Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 7

Separate the Klein-Gordon equation as $\varphi (t,x)=e^{\pm i\omega t}X(x)$. The spatial equation is $X^{\prime \prime }+k^{2}X=0$, with ${\omega }^2=k^2+m^2$. The first Dirichlet boundary condition removes the cosine solution, and the second requires $\sin (kL)=0$. ThusThere is no nonzero $n=0$ Dirichlet mode. The real field is built from both complex-conjugate time dependences, with mode functionsThe sine functions satisfy ${\int }_0^L\sin (k_{n}x)\sin (k_{m}x)dx=(L/2){\delta }_{nm}$. Consequently the factor $1/\sqrt{{\omega }_{n}L}$ gives unit Klein-Gordon inner product for each positive-frequency mode and the canonical equal-time field commutator after quantization. A general solution is a superposition of these modes, rather than a claim that a single mode exhausts all solutions.

The Hamiltonian density is $\mathcal{H}=[{\dot{\varphi }}^2+({\partial }_x\varphi {)}^2+m^2{\varphi }^2]/2$. In the vacuum only $⟨a_n{a}_m^{†}⟩={\delta }_{nm}$ contributes. HenceSubstitution, using ${\omega }_n^2=k_n^2+m^2$, yieldsThis Dirichlet scalar vacuum energy density illustrates why a massive field's local unregularized energy density need not be uniform. The cosine integrates to zero over the interval, soThese divergent expressions are understood with a common regulator before exchanging sums, integrals or derivatives. Normal ordering the Hamiltonian operator separately in each geometry would erase the vacuum terms one is trying to compare; a common regulated subtraction is used instead in c and d.

For $m=0$, the frequencies are ${\omega }_n=n\pi /L$. The stipulated spectral regulator therefore supplies $e^{-n/(L\Lambda )}$. With $ϵ=1/(L\Lambda )$,The sum identity follows by differentiating the convergent geometric series ${\sum }_{n\ge 1}{z}^n=z/(1-z)$ with $z=e^{-ϵ}$. Its small-$ϵ$ expansion issoIn the frequency-regulated Dirichlet Casimir energy, the inverse-length scale $\Lambda$ suppresses modes with frequency much above $\pi \Lambda$ in units $c=1$. It is an ultraviolet cutoff, removed after comparing the same underlying vacuum convention in the two geometries. Physically an ideal perfectly reflecting boundary also ceases to be adequate at sufficiently high frequency, but that material scale is not specified here. The source calls this heat-kernel regularization; its explicit regulator is exponential in frequency, rather than the alternative exponential in frequency squared.

On the unbounded line the momentum density of states in a reference length $L$ is $Ldk/(2\pi )$. Count both signs of $k$, each with zero-point energy $|k|/2$ and the same frequency regulator:The extensive divergent term agrees exactly with c. Subtract it before taking the cutoff away:The force tending to change the interval length isIts negative sign means attraction: separating the boundaries increases this negative vacuum-energy difference toward zero. The bulk subtraction must be made before interpreting the force, otherwise differentiating an unphysical cutoff-dependent bulk energy would contaminate it.

This is the one-dimensional scalar Dirichlet Casimir force. The source's name “Casimir-Polder” normally refers to a retarded atom–atom or atom–surface dispersion force, whose geometry and power law differ. No atom polarizability or electromagnetic mode multiplicity enters the scalar result derived here.