- #1

- 1,497

- 3

## Homework Statement

I'm trying to derive the Klein-Gordon equation from its lagrangian density

[tex] \mathcal{L} = - \frac{1}{2} \partial^{\mu} \varphi \partial_{\mu} \varphi - \frac{1}{2} m^2 \varphi^2 + \Omega_0 [/tex]

(Srednicki p.24)

## Homework Equations

[tex] S = \int d^4x \mathcal{L} [/tex]

[tex] \delta S = 0 [/tex]

## The Attempt at a Solution

So here is what I got so far,

[tex] \int d^4x \left[ -\frac{1}{2} \partial^{\mu} (\varphi + \delta\varphi) \partial_{\mu} (\varphi + \delta\varphi)- \frac{1}{2} m^2 (\varphi+\delta\varphi)^2 + \Omega_0 \right] = 0[/tex]

[tex] \int d^4x \left[ -\frac{1}{2} \partial^{\mu} \delta\varphi \partial_{\mu} \varphi -\frac{1}{2} \partial^{\mu} \varphi \partial_{\mu} \delta\varphi - m^2 \varphi \delta\varphi \right] + \int d^4x \left[\partial^{\mu}\varphi \partial_{\mu} \varphi + \partial^{\mu}\delta\varphi \partial_{\mu} \delta\varphi + \Omega_0 \right] = 0[/tex]

The answer in the book is the first integral on the left. But that would mean that the second integral has to vanish. If this is correct why does the second integral is zero?