, we get:

$$\delta(x^0-y^0)\partial_t\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle$$

I think I have some troubles in dealing with $$\partial_t\langle0\rvert[\phi(x),\phi(y)]\rvert0\rangle$$, I think it should be:

$$\langle0\rvert[\partial_t\phi(x),\phi(y)]\rvert0\rangle+\langle0\rvert[\phi(x),\partial_t\phi(y)]\rvert0\rangle=\langle0\rvert[\pi(x),\phi(y)]\rvert0\rangle+\langle0\rvert[\phi(x),\pi(y)]\rvert0\rangle$$

but obviously, the authors do not really think so, where I am wrong? Thank you very much.

The derivatives in $$\partial^2+m^2$$ are only with respect to the $$x$$ argument, not the $$y$$ (clearly it can't be both). So in all your steps you have the right idea but only the term with $$\pi(x)$$ appears.