Casimir Effect - Derivation of Casimir Effect Assuming Zeta-regularization | Derivation Casimir Effect Assuming Zeta-regularization

Derivation of Casimir Effect Assuming Zeta-regularization

In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates at distance apart. In this case, the standing waves are particularly easy to calculate, since the transverse component of the electric field and the normal component of the magnetic field must vanish on the surface of a conductor. Assuming the parallel plates lie in the xy-plane, the standing waves are

where stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, and are the wave vectors in directions parallel to the plates, and

is the wave-vector perpendicular to the plates. Here, n is an integer, resulting from the requirement that ψ vanish on the metal plates. The energy of this wave is

where c is the speed of light. The vacuum energy is then the sum over all possible excitation modes

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

where A is the area of the metal plates, and a factor of 2 is introduced for the two possible polarizations of the wave. This expression is clearly infinite, and to proceed with the calculation, it is convenient to introduce a regulator (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The zeta-regulated version of the energy per unit-area of the plate is

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

In the end, the limit is to be taken. Here s is just a complex number, not to be confused with the shape discussed previously. This integral/sum is finite for s real and larger than 3. The sum has a pole at s = 3, but may be analytically continued to s = 0, where the expression is finite. The above expression is easily simplified to:

$\frac{\langle E(s) \rangle}{A} = \frac{\hbar c^{1-s}}{4\pi^2} \sum_n \int_0^\infty 2\pi qdq \left \vert q^2 + \frac{\pi^2 n^2}{a^2} \right\vert^{(1-s)/2}$

where polar coordinates were introduced to turn the double integral into a single integral. The in front is the Jacobian, and the comes from the angular integration. The integral is easily performed and converges if Re > 3, resulting in

$\frac{\langle E(s) \rangle}{A} = -\frac {\hbar c^{1-s} \pi^{2-s}}{2a^{3-s}} \frac{1}{3-s} \sum_n \vert n\vert ^{3-s}.$

The sum clearly diverges at s in the neighborhood of zero, but if the damping of large-frequency excitations corresponding to analytic continuation of the Riemann zeta function to s = 0 is assumed to make sense physically in some way, then one has

$\frac{\langle E \rangle}{A} = \lim_{s\to 0} \frac{\langle E(s) \rangle}{A} = -\frac {\hbar c \pi^{2}}{6a^{3}} \zeta (-3).$

But and so one obtains

$\frac{\langle E \rangle}{A} = \frac {-\hbar c \pi^{2}}{3 \cdot 240 a^{3}}.$

The analytic continuation has evidently lost an additive positive infinity, somehow exactly accounting for the zero-point energy (not included above) outside the slot between the plates, but which changes upon plate movement within a closed system. The Casimir force per unit area for idealized, perfectly conducting plates with vacuum between them is

${F_c \over A} = - \frac{d}{da} \frac{\langle E \rangle}{A} = -\frac {\hbar c \pi^2} {240 a^4}$

where

(hbar, ħ) is the reduced Planck constant,

is the speed of light,

is the distance between the two plates.

The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of shows that the Casimir force per unit area is very small, and that furthermore, the force is inherently of quantum-mechanical origin.

NOTE: In Casimir's original derivation, a moveable conductive plate is positioned at a short distance a from one of two widely separated plates (distance L apart). The 0-point energy on both sides of the plate is considered. Instead of the above ad hoc analytic continuation assumption, non-convergent sums and integrals are computed using Euler-Maclaurin summation with a regularizing function (e.g., exponential regularization) not so anomalous as in the above.

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