# QFT question

1. Nov 16, 2007

### noospace

1. The problem statement, all variables and given/known data

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

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

to

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

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

2. Nov 16, 2007

### noospace

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

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

$\int d^4 x\, (\partial \varphi)^2 = -\int d^4x\, \varphi \partial^2\varphi$.