QFT question

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Homework Statement



I'm studying from Zee's QFT in a nutshell. On page 21, I don't understand how he uses integration by parts to get from Eq (14) to Eq (15), ie from

[itex]Z = \int D \varphi e^{i \int d^4 x \{ \frac{1}{2}[(\partial \varphi)^2 - m^2 \varphi^2] + J\varphi \}}[/itex]

to

[itex]Z = \int D \varphi e^{i \int d^4 x [-\frac{1}{2}(\partial^2+m^2)\varphi + J\varphi]}[/itex].

Is he suggesting that [itex]\int d^4x \varphi^2 = \int d^4x \varphi[/itex] and [itex]\int d^4x (\partial\varphi)^2 = \int d^4x( -\partial^2 \varphi)[/itex]? If so, I'm failing to see why this should be the case.
 

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  • #2
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Sigh. Nevermind, there was a typo in my second integral, Eq. (15) is actually

[itex]Z = \int D \varphi e^{i \int d^4 x [-\frac{1}{2}\varphi(\partial^2+m^2)\varphi + J\varphi]}[/itex] which can be obtained easily by integration by parts on the [itex](\partial \varphi)^2[/itex] term:

[itex]\int d^4 x\, (\partial \varphi)^2 = -\int d^4x\, \varphi \partial^2\varphi[/itex].
 

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