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Both in the Wald and Parker/Toms texts on QFT in curved space time, when introducing QFT in flat space time first, they solve the Klein Gordon equation over the whole real line by placing the “field in a box” of side L and use periodic boundary conditions. They then take the limit as $L \to \infty$ to get the solution over all space.

I’m a bit confused as to how this works. If you have periodic boundary conditions you’ll have eigenfunctions that will have terms like $\sin(a x / L)$ which will all vanish as $L \to \infty$ so I’m wondering how this trick works.

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In a box of side $L$ the solutions to Klein-Gordon are plane waves of the form $exp(i \mathbf k \cdot \mathbf x \pm i \omega t)$, where $\omega = +(k^2 + m^2)^{1/2}$ and $k_i = (2 \pi / L) n_i$ with $n_i = - \infty, ..., -2, -1, 0, 1, 2, ... + \infty$.

When $L \to \infty$, the infinitely denumerable $n_i$ allow to cover a continuous range of $k_i$ from $- \infty$ to $+ \infty$.

The trick works because conceptually you assume $L$ very large and then you let the $n_i$ to run over their infinite range.

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