# Green's function of the Klein-Gordon operator

Again, from the Peskin and Schroeder's book, I can't quite see how this computation goes:

See file attached

The thing I don't get is how the term with $(\partial^{2}+m^{2})\langle 0| [\phi(x),\phi(y)] | 0 \rangle$ vanishes, and also why they only get a $\langle 0 | [\pi(x),\phi(y)] | 0 \rangle$ from the $\partial_{t}\langle 0 | [\phi(x),\phi(y)] | 0 \rangle$ and not also a $\langle 0 | [\phi(x),\pi(y)] | 0 \rangle$

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vanhees71
Gold Member
Me neither. Peskin/Schroeder is pretty unclear concerning propagators and its analytic properties.

First of all, one has to specify which propagator one is talking about, and this depends on what you want to do with it. In the case of perturbation theory in vacuum qft you need the time-ordered propagator, which is defined as the vacuum-expectation value of free field operators (here for an uncharged Klein-Gordon field)
$$\mathrm{i} D(x-y)=\langle 0|\mathcal{T}_c \hat{\phi}(x) \hat{\phi}(y)|0\rangle.$$
Now you plug in the expansion of the field operator in terms of creation and annihilation operators
$$\hat{\phi}(x)=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{\sqrt{(2 \pi)^3 2 \omega(\vec{p})}} [\hat{a}(\vec{p}) \exp(-p \cdot x) + \hat{a}^{\dagger}(\vec{p}) \exp(+p \cdot x) ]_{p^0=\omega(\vec{p})}.$$
Then you can write the propgator after some algebra with vacuum expectation values of annihilation and creation operator products as
$$\mathrm{i} D(x-y)=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3 2 \omega(\vec{p})} \left [\Theta(\xi^0) \exp(-\mathrm{i} p \cdot \xi) + \Theta(-\xi) \exp(+\mathrm{i} p \cdot \xi) \right]_{p^0=\omega(\vec{p}),\xi=x-y}.$$
Now you take the Fourier transform of this wrt. $\xi$ with a regulating factor $\exp(-\epsilon |\xi^0|)$, which leads you to
$$\tilde{D}(p)=\int_{\mathbb{R}^4} \mathrm{d} \xi D(\xi) \exp(+\mathrm{i} p \cdot \ \xi)=\frac{1}{p^2-m^2+\mathrm{i} \epsilon}.$$
The $\mathrm{i} \epsilon$ has to be understood to be taken in the weak limit $\epsilon \rightarrow 0^+$.

For a more detailed explanation, why one has to use this time-ordered propagator, and also this derivation, see my QFT manuscript,

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

Chapter 3.

The thing I don't get is how the term with $(\partial^{2}+m^{2})\langle 0| [\phi(x),\phi(y)] | 0 \rangle$ vanishes, and also why they only get a $\langle 0 | [\pi(x),\phi(y)] | 0 \rangle$ from the $\partial_{t}\langle 0 | [\phi(x),\phi(y)] | 0 \rangle$ and not also a $\langle 0 | [\phi(x),\pi(y)] | 0 \rangle$
It is because of the free field eqn. (∂2+m2)$\phi$=0,the last term vanishes.Note also that ∂μ will act only on operators whose argument is x,not y.Also the first term will involve a by part to get the result.Rest is just simple calculation based on commutator relation and some property of step function.

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Note also that ∂μ will act only on operators whose argument is x,not y.

That was the key I was missing, probably missed due to writting it so abstractly :P. Thanks!

vanhees71
$$\partial_{t_1} \Theta(t_1-t_2)=-\partial_{t_2} \Theta(t_2-t_1)=\delta(t_1-t_2)$$