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We have a field [itex]\phi(x)[/itex], which obeys the CCR, and a renormalized field [itex]\phi'(x)=Z^{1/2}_3\phi(x)[/itex], which obeys the equation [itex]\langle k|\phi'(0)|0\rangle=1[/itex]. We now use the Lehmann-Kallen decomposition to determine that:

[tex]\langle 0|\phi'(x)\phi'(y)|0\rangle = \Delta_+(x-y,\mu^2) + \int^\infty_0{da^2\sigma(a^2)\Delta_+(x-y,a^2)}[/tex]

The text then goes on to use this to make a statement about [itex]Z_3[/itex] using the equal-time commutators, by saying that:

(1) [itex]\langle 0|[\phi'(x),\dot{\phi'}(y)]|0\rangle = iZ^{-1}_3\delta^{(3)}(x-y)[/itex]

(2) [itex]\langle 0|[\phi'(x),\dot{\phi'}(y)]|0\rangle = \delta^{(3)}(x-y) + \int^\infty_0{da^2\sigma(a^2)i\delta^{(3)}(x-y)}[/itex]

where (1) is by the definition of [itex]\phi'(x)[/itex], and (2) is by manipulation of the decomposition derived above. My question is about this second statement--the text essentially handwaves its derivation, but I'm not quite following it. It sounds like they're just differentiating the RHS of it, but I don't understand how that gives you [itex]\langle 0|[\phi'(x),\dot{\phi'}(y)]|0\rangle[/itex] on the left.

I'm sure it's just some kind of silly property of commutators or propagators or something that I'm not getting, but I'd appreciate if somebody could help me connect the dots. Can anybody show how to go through this derivation in slow motion?