I am trying to understand the derivation of the Casimir energy from https://en.wikipedia.org/wiki/Casim...f_Casimir_effect_assuming_zeta-regularization.(adsbygoogle = window.adsbygoogle || []).push({});

At one point, the derivation writes the following:

The vacuum energy is then the sum over all possible excitation modes ##\omega_{n}##. Since the area of the plates is large, we may sum by integrating over two of the dimensions in ##k##-space. The assumption of periodic boundary conditions yields

$$\langle E \rangle=\frac{\hbar}{2} \cdot 2 \int \frac{A dk_x dk_y}{(2\pi)^2} \sum_{n=1}^\infty \omega_n$$

What does the size of the plates have to do with being able to integrate over two of the dimensions in ##k##-space? I don't quite see the connection.

Thank you so much in advance for any comments.

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# A Derivation of the Casimir energy in flat space

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