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In Peskin and Schroeder, one reaches the following equation for the spacetime Klein-Gordon field:

$$\phi(x,t)=\int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}\Big(a(p)e^{-ip\cdot x}+a^\dagger(p)e^{ip\cdot x}\Big)$$

Then they say that the propagation amplitude for a particle to go from a spacetime point x to a spacetime point y is [itex]\langle 0\vert\phi(x)\phi(y)\vert 0\rangle[/itex] where the ket |0> is the vacuum. I understand this up to here. But then they start computing it as follows (equation 2.50):

$$\langle 0\vert\phi(x)\phi(y)\vert 0\rangle = \int\frac{d^3p\,d^3q}{(2\pi)^6}\frac{1}{2\sqrt{E_{p}E_{p'}}}\Big(a(p)e^{-ipx}+a^\dagger(p)e^{ipx}\Big)\Big(a(q)e^{-iqy}+a^\dagger(q)e^{iqy}\Big)$$

Clearly there will now be four terms when you expand the parentheses, and the book claims that all of these vanish except for the term with [itex]\langle 0\vert a(p)a^\dagger(q)\vert 0\rangle[/itex]. Two questions:

a) Wouldn't this term vanish also since a(p) kills the vacuum bra, producing a zero?

b) Why doesn't the term with [itex]a^\dagger(p)a^\dagger(q)[/itex] stay? In that term there are no annihilation operators to kill the vacuum, so surely the term should vanish.