The Casimir effect: Illustration of the Riemann zeta function regularization

For simplicity, we will consider:
- A one-dimensional space instead of our usual three-dimensional space.
- The space is a box of finite size $L$.
- The space is periodic so it is really a circle (this will eliminate edge effects).
- An electromagnetic field of particles (photons) which is scalar (with no spin) which permeates this space.
A mode of this field is a plane wave with a  quantized (because of the periodic  boundary condition) frequency or energy  given by
\begin{eqnarray}
\omega_n(L)=\frac{\pi}{L}n~,~n=0,1,2,...
\end{eqnarray}
These modes represent virtual particles which are constantly created and annihilated in the vacuum and thus the energy of the quantum vacuum is given by the formula (we set $\hbar=1$)
\begin{eqnarray}
E(L)=\frac{1}{2}\sum_n\omega_n(L)=\frac{\pi}{2L}\sum_{n=1}^{\infty}n.
\end{eqnarray}
This is the Ramanujan series and there is no doubt that it is divergent quadratically with a sharp cutoff $N$.
Let us now imagine two metal parallel plates placed at $x=0$ and $x=a$ inside this space.
The field modes in the space between the two plates are still plane waves but now with quantized frequencies given by
\begin{eqnarray}
\omega_n(a)=\frac{\pi}{a}n~,~n=0,1,2,...
\end{eqnarray}
The energy of the quantum vacuum inside the plates is then given by
\begin{eqnarray}
E(a)=\frac{1}{2}\sum_n\omega_n(a)=\frac{\pi}{2a}\sum_{n=1}^{\infty}n.
\end{eqnarray}
We regularize this expression by a smooth cutoff function $\eta(x)$. As we have shown (in our previous post on the Riemann zeta function) the smoothed Ramanujan partial series is given by
\begin{eqnarray}
\sum_{n=1}^{\infty}n\eta(\frac{n}{N})=-\frac{1}{12}+N^2C_{\eta,1}+O(\frac{1}{N^2}).
\end{eqnarray}
The cutoff $N$ here is quite physical and represents the frequency at which the metal of the plates stops from being a conductor due to the so-called skin effect.  This cutoff $N$ is related to the energy cutoff $\Lambda$ by
\begin{eqnarray}
\Lambda=\frac{\pi N}{a}.
\end{eqnarray}
The energy of the quantum vacuum inside the plates is then given by
\begin{eqnarray}
E(a)=-\frac{\pi}{24 a}+\frac{a\Lambda^2 C_{\eta,1}}{2\pi}.
\end{eqnarray}

In the rest of the universe (outside the plates which is also a box but of size $b=L-a$) the energy of the quantum vacuum is  given similarly by
\begin{eqnarray}
E(b)=\frac{1}{2}\sum_n\omega_n(b)=\frac{\pi}{2b}\sum_{n=1}^{\infty}n=-\frac{\pi}{24 b}+\frac{b\Lambda^2 C_{\eta,1}}{2\pi}.
\end{eqnarray}
The total energy of the quantum vacuum is then 

\begin{eqnarray}
E=E(a)+E(b)=\frac{L\Lambda^2 C_{\eta,1}}{2\pi}-\frac{\pi}{24 a}-\frac{\pi}{24 b}+O(\frac{1}{{\Lambda}^2}).
\end{eqnarray}
This is still quadratically divergent as it should be.  But the coefficient of the quadratic divergence is proportional to $L$ which is a fixed quantity.

Now, the force between the two plates will cause their separation $a$ to vary by an amount $da$ causing therefore a variation in their energy by the amount $dE$. Saying it differently, the force between the two plates is given by
\begin{eqnarray}
F=-\frac{dE}{da}=-\frac{\pi}{24 a^2}.
\end{eqnarray}
In this formula we have also taken the continuum limit $L\longrightarrow \infty$ which made the $b-$term vanishes identically.

This force is measured experimentally yet it remains one of the most mysterious effect in physics. Indeed,   this is a force due to the quantum vacuum and as such it is intimately connected to the vacuum energy and dark energy. However, the regularization scheme here is physically quite motivated and very transparent and the role of the Riemann zeta function is illustrated in a spectacular way. But this does not mean that the Riemann zeta function is less mysterious but only means that by such considerations this function is at least  brought closer to logical and physical intuitions.