Proof that Casimir force does not originate from vacuum energy

Abstract

We present a simple general proof that Casimir force cannot originate from the vacuum energy of electromagnetic (EM) field. The full QED Hamiltonian consists of 3 terms: the pure electromagnetic term $H_{\mathrm{em}}$, the pure matter term $H_{\mathrm{matt}}$ and the interaction term $H_{\mathrm{int}}$. The $H_{\mathrm{em}}$-term commutes with all matter fields because it does not have any explicit dependence on matter fields. As a consequence, $H_{\mathrm{em}}$ cannot generate any forces on matter. Since it is precisely this term that generates the vacuum energy of EM field, it follows that the vacuum energy does not generate the forces. The misleading statements in the literature that vacuum energy generates Casimir force can be boiled down to the fact that $H_{\mathrm{em}}$ attains an implicit dependence on matter fields by the use of the equations of motion and the illegitimate treatment of the implicit dependence as if it was explicit. The true origin of the Casimir force is van der Waals force generated by $H_{\mathrm{int}}$.

14 pages, revised, accepted for publication in Phys. Lett. B

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